**3. Bifurcation analysis**

Consider a family of ODE's that depend on one parameter *λ*

$$\mathbf{x}' = f(\mathbf{x}, \boldsymbol{\lambda}), \tag{5}$$

*3.1.1 Saddle-node bifurcation*

Now consider the dynamical system defined by

if *a <* 0, then we have no real solutions, if *a >* 0, then we have two real solutions.

usual way. First, we add a small perturbation:

Substituting this into the equation yields

which has the solution

other is linearly unstable.

*3.1.2 Transcritical bifurcation*


Now consider the dynamical system

fixed point goes from being unstable to being stable.

*dx*

An equilibrium solution (where *<sup>x</sup>*� <sup>=</sup> <sup>0</sup>) is simply *<sup>x</sup>* <sup>=</sup> <sup>±</sup>√*a*. Therefore,

*dε*

and since the term in brackets on the RHS is trivially zero, therefore

*dt* = (*<sup>a</sup>* <sup>−</sup> *<sup>x</sup>*¯

*dε*

*dt* <sup>=</sup> <sup>−</sup>2*x*¯*ε*,

*ε*(*t*) = *A* exp(−2*xt*¯ ). From this, we see that for *<sup>x</sup>* = +√*a*, <sup>|</sup>*x*| → 0 as *<sup>t</sup>* <sup>→</sup> <sup>∞</sup> (linear stability); for *<sup>x</sup>* <sup>=</sup> <sup>−</sup>√*a*,

As sketched in the "bifurcation diagram" below, therefore, the saddle node bifurcation at *a* = 0 corresponds to the creation of two new solution branches. One of these is linearly stable, the

In a transcritical bifurcation, two families of fixed points collide and exchange their stability properties. The family that was stable before the bifurcation is unstable after it. The other

*dt* <sup>=</sup> *az* <sup>−</sup> *bx*2, for *<sup>x</sup>*, *<sup>a</sup>*, *<sup>b</sup>* real.

bifurcation.

A saddle-node bifurcation or tangent bifurcation is a collision and disappearance of two equilibria in dynamical systems. In autonomous systems, this occurs when the critical equilibrium has one zero eigenvalue. This phenomenon is also called fold or limit point

We now consider each of the two solutions for *a >* 0, and examine their linear stability in the

*x* = *x*¯ + *ε*.

<sup>2</sup>) <sup>−</sup> <sup>2</sup>*x*¯*<sup>ε</sup>* <sup>−</sup> *<sup>ε</sup>*

2,

*<sup>x</sup>*� <sup>=</sup> *<sup>a</sup>* <sup>−</sup> *<sup>x</sup>*2, for *<sup>a</sup>* is real. (6)

Bifurcation Analysis and Its Applications 11

where *<sup>f</sup>* : **<sup>R</sup>***n*+<sup>1</sup> <sup>→</sup> **<sup>R</sup>***<sup>n</sup>* is analytic for *λ�***R**, *<sup>x</sup>�***R***<sup>n</sup>* . Let *<sup>x</sup>* <sup>=</sup> *<sup>x</sup>*0(*λ*) be a family of equilibrium points of equation (5), i.e., *f*(*x*0(*λ*), *λ*) = 0. Now let's set

$$z = \mathbf{x} - \mathbf{x}\_0(\lambda).$$

Then

$$z' = A(\lambda)z + O(|z|^2)\_{\prime}$$

where *A*(*λ*) = *<sup>∂</sup> <sup>f</sup> <sup>∂</sup><sup>x</sup>* (*x*0(*λ*), *λ*).

Let *λ*1, *λ*2, ...*λn*(*λ*) be the eigenvalues of *A*(*λ*). If, for some *i*, *Reλi*(*λ*) changes sign at *λ* = *λ*0, we say that *λ*<sup>0</sup> is a bifurcation point of equation (5).

#### **3.1 Bifurcation in one dimension**

We may assume that *<sup>n</sup>* <sup>=</sup> 1 so that *<sup>f</sup>* : **<sup>R</sup>**<sup>2</sup> <sup>→</sup> **<sup>R</sup>**1, and *<sup>x</sup>*0(*λ*) is a real valued analytic function of *λ* provided

$$
\lambda\_1(\lambda) = \frac{\partial f}{\partial \mathbf{x}}(\mathbf{x}\_0(\lambda), \lambda) = A(\lambda) = 0.
$$

Therefore, the equilibrium point is asymptotically stable if *λ*1(*λ*) *<* 0, and unstable if *λ*1(*λ*) *>* 0. This implies that *λ*<sup>0</sup> is a bifurcation point if *λ*1(*λ*0) = 0. Hence, bifurcation points (*x*0(*λ*), *λ*)

are solutions of

$$f(\mathfrak{x}, \lambda) = 0 \quad \text{and} \ \frac{\partial f}{\partial \mathfrak{x}}(\mathfrak{x}, \lambda) = 0.$$

The most common bifurcation types are illustrated by the following examples.

#### *3.1.1 Saddle-node bifurcation*

8 Will-be-set-by-IN-TECH

*det*(*J* − *λI*) = 0.

Since the point is non-hyperbolic, the linearized system can not tell about the stability. Later

*λ*<sup>1</sup> = 1 and *λ*<sup>2</sup> = −1,

where one of the eigenvalues is strictly positive). Since it is a hyperbolic equilibrium point, the stability of fixed point is the same as in the linearized system. So it is also unstable.

where *<sup>f</sup>* : **<sup>R</sup>***n*+<sup>1</sup> <sup>→</sup> **<sup>R</sup>***<sup>n</sup>* is analytic for *λ�***R**, *<sup>x</sup>�***R***<sup>n</sup>* . Let *<sup>x</sup>* <sup>=</sup> *<sup>x</sup>*0(*λ*) be a family of equilibrium

*z* = *x* − *x*0(*λ*).

*z*� = *A*(*λ*)*z* + *O*(|*z*|

Let *λ*1, *λ*2, ...*λn*(*λ*) be the eigenvalues of *A*(*λ*). If, for some *i*, *Reλi*(*λ*) changes sign at *λ* = *λ*0,

We may assume that *<sup>n</sup>* <sup>=</sup> 1 so that *<sup>f</sup>* : **<sup>R</sup>**<sup>2</sup> <sup>→</sup> **<sup>R</sup>**1, and *<sup>x</sup>*0(*λ*) is a real valued analytic function

Therefore, the equilibrium point is asymptotically stable if *λ*1(*λ*) *<* 0, and unstable if *λ*1(*λ*) *>* 0. This implies that *λ*<sup>0</sup> is a bifurcation point if *λ*1(*λ*0) = 0. Hence, bifurcation points (*x*0(*λ*), *λ*)

*∂x*

(*x*, *λ*) = 0.

(*x*0(*λ*), *λ*) = *A*(*λ*) = 0.

2),

*x*� = *f*(*x*, *λ*), (5)

thus the point is locally

by solving the characteristic equation

on we will show that this is a center.

**3. Bifurcation analysis**

unstable (as

Then

where *A*(*λ*) = *<sup>∂</sup> <sup>f</sup>*

of *λ* provided

are solutions of

For the equilibrium point (1, 2), the Jacobian *J*(1, 2) =

Consider a family of ODE's that depend on one parameter *λ*

points of equation (5), i.e., *f*(*x*0(*λ*), *λ*) = 0. Now let's set

*<sup>∂</sup><sup>x</sup>* (*x*0(*λ*), *λ*).

**3.1 Bifurcation in one dimension**

we say that *λ*<sup>0</sup> is a bifurcation point of equation (5).

*<sup>λ</sup>*1(*λ*) = *<sup>∂</sup> <sup>f</sup>*

*∂x*

*<sup>f</sup>*(*x*, *<sup>λ</sup>*) = 0 and *<sup>∂</sup> <sup>f</sup>*

The most common bifurcation types are illustrated by the following examples.

A saddle-node bifurcation or tangent bifurcation is a collision and disappearance of two equilibria in dynamical systems. In autonomous systems, this occurs when the critical equilibrium has one zero eigenvalue. This phenomenon is also called fold or limit point bifurcation.

Now consider the dynamical system defined by

$$\mathbf{x}' = a - \mathbf{x}^2, \quad \text{for} \quad a \quad \text{is real.} \tag{6}$$

An equilibrium solution (where *<sup>x</sup>*� <sup>=</sup> <sup>0</sup>) is simply *<sup>x</sup>* <sup>=</sup> <sup>±</sup>√*a*. Therefore,

if *a <* 0, then we have no real solutions,

if *a >* 0, then we have two real solutions.

We now consider each of the two solutions for *a >* 0, and examine their linear stability in the usual way. First, we add a small perturbation:

$$
\mathfrak{x} = \mathfrak{x} + \mathfrak{e}.
$$

Substituting this into the equation yields

$$\frac{d\varepsilon}{dt} = (a - \bar{\mathbf{x}}^2) - 2\bar{\mathbf{x}}\varepsilon - \varepsilon^2\omega$$

and since the term in brackets on the RHS is trivially zero, therefore

$$\frac{d\varepsilon}{dt} = -2\overline{x}\varepsilon\_{\prime}$$

which has the solution

$$
\varepsilon(t) = A \exp(-2\overline{x}t).
$$

From this, we see that for *<sup>x</sup>* = +√*a*, <sup>|</sup>*x*| → 0 as *<sup>t</sup>* <sup>→</sup> <sup>∞</sup> (linear stability); for *<sup>x</sup>* <sup>=</sup> <sup>−</sup>√*a*, |*x*| → 0 as *t* → ∞ (linear stability).

As sketched in the "bifurcation diagram" below, therefore, the saddle node bifurcation at *a* = 0 corresponds to the creation of two new solution branches. One of these is linearly stable, the other is linearly unstable.

#### *3.1.2 Transcritical bifurcation*

In a transcritical bifurcation, two families of fixed points collide and exchange their stability properties. The family that was stable before the bifurcation is unstable after it. The other fixed point goes from being unstable to being stable.

Now consider the dynamical system

$$\frac{d\mathbf{x}}{dt} = az - b\mathbf{x}^2 \quad \text{for} \quad \mathbf{x}, a, b \quad \text{real.}$$

Now for the state *x*¯2, we add a small perturbation

Therefore, perturbations grow for a *>* 0 and decay for a *<* 0. So

the state *<sup>x</sup>*¯2 <sup>=</sup> *<sup>a</sup>*

the state *<sup>x</sup>*¯2 <sup>=</sup> *<sup>a</sup>*

**Figure 2.** Bifurcation diagram corresponding to the transcritical bifurcation

In pitchfork bifurcation one family of fixed points transfers its stability properties to two families after or before the bifurcation point. If this occurs after the bifurcation point, then pitchfork bifurcation is called supercritical. Similarly, a pitchfork bifurcation is called subcritical if the nontrivial fixed points occur for values of the parameter lower than the

which yields the linearized form

between the two solution branches.

*3.1.3 The pitchfork bifurcation*

has the solution

*x* = *x*¯2 + *ε*,

*e*(*t*) = *A* exp(−*at*).

It can be easily seen that the bifurcation point *a* = 0 corresponds to an exchange of stabilities

*<sup>b</sup>* is linearly stable if *<sup>a</sup> <sup>&</sup>gt;* 0,

*<sup>b</sup>* is linearly unstable if *<sup>a</sup> <sup>&</sup>lt;* 0.

Bifurcation Analysis and Its Applications 13

*dε dt* <sup>=</sup> <sup>−</sup>*a<sup>ε</sup>*

**Figure 1.** Bifurcation diagram corresponding to the saddle-node bifurcation

Again, *a* and *b* are control parameters. We can find two steady states (*x*� = 0) to this system

$$\begin{aligned} \mathfrak{x} &= \mathfrak{x}\_1 = 0 \, \forall a \, b \\ \mathfrak{x} &= \mathfrak{x}\_2 = \frac{a}{b} \, \forall a \, b \, b \neq 0. \end{aligned}$$

We now examine the linear stability of each of these states in turn, following the usual procedure.

For the state *x*¯1, we add a small perturbation

$$\mathfrak{x} = \mathfrak{x}\_1 + \varepsilon\_\prime$$

which yields

$$\frac{d\varepsilon}{dt} = a\varepsilon - b\varepsilon^2$$

with the linearized form

$$\frac{d\varepsilon}{dt} = a\varepsilon$$

has the solution

$$e(t) = A \exp(at).$$

Therefore, perturbations grow for a *>* 0 and decay for a *<* 0. So

the state *x*¯1 is unstable if *a >* 0, the state *x*¯1 is stable if *a <* 0. Now for the state *x*¯2, we add a small perturbation

$$\mathfrak{x} = \mathfrak{x}\_2 + \varepsilon\_\prime$$

which yields the linearized form

$$\frac{d\varepsilon}{dt} = -a\varepsilon$$

has the solution

10 Will-be-set-by-IN-TECH

**Figure 1.** Bifurcation diagram corresponding to the saddle-node bifurcation

For the state *x*¯1, we add a small perturbation

procedure.

which yields

has the solution

with the linearized form

Again, *a* and *b* are control parameters. We can find two steady states (*x*� = 0) to this system

*b*

We now examine the linear stability of each of these states in turn, following the usual

*x* = *x*¯1 + *ε*,

*dt* <sup>=</sup> *<sup>a</sup><sup>ε</sup>* <sup>−</sup> *<sup>b</sup><sup>ε</sup>*

*e*(*t*) = *A* exp(*at*).

the state *x*¯1 is unstable if *a >* 0, the state *x*¯1 is stable if *a <* 0.

*dε dt* <sup>=</sup> *<sup>a</sup><sup>ε</sup>*

*dε*

Therefore, perturbations grow for a *>* 0 and decay for a *<* 0. So

, ∀*a*, *b*, *b* �= 0.

2

*x* = *x*¯1 = 0, ∀*a*, *b <sup>x</sup>* <sup>=</sup> *<sup>x</sup>*¯2 <sup>=</sup> *<sup>a</sup>*

$$e(t) = A \exp(-at).$$

Therefore, perturbations grow for a *>* 0 and decay for a *<* 0. So

$$\begin{aligned} \text{the state } \vec{x}\_2 &= \frac{a}{b} \text{ is linearly stable if } a > 0, \\ \text{the state } \vec{x}\_2 &= \frac{a}{b} \text{ is linearly unstable if } a < 0. \end{aligned}$$

It can be easily seen that the bifurcation point *a* = 0 corresponds to an exchange of stabilities between the two solution branches.

**Figure 2.** Bifurcation diagram corresponding to the transcritical bifurcation

#### *3.1.3 The pitchfork bifurcation*

In pitchfork bifurcation one family of fixed points transfers its stability properties to two families after or before the bifurcation point. If this occurs after the bifurcation point, then pitchfork bifurcation is called supercritical. Similarly, a pitchfork bifurcation is called subcritical if the nontrivial fixed points occur for values of the parameter lower than the bifurcation value. In other words, the cases in which the emerging nontrivial equilibria are stable are called supercritical whereas the cases in which these equilibria are called subcritical.

Consider the dynamical system

$$x' = a\mathbf{x} - bx^3 \text{, for } \quad a, b \quad \text{real.}$$

As usual, *a* and *b* are external control parameters. Steady states, for which *x*� = 0 are as follows:

$$\begin{aligned} \mathfrak{x} &= \mathfrak{x}\_1 = 0, \forall a \, b \, \\ \mathfrak{x} &= \mathfrak{x}\_2 = -\sqrt{a/b}, \text{ for } a/b > 0, \\ \mathfrak{x} &= \mathfrak{x}\_3 = -\sqrt{a/b}, \text{ for } a/b > 0. \end{aligned}$$

Note that the equilibrium points *x*¯2 and *x*¯3 only exist when *a >* 0 if *b >* 0 and for *a <* 0 if *b <* 0.

As usual, we now examine the linear stability of each of these steady states in turn. (This can be done for a general *b*). First we write the perturbation for *x*¯1 = 0,

$$\mathfrak{x} = \bar{\mathfrak{x}}\_1 + \varepsilon$$

that yields the linearized equation

$$\frac{d\varepsilon}{dt} = a\varepsilon\_{\prime}$$

**Figure 3.** Bifurcation diagram corresponding to the pitchfork bifurcation

**Definition :** A Hopf or Poincare-Andronov-Hopf bifurcation is a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of

*dt* <sup>=</sup> *<sup>g</sup>* (*x*, *<sup>y</sup>*, *<sup>τ</sup>*),

where *τ* is the parameter and suppose that (*x* (*τ*), *y* (*τ*)) is the equilibrium point and *α* (*τ*) ± *iβ* (*τ*) are the eigenvalues of the Jacobian matrix which is evaluated at the equilibrium point. In addition let's assume that the change in the stability of the equilibrium point occurs at *τ* =

*dt* <sup>=</sup> *<sup>f</sup>* (*x*, *<sup>y</sup>*, *<sup>τ</sup>*), (7)

Bifurcation Analysis and Its Applications 15

linearization around the fixed point cross the imaginary axis of the complex plane.

*dx*

*dy*

**3.2 Bifurcation in two dimension**

**3.3 Hopf bifurcation theorem**

Consider the two dimensional system

*3.2.1 Hopf bifurcation*

*τ*∗ where *α* (*τ*∗) = 0.

with the solution

$$e(t) = A \exp(at).$$

So we see that

the state *x*¯1 = 0 is linearly unstable if *a >* 0,

the state *x*¯2 = 0 is linearly stable if *a <* 0.

For the states *x* = *x*¯2 and *x* = *x*¯3, setting

$$\begin{aligned} \vec{x} &= \pm \sqrt{a/b} + \varepsilon\_r \\\\ \frac{d\varepsilon}{dt} &= a\varepsilon - 3b\vec{x}^2 \varepsilon\_r \end{aligned}$$

with the solution

$$e(t) = A \exp(ct) \text{ where } c = -2a.$$

Thus it is obvious that

the states *x*¯2 and *x*¯3 are linearly stable if *a >* 0, the states *x*¯2 and *x*¯3 are linearly unstable if *a <* 0,

**Figure 3.** Bifurcation diagram corresponding to the pitchfork bifurcation

## **3.2 Bifurcation in two dimension**

#### *3.2.1 Hopf bifurcation*

12 Will-be-set-by-IN-TECH

bifurcation value. In other words, the cases in which the emerging nontrivial equilibria are stable are called supercritical whereas the cases in which these equilibria are called subcritical.

*<sup>x</sup>*� <sup>=</sup> *ax* <sup>−</sup> *bx*3, for *<sup>a</sup>*, *<sup>b</sup>* real.

As usual, *a* and *b* are external control parameters. Steady states, for which *x*� = 0 are as

*<sup>x</sup>* <sup>=</sup> *<sup>x</sup>*¯3 <sup>=</sup> <sup>√</sup>*a*/*b*, for *<sup>a</sup>*/*<sup>b</sup> <sup>&</sup>gt;* 0.

Note that the equilibrium points *x*¯2 and *x*¯3 only exist when *a >* 0 if *b >* 0 and for *a <* 0 if

As usual, we now examine the linear stability of each of these steady states in turn. (This can

*x* = *x*¯1 + *ε*

*e*(*t*) = *A* exp(*at*).

the state *x*¯1 = 0 is linearly unstable if *a >* 0,

<sup>√</sup>*a*/*<sup>b</sup>* <sup>+</sup> *<sup>ε</sup>*,

2*ε*,

the state *x*¯2 = 0 is linearly stable if *a <* 0.

*dt* <sup>=</sup> *<sup>a</sup><sup>ε</sup>* <sup>−</sup> <sup>3</sup>*bx*¯

*e*(*t*) = *A* exp(*ct*) where *c* = −2*a*.

the states *x*¯2 and *x*¯3 are linearly stable if *a >* 0, the states *x*¯2 and *x*¯3 are linearly unstable if *a <* 0,

*x*¯ = ±

*dε*

*dε dt* <sup>=</sup> *<sup>a</sup>ε*,

<sup>√</sup>*a*/*b*, for *<sup>a</sup>*/*<sup>b</sup> <sup>&</sup>gt;* 0,

*x* = *x*¯1 = 0, ∀*a*, *b*,

*x* = *x*¯2 = −

be done for a general *b*). First we write the perturbation for *x*¯1 = 0,

Consider the dynamical system

that yields the linearized equation

For the states *x* = *x*¯2 and *x* = *x*¯3, setting

with the solution

So we see that

with the solution

Thus it is obvious that

follows:

*b <* 0.

**Definition :** A Hopf or Poincare-Andronov-Hopf bifurcation is a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of linearization around the fixed point cross the imaginary axis of the complex plane.

#### **3.3 Hopf bifurcation theorem**

Consider the two dimensional system

$$\begin{aligned} \frac{d\mathbf{x}}{dt} &= f\left(\mathbf{x}, \mathbf{y}, \tau\right), \\ \frac{d\mathbf{y}}{dt} &= \mathbf{g}\left(\mathbf{x}, \mathbf{y}, \tau\right), \end{aligned} \tag{7}$$

where *τ* is the parameter and suppose that (*x* (*τ*), *y* (*τ*)) is the equilibrium point and *α* (*τ*) ± *iβ* (*τ*) are the eigenvalues of the Jacobian matrix which is evaluated at the equilibrium point.

In addition let's assume that the change in the stability of the equilibrium point occurs at *τ* = *τ*∗ where *α* (*τ*∗) = 0.

**Figure 4.** Hopf bifurcation diagram

First the system is transformed so that the equilibrium is at the origin and the parameter *τ* at *τ*∗ = 0 gives purely imaginary eigenvalues. System (7) is rewritten as follows;

$$\frac{d\mathbf{x}}{dt} = a\_{11}\left(\tau\right)\mathbf{x} + a\_{12}\left(\tau\right)y + f\_1(\mathbf{x}, y, \tau),\tag{8}$$

$$\frac{dy}{dt} = a\_{21}\left(\tau\right)\mathbf{x} + a\_{22}\left(\tau\right)y + g\_1\left(\mathbf{x}, y, \tau\right).$$

with nonzero speed, i.e.,

in *U*. (Allen, L.J.S).

*t* → −∞) when *τ* = 0.

to appear.

below)

*dα*

*<sup>d</sup><sup>τ</sup>* <sup>|</sup>*τ*=0�<sup>=</sup> 0.

Bifurcation Analysis and Its Applications 17

Then in any open set *U* containing the origin in **R**<sup>2</sup> and for any *τ*<sup>0</sup> *>* 0, there exists a value *τ*¯, |*τ*¯| *< τ*<sup>0</sup> such that the system of differential equations (8) has a periodic solution for *τ* = *τ*¯

**Note:** The Hopf bifurcation requires at least a two dimensional differential equation system

**Definition:** The bifurcation stated in the Hopf bifurcation theorem is called "**supercritical**" if the equilibrium point (0, 0) is asymptotically stable when *τ* = 0 (at the bifurcation point) and it is called "**subcritical"** if the equilibrium point (0, 0) is negatively asymptotically stable (as

In a supercritical Hopf bifurcation, the limit cycle grows out of the equilibrium point. In other words, right at the parameters of the Hopf bifurcation, the limit cycle has zero amplitude, and this amplitude grows as the parameters move further into the limit-cycle. (See the figure

However in a subcritical Hopf bifurcation, there is an unstable limit cycle surrounding the equilibrium point, and a stable limit cycle surrounding that. The unstable limit cycle shrinks down to the equilibrium point, which becomes unstable in the process. For systems started near the equilibrium point, the result is a sudden change in behavior from approach to a stable

*dt* <sup>=</sup> <sup>−</sup>(*x*<sup>2</sup> <sup>+</sup> <sup>1</sup>)*y*,

where *α* is a parameter. When we compute the equilibrium points depending on parameter *α*;

*dt* <sup>=</sup> *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>α</sup>*, (9)

<sup>√</sup>*α*, 0).

**Figure 5.** Bifurcation diagram corresponding to Supercritical Hopf bifurcation

*dx*

*dy*

If *α <* 0, then there is no x-nullclines, hence the system has no equilibrium points.

If *α* = 0, then the system has exactly one equilibrium point at (0, 0). If *<sup>α</sup> <sup>&</sup>gt;* 0, then the system has two equilibrium points (−√*α*, 0) and (

focus, to large-amplitude oscillations.(See the figure below).

**Example:** Consider the two dimensional system

The linearization of the system (7) about the origin is given by *dX dt* = *J*(*τ*)*X*, where *X* = *x y* and

$$J(\tau) = \begin{bmatrix} a\_{11} \left(\tau\right) a\_{12} \left(\tau\right) \\\\ a\_{21} \left(\tau\right) a\_{22} \left(\tau\right) \end{bmatrix}$$

is the Jacobian matrix evaluated at origin.

#### **Theorem (Hopf bifurcation theorem)**

Let *f*<sup>1</sup> and *g*1, in system (8) have continuous third order partial derivatives in *x* and *y*. Suppose that the origin is an equilibrium point of (8) and that the Jacobian matrix *J* (*τ*) as above, is valid for all sufficiently small |*τ*|. Moreover, assume that the eigenvalues of matrix *J* (*τ*) are *α* (*τ*) ± *iβ* (*τ*) where *α* (0) = 0, *β* (0) �= 0 such that the eigenvalues cross the imaginary axis with nonzero speed, i.e.,

14 Will-be-set-by-IN-TECH

First the system is transformed so that the equilibrium is at the origin and the parameter *τ* at

*dt* <sup>=</sup> *<sup>a</sup>*<sup>21</sup> (*τ*) *<sup>x</sup>* <sup>+</sup> *<sup>a</sup>*<sup>22</sup> (*τ*) *<sup>y</sup>* <sup>+</sup> *<sup>g</sup>*<sup>1</sup> (*x*, *<sup>y</sup>*, *<sup>τ</sup>*).

 *a*<sup>11</sup> (*τ*) *a*<sup>12</sup> (*τ*) *a*<sup>21</sup> (*τ*) *a*<sup>22</sup> (*τ*)

Let *f*<sup>1</sup> and *g*1, in system (8) have continuous third order partial derivatives in *x* and *y*. Suppose that the origin is an equilibrium point of (8) and that the Jacobian matrix *J* (*τ*) as above, is valid for all sufficiently small |*τ*|. Moreover, assume that the eigenvalues of matrix *J* (*τ*) are *α* (*τ*) ± *iβ* (*τ*) where *α* (0) = 0, *β* (0) �= 0 such that the eigenvalues cross the imaginary axis

*dt* <sup>=</sup> *<sup>a</sup>*<sup>11</sup> (*τ*) *<sup>x</sup>* <sup>+</sup> *<sup>a</sup>*<sup>12</sup> (*τ*) *<sup>y</sup>* <sup>+</sup> *<sup>f</sup>*1(*x*, *<sup>y</sup>*, *<sup>τ</sup>*), (8)

*dt* = *J*(*τ*)*X*, where *X* =

 *x y* 

*τ*∗ = 0 gives purely imaginary eigenvalues. System (7) is rewritten as follows;

*dx*

*dy*

is the Jacobian matrix evaluated at origin. **Theorem (Hopf bifurcation theorem)**

The linearization of the system (7) about the origin is given by *dX*

*J*(*τ*) =

**Figure 4.** Hopf bifurcation diagram

and

$$\frac{d\alpha}{d\tau} \mid\_{\tau=0} \neq 0.$$

Then in any open set *U* containing the origin in **R**<sup>2</sup> and for any *τ*<sup>0</sup> *>* 0, there exists a value *τ*¯, |*τ*¯| *< τ*<sup>0</sup> such that the system of differential equations (8) has a periodic solution for *τ* = *τ*¯ in *U*. (Allen, L.J.S).

**Note:** The Hopf bifurcation requires at least a two dimensional differential equation system to appear.

**Definition:** The bifurcation stated in the Hopf bifurcation theorem is called "**supercritical**" if the equilibrium point (0, 0) is asymptotically stable when *τ* = 0 (at the bifurcation point) and it is called "**subcritical"** if the equilibrium point (0, 0) is negatively asymptotically stable (as *t* → −∞) when *τ* = 0.

In a supercritical Hopf bifurcation, the limit cycle grows out of the equilibrium point. In other words, right at the parameters of the Hopf bifurcation, the limit cycle has zero amplitude, and this amplitude grows as the parameters move further into the limit-cycle. (See the figure below)

**Figure 5.** Bifurcation diagram corresponding to Supercritical Hopf bifurcation

However in a subcritical Hopf bifurcation, there is an unstable limit cycle surrounding the equilibrium point, and a stable limit cycle surrounding that. The unstable limit cycle shrinks down to the equilibrium point, which becomes unstable in the process. For systems started near the equilibrium point, the result is a sudden change in behavior from approach to a stable focus, to large-amplitude oscillations.(See the figure below).

**Example:** Consider the two dimensional system

$$\frac{d\mathbf{x}}{dt} = \mathbf{x}^2 - \mathbf{a}\_\prime \tag{9}$$

$$\frac{dy}{dt} = -(x^2 + 1)y\_\prime$$

where *α* is a parameter. When we compute the equilibrium points depending on parameter *α*; If *α <* 0, then there is no x-nullclines, hence the system has no equilibrium points. If *α* = 0, then the system has exactly one equilibrium point at (0, 0).

If *<sup>α</sup> <sup>&</sup>gt;* 0, then the system has two equilibrium points (−√*α*, 0) and ( <sup>√</sup>*α*, 0).

**Figure 6.** Supercritical Hopf bifurcation

**Figure 7.** Diagram for Subcritical Hopf bifurcation

Then the Jacobian matrix is

$$J = \begin{bmatrix} 2x & 0\\ -2xy & -(x^2+1) \end{bmatrix}'$$
 
$$\text{where at the equilibrium points } J(0.0) \text{ } = \begin{bmatrix} 0 & 0\\ 0 & -1 \end{bmatrix}, J(\sqrt{\pi}, 0) \text{ } = \begin{bmatrix} -2\sqrt{a} & 0\\ 0 & -a-1 \end{bmatrix} \text{ and}$$

**Figure 8.** Subcritical Hopf bifurcation

**4. Center Manifold teorem**

Let *f*(0) = 0, for the dynamical system

origin.

**Example:** Consider the two dimensional system

±1.It follows that Re*λ*(0) = 0 and Im*λ*(0) �= 0. and also

*dx*

*dy*

*J* =

*dReλ*(*α*)

*dt* <sup>=</sup> <sup>−</sup>*<sup>x</sup>* <sup>+</sup> *<sup>α</sup>y*,

where *α* is the bifurcation parameter. We can easily show that the conditions of the Hopf Bifurcation theorem hold. In this system *f*<sup>1</sup> and *f*<sup>2</sup> are zero. Then the Jacobian matrix is

> *α* 1 −1 *α*

for which the eigenvalues are *λ*1,2 = *α* ± *i* where Re*λ*(*α*) = *α* and the imaginary part Im*λ*(*α*) =

*<sup>d</sup><sup>α</sup>* <sup>|</sup>*α*=0<sup>=</sup> <sup>1</sup> �<sup>=</sup> 0.

Hence, we conclude that there exists a periodic solution for *α* = 0 in every neighborhood of

*dt* <sup>=</sup> *<sup>y</sup>* <sup>+</sup> *<sup>α</sup>x*, (10)

Bifurcation Analysis and Its Applications 19

*<sup>x</sup>*� <sup>=</sup> *<sup>f</sup>*(*x*), *<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***n*, (11)

$$J(\sqrt{\alpha},0) = \begin{bmatrix} 2\sqrt{\alpha} & 0\\ 0 & -\alpha - 1 \end{bmatrix}.$$

For *α* = 0, there will be a line equilibrium (since one of the eigenvalues is zero) and for *α >* 0, the point (−√*α*, 0) is a sink and ( <sup>√</sup>*α*, 0) is a saddle point so that *<sup>α</sup>* <sup>=</sup> 0 is the bifurcation point for this differential equation system.

**Figure 8.** Subcritical Hopf bifurcation

16 Will-be-set-by-IN-TECH

**Figure 6.** Supercritical Hopf bifurcation

**Figure 7.** Diagram for Subcritical Hopf bifurcation

where at the equilibrium points *J*(0.0) =

 . *J* =

 2*x* 0 <sup>−</sup>2*xy* <sup>−</sup>(*x*<sup>2</sup> <sup>+</sup> <sup>1</sup>)

For *α* = 0, there will be a line equilibrium (since one of the eigenvalues is zero) and for *α >* 0,

 ,

<sup>√</sup>*α*, 0) =

<sup>√</sup>*α*, 0) is a saddle point so that *<sup>α</sup>* <sup>=</sup> 0 is the bifurcation point

 −2

<sup>√</sup>*<sup>α</sup>* <sup>0</sup> 0 −*α* − 1

 and

 , *J*(

Then the Jacobian matrix is

 2 <sup>√</sup>*<sup>α</sup>* <sup>0</sup> 0 −*α* − 1

the point (−√*α*, 0) is a sink and (

for this differential equation system.

*J*(

<sup>√</sup>*α*, 0) =

**Example:** Consider the two dimensional system

$$\begin{aligned} \frac{d\mathbf{x}}{dt} &= y + a\mathbf{x}\_{\prime} \\ \frac{dy}{dt} &= -\mathbf{x} + ay\_{\prime} \end{aligned} \tag{10}$$

where *α* is the bifurcation parameter. We can easily show that the conditions of the Hopf Bifurcation theorem hold. In this system *f*<sup>1</sup> and *f*<sup>2</sup> are zero. Then the Jacobian matrix is

$$J = \begin{bmatrix} \alpha & 1\\ -1 & \alpha \end{bmatrix}$$

for which the eigenvalues are *λ*1,2 = *α* ± *i* where Re*λ*(*α*) = *α* and the imaginary part Im*λ*(*α*) = ±1.It follows that Re*λ*(0) = 0 and Im*λ*(0) �= 0. and also

$$\frac{d\mathrm{Re}\lambda(\alpha)}{d\alpha} \mid\_{\alpha=0} = 1 \neq 0.$$

Hence, we conclude that there exists a periodic solution for *α* = 0 in every neighborhood of origin.

### **4. Center Manifold teorem**

Let *f*(0) = 0, for the dynamical system

$$\mathbf{x}' = f(\mathbf{x}), \; \mathbf{x} \in \mathbb{R}^n,\tag{11}$$

#### 18 Will-be-set-by-IN-TECH 20 Numerical Simulation – From Theory to Industry Bifurcation Analysis and Its Applications <sup>19</sup>

and let the eigenvalues of the Jacobian matrix be *λ*1, *λ*2, ..., *λn*. Suppose that, the real parts of the eigenvalues are zero and if not, suppose there are *n*+ numbers of eigenvalues with Re *λ >* 0, *n*<sup>0</sup> number of eigenvalues with Re *λ* = 0 and *n*<sup>−</sup> number of eigenvalues with Re *λ <* 0. Let *T<sup>c</sup>* be the eigenspace on imaginary axis corresponding to *n*<sup>0</sup> eigenvalues. The eigenvalues on the imaginary axis (Re *λ* = 0) are called the critical eigenvalues as on the eigenspace *Tc*. And suppose the function *ϕ<sup>t</sup>* denote the flow corresponding to the equation (11).

With these assumption, we state the Center Manifold theorem as follows;

#### **Theorem: (Center Manifold theorem)**

There exists a locally invariant *C*<sup>∞</sup> center manifold *W<sup>c</sup> loc* (0) such that

$$\mathcal{W}\_{loc}^{\mathbb{C}}(0) = \{ (\mathfrak{x}, y) : y = h(\mathfrak{x}); |\mathfrak{x}| < \delta, h(0) = 0; Df(0) = 0 \}$$

such that the dynamics of the system

$$\begin{aligned} \mathbf{x}' &= A^c \mathbf{x} + r\_1(\mathbf{x}, \mathbf{y})\_{\prime} \\ \mathbf{y}' &= A^s \mathbf{y} + r\_2(\mathbf{x}, \mathbf{y})\_{\prime} \end{aligned}$$

(where *A<sup>c</sup>* and *A<sup>s</sup>* are are the blocks in the canonical form whose diagonals contain the eigenvalues with Re *λ* = 0 and Re *λ <* 0; respectively) restricted to the center manifold are given by

$$\mathbf{x}' = A^c(\mathbf{x}) + r\_1(\mathbf{x}, h(\mathbf{x})).$$

And the manifold *W<sup>c</sup> loc* is called center manifold.

Remark: Center manifolds are not unique.

#### **4.1 Center Manifold reduction for two dimensional systems**

**Example:** Consider the two dimensional system of differential equations

$$x' = xy,$$

$$y' = -y + x^2.$$

and on the other hand,

manifold reduction takes the form

stable for the original system.

and

system.

**Example:** Consider the two dimensional system

*y*� = *h*�

from which we deduce that *a* = 1 and *b* = 0 and

Hopf bifurcations in delay differential equations

**4.2 Center Manifold reduction for Hopf bifurcation**

(*x*)*x*�

*<sup>y</sup>*� <sup>=</sup> <sup>−</sup>*h*(*x*) + *<sup>x</sup>*<sup>2</sup>

*<sup>y</sup>*� <sup>=</sup> <sup>−</sup>*h*(*x*) <sup>−</sup> *<sup>x</sup>*<sup>2</sup>

<sup>=</sup> <sup>−</sup>(*<sup>a</sup>* <sup>+</sup> <sup>1</sup>)*x*<sup>2</sup> <sup>−</sup> *bx*<sup>3</sup> <sup>−</sup> *cx*<sup>4</sup> <sup>−</sup> *dx*<sup>5</sup> <sup>+</sup> *<sup>O</sup>*(*x*6).

Comparing the two expressions, we deduce that *a* = −1, *b* = 0, *c* = −2, *d* = 0 and the center

*<sup>y</sup>* <sup>=</sup> *<sup>h</sup>*(*x*) = <sup>−</sup>*x*<sup>2</sup> <sup>−</sup> <sup>2</sup>*x*<sup>4</sup> <sup>+</sup> *<sup>O</sup>*(*x*7).

Hence for the last equation *x* = 0 is asymptotically stable and therefore (0, 0) is asymptotically

*<sup>y</sup>*� <sup>=</sup> <sup>−</sup>*<sup>y</sup>* <sup>+</sup> *<sup>x</sup>*2.

<sup>=</sup> <sup>2</sup>*a*2*x*<sup>5</sup> + [2*a*(*<sup>b</sup>* <sup>−</sup> <sup>1</sup>) + <sup>3</sup>*ab*]*x*<sup>6</sup> <sup>+</sup> *<sup>O</sup>*(*x*7),

<sup>=</sup> <sup>−</sup>(*<sup>a</sup>* <sup>+</sup> <sup>1</sup>)*x*<sup>2</sup> <sup>−</sup> *bx*<sup>3</sup> <sup>+</sup> *<sup>O</sup>*(*x*4) <sup>=</sup> <sup>2</sup>*a*2*x*<sup>4</sup> <sup>+</sup> <sup>5</sup>*<sup>a</sup>* <sup>−</sup> *bxbx*<sup>5</sup> <sup>+</sup> *<sup>O</sup>*(*x*6),

*x*� = *x*<sup>4</sup> + *O*(*x*5).

For this reduced equation, *x* = 0 is unstable and hence, (0, 0) is also unstable for the original

The aim of this section is to give a formal framework for the analytical bifurcation analysis of

 .

Again (0, 0) is an equilibrium point and the jacobian matrix for the linearized system is

*<sup>J</sup>*(0, 0) =

Consider the transformation *y* = *h*(*x*) = *ax*<sup>2</sup> + *bx*<sup>3</sup> + *cx*<sup>4</sup> + *dx*<sup>5</sup> + *O*(*x*6), which leads

*<sup>x</sup>*� <sup>=</sup> *<sup>x</sup>*2*<sup>y</sup>* <sup>−</sup> *<sup>x</sup>*5, (13)

Bifurcation Analysis and Its Applications 21

*x*� = *f*(*x*, *τ*), *x�***R**<sup>3</sup> (14)

The only equilibrium point is (0, 0), we linearize around that and obtain

$$J(0,0) = \begin{bmatrix} 0 & 0\\ 0 & -1 \end{bmatrix}.$$

Now we look for *y* = *h*(*x*) = *ax*<sup>2</sup> + *bx*<sup>3</sup> + *cx*<sup>4</sup> + *dx*<sup>5</sup> + *O*(*x*6). Then,

$$y' = h'(x)x' = xh'(x)h(x)$$

$$= 2a^2x^4 + 5abx^5 + O(x^6)$$

and on the other hand,

18 Will-be-set-by-IN-TECH

and let the eigenvalues of the Jacobian matrix be *λ*1, *λ*2, ..., *λn*. Suppose that, the real parts of the eigenvalues are zero and if not, suppose there are *n*+ numbers of eigenvalues with Re *λ >* 0, *n*<sup>0</sup> number of eigenvalues with Re *λ* = 0 and *n*<sup>−</sup> number of eigenvalues with Re *λ <* 0. Let *T<sup>c</sup>* be the eigenspace on imaginary axis corresponding to *n*<sup>0</sup> eigenvalues. The eigenvalues on the imaginary axis (Re *λ* = 0) are called the critical eigenvalues as on the eigenspace *Tc*. And

*loc* (0) = {(*x*, *y*) : *y* = *h*(*x*); |*x*| *< δ*, *h*(0) = 0; *D J*(0) = 0}

*x* + *r*1(*x*, *y*),

*y* + *r*2(*x*, *y*),

(*x*) + *r*1(*x*, *h*(*x*)).

(where *A<sup>c</sup>* and *A<sup>s</sup>* are are the blocks in the canonical form whose diagonals contain the eigenvalues with Re *λ* = 0 and Re *λ <* 0; respectively) restricted to the center manifold are

*<sup>y</sup>*� <sup>=</sup> <sup>−</sup>*<sup>y</sup>* <sup>+</sup> *<sup>x</sup>*2.

(*x*)*x*� = *xh*�

= 2*a*2*x*<sup>4</sup> + 5*abx*<sup>5</sup> + *O*(*x*6)

 .

(*x*)*h*(*x*)

*loc* (0) such that

*x*� = *xy*, (12)

suppose the function *ϕ<sup>t</sup>* denote the flow corresponding to the equation (11).

*x*� = *A<sup>c</sup>*

*y*� = *A<sup>s</sup>*

*x*� = *A<sup>c</sup>*

*loc* is called center manifold.

**4.1 Center Manifold reduction for two dimensional systems**

**Example:** Consider the two dimensional system of differential equations

The only equilibrium point is (0, 0), we linearize around that and obtain

Now we look for *y* = *h*(*x*) = *ax*<sup>2</sup> + *bx*<sup>3</sup> + *cx*<sup>4</sup> + *dx*<sup>5</sup> + *O*(*x*6). Then,

*y*� = *h*�

*J*(0, 0) =

With these assumption, we state the Center Manifold theorem as follows;

**Theorem: (Center Manifold theorem)**

*W<sup>c</sup>*

such that the dynamics of the system

Remark: Center manifolds are not unique.

given by

And the manifold *W<sup>c</sup>*

There exists a locally invariant *C*<sup>∞</sup> center manifold *W<sup>c</sup>*

$$\begin{aligned} y' &= -h(\mathbf{x}) - \mathbf{x}^2 \\ &= -(a+1)\mathbf{x}^2 - b\mathbf{x}^3 - c\mathbf{x}^4 - d\mathbf{x}^5 + O(\mathbf{x}^6). \end{aligned}$$

Comparing the two expressions, we deduce that *a* = −1, *b* = 0, *c* = −2, *d* = 0 and the center manifold reduction takes the form

$$y = h(\mathbf{x}) = -\mathbf{x}^2 - 2\mathbf{x}^4 + O(\mathbf{x}^7).$$

Hence for the last equation *x* = 0 is asymptotically stable and therefore (0, 0) is asymptotically stable for the original system.

**Example:** Consider the two dimensional system

$$\begin{aligned} \mathbf{x'} &= \mathbf{x}^2 \mathbf{y} - \mathbf{x}^5, \\ \mathbf{y'} &= -\mathbf{y} + \mathbf{x}^2. \end{aligned} \tag{13}$$

Again (0, 0) is an equilibrium point and the jacobian matrix for the linearized system is

$$J(0,0) = \begin{bmatrix} 0 & 0\\ 0 & -1 \end{bmatrix}$$

.

Consider the transformation *y* = *h*(*x*) = *ax*<sup>2</sup> + *bx*<sup>3</sup> + *cx*<sup>4</sup> + *dx*<sup>5</sup> + *O*(*x*6), which leads

$$\begin{aligned} y' &= h'(x)x' \\ &= 2a^2x^5 + [2a(b-1) + 3ab]x^6 + O(x^7), \end{aligned}$$

and

$$\begin{aligned} y' &= -h(\mathfrak{x}) + \mathfrak{x}^2 \\ &= -(a+1)\mathfrak{x}^2 - b\mathfrak{x}^3 + O(\mathfrak{x}^4) \\ &= 2a^2\mathfrak{x}^4 + 5a - b\mathfrak{x}b\mathfrak{x}^5 + O(\mathfrak{x}^6) \end{aligned}$$

from which we deduce that *a* = 1 and *b* = 0 and

$$\mathbf{x}' = \mathbf{x}^4 + O(\mathbf{x}^5).$$

For this reduced equation, *x* = 0 is unstable and hence, (0, 0) is also unstable for the original system.

#### **4.2 Center Manifold reduction for Hopf bifurcation**

The aim of this section is to give a formal framework for the analytical bifurcation analysis of Hopf bifurcations in delay differential equations

$$\mathbf{x}' = f(\mathbf{x}, \tau), \mathbf{x} \epsilon \mathbb{R}^3 \tag{14}$$

#### 20 Will-be-set-by-IN-TECH 22 Numerical Simulation – From Theory to Industry Bifurcation Analysis and Its Applications <sup>21</sup>

with a single fixed time delay *τ* to be chosen as a bifurcation parameter. Characteristic equations of the delay differential equation form (14) are often studied in order to understand changes in the local stability of equilibria of certain delay differential equations. It is therefore important to determine the values of the delay at which there are roots with zero part. We give a general formalization of these calculations and determine closed form algebraic equations where the stability and amplitude of periodic solutions close to bifurcation can be calculated.

We shall determine the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions by applying the normal form theory and the center manifold theorem by Hassard et al.,[10], and throughout this section, we assume that the three dimensional system of delay differential equations (14) undergoes Hopf bifurcations at the positive equilibrium (*N*∗ <sup>0</sup> , *P*<sup>∗</sup> <sup>0</sup> , *S*<sup>∗</sup> <sup>0</sup> ) at *τ* = *τk*, and *iω*<sup>1</sup> is the corresponding purely imaginary root of the characteristic equation at the positive equilibrium (*N*∗ <sup>0</sup> , *P*<sup>∗</sup> <sup>0</sup> , *S*<sup>∗</sup> <sup>0</sup> ). For the sake of simplicity, we use the notation *iω* for *iω*1.

We first consider the system (14) by the transformation

$$\mathbf{x}\_1 = \mathbf{N} - \mathbf{N}\_{0\prime}^\* \ x\_2 = P - P\_{0\prime}^\* \ x\_3 = \mathbf{S} - \mathbf{S}\_{0\prime}^\* \ t = \frac{\mathbf{t}}{\mathbf{\tau}} \ \ \mathbf{\tau} = \mathbf{\tau}\_k + \mu$$

which is equivalent to the following Functional Differential Equation(FDE) system in *C* = *<sup>C</sup>*([−1, 0], **<sup>R</sup>**3) ,

$$
\dot{\mathbf{x}}(t) = L\_{\mu}(\mathbf{x}\_{t}) + f(\mu, \mathbf{x}\_{t}), \tag{15}
$$

where *<sup>δ</sup>* is the Dirac delta function. For *φ�C*1([−1, 0], **<sup>R</sup>**3), define

*A*(*μ*)*φ* =

Then the system (15) is equivalent to

and a bilinear inner product

On the center manifold *C*0, we have

and (15), we have

where

*R*(*μ*)*φ* =

where *xt*(*θ*) = *<sup>x</sup>*(*<sup>t</sup>* <sup>+</sup> *<sup>θ</sup>*) for *θ�*[−1, 0). For *ψ�C*1([−1, 0],(**R**3)∗), define

*<sup>A</sup>*∗*ψ*(*s*) =

�*ψ*(*s*), *<sup>φ</sup>*(*θ*)� <sup>=</sup> *<sup>ψ</sup>*¯(0)*φ*(0) <sup>−</sup>

coordinates describing center manifold *C*<sup>0</sup> at *μ* = 0. Define

*W*(*t*, *θ*) = *W*(*z*(*t*), *z*¯(*t*), *θ*) = *W*20(*θ*)

= *iωτkz* + *q*¯

= *iωτkz* + *q*¯

= *iωτkz* + *g*(*z*, *z*¯),

<sup>∗</sup>(0)*F*0(*z*, *z*¯) = *g*<sup>20</sup>

*de f*

*g*(*z*, *z*¯) = *q*¯

and

*dφ*(*θ*)

*<sup>d</sup><sup>θ</sup> θ�*[−1, 0), <sup>0</sup> <sup>−</sup><sup>1</sup> *<sup>d</sup>η*(*μ*,*s*)*φ*(*s*) *<sup>θ</sup>* <sup>=</sup> <sup>0</sup>

> <sup>0</sup> *θ�*[−1, 0), *f*(*μ*, *φ*), *θ* = 0.

*x*˙*<sup>t</sup>* = *A*(*μ*)*xt* + *R*(*μ*)*xt*,

<sup>−</sup> *<sup>d</sup>ψ*(*s*) *ds <sup>s</sup>�*(0, 1], <sup>0</sup> <sup>−</sup><sup>1</sup> *<sup>d</sup>ηT*(*t*, 0)*ψ*(−*t*) *<sup>s</sup>* <sup>=</sup> <sup>0</sup>

> 0 −1

where *η*(*θ*) = *η*(*θ*, 0). Then *A*(0) and *A*∗ are adjoint operators. Suppose that *q*(*θ*) and *q*∗(*s*) are eigenvectors of *A* and *A*<sup>∗</sup> corresponding to *iωτ<sup>k</sup>* and −*iωτk*, respectively. Then suppose that *q*(*θ*)=(1, *β*, *γ*)*Teiωτ<sup>k</sup> <sup>θ</sup>* is the eigenvector of *A*(0) corresponding to *iωτk*, then *A*(0)*q*(*θ*) = *iωτkq*(*θ*). Then in the following, we use the theory by Hassard et al.,[10], to compute the

 *θ ξ*=0

*z*2

where *z* and *z*¯ are local coordinates for center manifold *C*<sup>0</sup> in the direction of *q* and *q*¯

<sup>∗</sup>(0)*F*0(*z*, *z*¯)

*z*2

that *W* is real if *xt* is real. We consider only real solutions. For the solution *xt�C*0, since *μ* = 0

*z*˙ = *iωτkz* + �*q*∗(*θ*), *F*(0, *W*(*z*, *z*¯, *θ*) + 2*Rezq*(*θ*))�

∗(0)*F*(0, *W*(*z*, *z*¯, 0) + 2*Rezq*(0))

<sup>2</sup> <sup>+</sup> *<sup>g</sup>*11*zz*¯ <sup>+</sup> *<sup>g</sup>*<sup>02</sup>

*z*(*t*) = �*q*∗, *xt*�, *W*(*t*, *θ*) = *xt* − 2*Rez*(*t*)*q*(*θ*). (17)

<sup>2</sup> <sup>+</sup> *<sup>W</sup>*11(*θ*)*zz*¯ <sup>+</sup> *<sup>W</sup>*02(*θ*)

*z*¯ 2 <sup>2</sup> <sup>+</sup> *<sup>g</sup>*<sup>21</sup> *z*2*z*¯

<sup>2</sup> <sup>+</sup> .... (18)

*<sup>ψ</sup>*¯(*<sup>ξ</sup>* <sup>−</sup> *<sup>θ</sup>*)*dη*(*θ*)*φ*(*ξ*)*dξ*, (16)

Bifurcation Analysis and Its Applications 23

*z*¯2 <sup>2</sup> <sup>+</sup> ...,

∗. Note

where *<sup>x</sup>*(*t*)=(*x*1(*t*), *<sup>x</sup>*2(*t*), *<sup>x</sup>*3(*t*))*T�***R**3, and *<sup>L</sup><sup>μ</sup>* : *<sup>C</sup>* <sup>→</sup> **<sup>R</sup>**3, *<sup>f</sup>* : **<sup>R</sup>** <sup>×</sup> *<sup>C</sup>* <sup>→</sup>**R**<sup>3</sup> are given

respectively, by

$$L\_{\mu}(\phi) = (\tau\_k + \mu) \begin{bmatrix} a\_1 \ a\_2 \ a\_3 \\ a\_4 \ a\_5 \ a\_6 \\ a\_7 \ a\_8 \ a\_9 \end{bmatrix} \begin{bmatrix} \phi\_1(0) \\ \phi\_2(0) \\ \phi\_3(0) \end{bmatrix}$$

$$+ (\tau\_k + \mu) \begin{bmatrix} b\_1 \ b\_2 \ b\_3 \\ b\_4 \ b\_5 \ b\_6 \\ b\_7 \ b\_8 \ b\_9 \end{bmatrix} \begin{bmatrix} \phi\_1(-1) \\ \phi\_2(-1) \\ \phi\_3(-1) \end{bmatrix}'$$

and

$$f(\mu, \phi) = (\tau\_k + \mu) \begin{bmatrix} f\_{11} \\ f\_{12} \\ f\_{13} \end{bmatrix}.$$

By Riesz representation theorem, there exists a function *η*(*θ*, *μ*) of bounded variation for *θ�*[−1, 0], such that

$$L\_{\mu}\phi = \int\_{-1}^{0} d\eta \,(\theta \,\!\!/ 0)\phi(\theta) \,\!\!/ \,\!\!/ \,\!\!/ \,\!\!/ \,\!\!\/} \text{ for } \phi \in \mathbb{C} \,\!\!\/.$$

Indeed we may take

$$\eta(\theta,\mu) = (\tau\_k + \mu) \begin{bmatrix} a\_1 \ a\_2 \ a\_3 \\ a\_4 \ a\_5 \ a\_6 \\ a\_7 \ a\_8 \ a\_9 \end{bmatrix} \delta(\theta) - (\tau\_k + \mu) \begin{bmatrix} b\_1 \ b\_2 \ b\_3 \\ b\_4 \ b\_5 \ b\_6 \\ b\_7 \ b\_8 \ b\_9 \end{bmatrix} \delta(\theta + 1)\_\tau$$

where *<sup>δ</sup>* is the Dirac delta function. For *φ�C*1([−1, 0], **<sup>R</sup>**3), define

$$A(\mu)\phi = \begin{cases} \begin{array}{c} \frac{d\phi(\theta)}{d\theta} \quad \theta\epsilon[-1,0), \\ \int\_{-1}^{0} d\eta(\mu,s)\phi(s) \quad \theta = 0. \end{array} \end{cases}$$

and

20 Will-be-set-by-IN-TECH

with a single fixed time delay *τ* to be chosen as a bifurcation parameter. Characteristic equations of the delay differential equation form (14) are often studied in order to understand changes in the local stability of equilibria of certain delay differential equations. It is therefore important to determine the values of the delay at which there are roots with zero part. We give a general formalization of these calculations and determine closed form algebraic equations where the stability and amplitude of periodic solutions close to bifurcation can be calculated. We shall determine the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions by applying the normal form theory and the center manifold theorem by Hassard et al.,[10], and throughout this section, we assume that the three dimensional system of delay differential equations (14) undergoes Hopf bifurcations at the positive equilibrium

<sup>0</sup> ) at *τ* = *τk*, and *iω*<sup>1</sup> is the corresponding purely imaginary root of the characteristic

<sup>0</sup> , *x*<sup>3</sup> = *S* − *S*<sup>∗</sup>

*a*<sup>1</sup> *a*<sup>2</sup> *a*<sup>3</sup> *a*<sup>4</sup> *a*<sup>5</sup> *a*<sup>6</sup> *a*<sup>7</sup> *a*<sup>8</sup> *a*<sup>9</sup>

> ⎤ ⎦ ⎡ ⎣

> > ⎡ ⎣

*dη*(*θ*, 0)*φ*(*θ*), for *φ�C*.

⎦ *δ*(*θ*) − (*τ<sup>k</sup>* + *μ*)

*f*11 *f*12 *f*13

⎤ ⎦ .

> ⎡ ⎣

*b*<sup>1</sup> *b*<sup>2</sup> *b*<sup>3</sup> *b*<sup>4</sup> *b*<sup>5</sup> *b*<sup>6</sup> *b*<sup>7</sup> *b*<sup>8</sup> *b*<sup>9</sup> ⎤

⎦ *δ*(*θ* + 1),

⎤ ⎦ ⎡ ⎣

*φ*1(−1) *φ*2(−1) *φ*3(−1)

which is equivalent to the following Functional Differential Equation(FDE) system in *C* =

where *<sup>x</sup>*(*t*)=(*x*1(*t*), *<sup>x</sup>*2(*t*), *<sup>x</sup>*3(*t*))*T�***R**3, and *<sup>L</sup><sup>μ</sup>* : *<sup>C</sup>* <sup>→</sup> **<sup>R</sup>**3, *<sup>f</sup>* : **<sup>R</sup>** <sup>×</sup> *<sup>C</sup>* <sup>→</sup>**R**<sup>3</sup> are given

⎡ ⎣

*b*<sup>1</sup> *b*<sup>2</sup> *b*<sup>3</sup> *b*<sup>4</sup> *b*<sup>5</sup> *b*<sup>6</sup> *b*<sup>7</sup> *b*<sup>8</sup> *b*<sup>9</sup>

By Riesz representation theorem, there exists a function *η*(*θ*, *μ*) of bounded variation for

<sup>0</sup>, *<sup>t</sup>* <sup>=</sup> *<sup>t</sup>*

*x*˙(*t*) = *Lμ*(*xt*) + *f*(*μ*, *xt*), (15)

⎤ ⎦

*φ*1(0) *φ*2(0) *φ*3(0)

> ⎤ ⎦ ,

<sup>0</sup> ). For the sake of simplicity, we use the

*<sup>τ</sup>* , *<sup>τ</sup>* <sup>=</sup> *<sup>τ</sup><sup>k</sup>* <sup>+</sup> *<sup>μ</sup>*

<sup>0</sup> , *P*<sup>∗</sup> <sup>0</sup> , *S*<sup>∗</sup>

<sup>0</sup> , *x*<sup>2</sup> = *P* − *P*<sup>∗</sup>

*Lμ*(*φ*)=(*τ<sup>k</sup>* + *μ*)

⎡ ⎣

*f*(*μ*, *φ*)=(*τ<sup>k</sup>* + *μ*)

⎤

+(*τ<sup>k</sup>* + *μ*)

*Lμφ* =

⎡ ⎣ � 0 −1

*a*<sup>1</sup> *a*<sup>2</sup> *a*<sup>3</sup> *a*<sup>4</sup> *a*<sup>5</sup> *a*<sup>6</sup> *a*<sup>7</sup> *a*<sup>8</sup> *a*<sup>9</sup>

(*N*∗ <sup>0</sup> , *P*<sup>∗</sup> <sup>0</sup> , *S*<sup>∗</sup>

notation *iω* for *iω*1.

*<sup>C</sup>*([−1, 0], **<sup>R</sup>**3) ,

respectively, by

*θ�*[−1, 0], such that

Indeed we may take

*η*(*θ*, *μ*)=(*τ<sup>k</sup>* + *μ*)

and

equation at the positive equilibrium (*N*∗

We first consider the system (14) by the transformation

*x*<sup>1</sup> = *N* − *N*<sup>∗</sup>

$$\mathcal{R}(\mu)\phi = \begin{cases} 0 & \theta\varepsilon[-1,0), \\ f(\mu,\phi), \theta = 0. \end{cases}$$

Then the system (15) is equivalent to

$$\dot{\mathbf{x}}\_t = A(\mu)\mathbf{x}\_t + R(\mu)\mathbf{x}\_{t\prime}$$

where *xt*(*θ*) = *<sup>x</sup>*(*<sup>t</sup>* <sup>+</sup> *<sup>θ</sup>*) for *θ�*[−1, 0). For *ψ�C*1([−1, 0],(**R**3)∗), define

$$A^\*\psi(s) = \begin{cases} -\frac{d\psi(s)}{ds} & s\epsilon(0,1],\\ \int\_{-1}^0 d\eta^T(t,0)\psi(-t) & s=0 \end{cases}$$

and a bilinear inner product

$$
\langle \Psi(\mathfrak{s}), \phi(\theta) \rangle = \bar{\psi}(0)\phi(0) - \int\_{-1}^{0} \int\_{\mathfrak{f}=0}^{\theta} \bar{\psi}(\xi-\theta)d\eta(\theta)\phi(\xi)d\xi,\tag{16}
$$

where *η*(*θ*) = *η*(*θ*, 0). Then *A*(0) and *A*∗ are adjoint operators. Suppose that *q*(*θ*) and *q*∗(*s*) are eigenvectors of *A* and *A*<sup>∗</sup> corresponding to *iωτ<sup>k</sup>* and −*iωτk*, respectively. Then suppose that *q*(*θ*)=(1, *β*, *γ*)*Teiωτ<sup>k</sup> <sup>θ</sup>* is the eigenvector of *A*(0) corresponding to *iωτk*, then *A*(0)*q*(*θ*) = *iωτkq*(*θ*). Then in the following, we use the theory by Hassard et al.,[10], to compute the coordinates describing center manifold *C*<sup>0</sup> at *μ* = 0. Define

$$z(t) = \left< q^\*, \mathbf{x}\_l \right>, \qquad \mathcal{W}(t, \theta) = \mathbf{x}\_l - 2\text{Re}\, z(t)q(\theta). \tag{17}$$

On the center manifold *C*0, we have

$$W(t,\theta) = W(z(t),\bar{z}(t),\theta) = W\_{20}(\theta)\frac{z^2}{2} + W\_{11}(\theta)z\bar{z} + W\_{02}(\theta)\frac{\bar{z}^2}{2} + \dots$$

where *z* and *z*¯ are local coordinates for center manifold *C*<sup>0</sup> in the direction of *q* and *q*¯ ∗. Note that *W* is real if *xt* is real. We consider only real solutions. For the solution *xt�C*0, since *μ* = 0 and (15), we have

$$\begin{aligned} \dot{z} &= i\omega \tau\_k z + \langle q^\*(\theta), F(0, W(z, \overline{z}, \theta) + 2\operatorname{Re} z q(\theta)) \rangle \\ &= i\omega \tau\_k z + \dot{q}^\*(0) F(0, W(z, \overline{z}, 0) + 2\operatorname{Re} z q(0)) \\ &\stackrel{def}{=} i\omega \tau\_k z + \dot{q}^\*(0) F\_0(z, \overline{z}) \\ &= i\omega \tau\_k z + g(z, \overline{z}), \end{aligned}$$

where

$$\mathbf{g}(z,\bar{z}) = \bar{q}^\*(0)\mathbf{F}\_0(z,\bar{z}) = \mathbf{g}\_{20}\frac{z^2}{2} + \mathbf{g}\_{11}z\bar{z} + \mathbf{g}\_{02}\frac{z^2}{2} + \mathbf{g}\_{21}\frac{z^2}{2} + \dots \tag{18}$$

#### 22 Will-be-set-by-IN-TECH 24 Numerical Simulation – From Theory to Industry Bifurcation Analysis and Its Applications <sup>23</sup>

By using (17), we have *xt*(*x*1*t*(*θ*), *x*2*t*(*θ*), *x*3*t*(*θ*)) = *W*(*t*, *θ*) + *zq*(*θ*) + *zq*(*θ*), *q*(*θ*)=(1, *β*, *γ*)*Teiωτ<sup>k</sup> <sup>θ</sup>*, and

$$\begin{split} \mathbf{x}\_{1l}(0) &= z + \widetilde{z} + \mathcal{W}\_{20}^{(1)}(0)\frac{z^2}{2} + \mathcal{W}\_{11}^{(1)}(0)z\widetilde{z} + \mathcal{W}\_{02}^{(1)}(0)\frac{z^2}{2} + O(|(z,\widetilde{z})|^3), \\ \mathbf{x}\_{2l}(0) &= \beta\widetilde{z} + \widetilde{\beta}\widetilde{z} + \mathcal{W}\_{20}^{(2)}(0)\frac{z^2}{2} + \mathcal{W}\_{11}^{(2)}(0)z\widetilde{z} + \mathcal{W}\_{02}^{(2)}(0)\frac{\widetilde{z}^2}{2} + O(|(z,\widetilde{z})|^3), \\ \mathbf{x}\_{3l}(0) &= \gamma z + \overline{\gamma}\overline{a} + \mathcal{W}\_{20}^{(3)}(0)\frac{z^2}{2} + \mathcal{W}\_{11}^{(3)}(0)z\widetilde{z} + \mathcal{W}\_{02}^{(3)}(0)\frac{\widetilde{z}^2}{2} + O(|(z,\widetilde{z})|^3), \\ \mathbf{x}\_{1l}(-1) &= ze^{-i\omega\tau\_{l}\theta} + \overline{z}e^{i\omega\tau\_{l}\theta} + \mathcal{W}\_{20}^{(1)}(-1)\frac{z^2}{2} + \mathcal{W}\_{11}^{(1)}(-1)z\overline{z} + \mathcal{W}\_{02}^{(1)}(-1)\frac{z^2}{2} \\ &\quad + O(|(z,\widetilde{z})|^3). \end{split}$$

From the definition of *F*(*μ*, *xt*), we have

$$g(z,\overline{z}) = \overline{q}^\*(0)f\_0(z,\overline{z}) = \mathcal{D}\tau\_k(1,\overline{\beta}^\*,\overline{\gamma}^\*) \begin{bmatrix} f\_{11}^0 \\ f\_{12}^0 \\ f\_{13}^0 \end{bmatrix}$$

and we evaluate *g*(*z*, *z*¯).

To determine *g*21, we need to compute *W*20(*θ*) and *W*11(*θ*). By (15) and (18), we have

$$\begin{split} \dot{W} &= \dot{\mathbf{x}}\_{l} - \dot{\mathbf{z}}\dot{\boldsymbol{q}} + \dot{\overline{\boldsymbol{z}}\dot{\boldsymbol{q}}} \\ &= \begin{Bmatrix} AW - 2\text{Re}\{\dot{\boldsymbol{\eta}}^{\*}(0)f\_{0}q(\boldsymbol{\theta})\}, & \boldsymbol{\theta}\boldsymbol{\varepsilon}[-1, 0) \\ AW - 2\text{Re}\{\dot{\boldsymbol{\eta}}^{\*}(0)f\_{0}q(\boldsymbol{\theta})\} + f\_{0} & \boldsymbol{\theta} = \mathbf{0}, \end{Bmatrix} \end{split} \tag{19}$$
 
$$\stackrel{def}{=} AW + H(z, \overline{z}, \boldsymbol{\theta}),$$

where

$$H(z.\bar{z},\theta) = H\_{20}(\theta)\frac{z^2}{2} + H\_{11}(\theta)z\bar{z} + H\_{02}(\theta)\frac{\bar{z}^2}{2} + \dots \tag{20}$$

Note that on the center manifold *C*<sup>0</sup> near to the origin,

$$
\dot{W} = W\_{\overline{z}} \dot{z} + W\_{\overline{z}} \overline{z}. \tag{21}
$$

From (22) and (24) and the definition of *A*, we get

vector. From the definition of *A* and (22), we obtain 0 −1

and

in terms of *g*�

Noticing *<sup>q</sup>*(*θ*) = *<sup>q</sup>*(0)*eiωτ<sup>k</sup> <sup>θ</sup>*, we evaluate *<sup>W</sup>*20(*θ*) by *<sup>E</sup>*<sup>1</sup> = (*E*(1)

 0 −1

2*ωτ<sup>k</sup>*

*<sup>c</sup>*1(0) = *<sup>i</sup>*

and we state this as in the following theorem.

increases if *T*<sup>2</sup> *>* 0, it decreases if *T*<sup>2</sup> *<* 0.

*4.2.1 Numerical example of Center Manifold reduction*

Consider the following system with discrete time delay *τ*;

*dN*(*t*)

*dP*(*t*)

*dS*(*t*)

*<sup>W</sup>*˙ <sup>20</sup>(*θ*) = <sup>2</sup>*iωτkW*20(*θ*) <sup>−</sup> *<sup>g</sup>*20*q*(*θ*) <sup>−</sup> *<sup>g</sup>*¯02*q*¯(*θ*).

where *dη*(*θ*) = *η*(*θ*, 0). Next we compute *W*20(*θ*) and *W*11(*θ*) from (25) and (26) and determine the following values to investigate the qualities of bifurcating periodic solution in the center manifold at the critical value *τk*. For this purpose, we express the direction of Hopf bifurcation

(*g*<sup>20</sup> *g*<sup>11</sup> − 2|*g*11|

*<sup>μ</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup> *Re*{*c*1(0)} *Re*{*λ* � (*τk*)} ,

*<sup>T</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup> *Im*{*c*1(0)} <sup>+</sup> *<sup>μ</sup>*<sup>2</sup> *Im*{*<sup>λ</sup>*

*ωτ<sup>k</sup>*

**Theorem :** *μ*<sup>2</sup> determines the direction of Hopf bifurcation; if *μ*<sup>2</sup> *>* 0, then the Hopf bifurcation is supercritical and the bifurcating periodic solutions exist for *τ > τ*0, if *μ*<sup>2</sup> *<* 0, then the Hopf bifurcation is subcritical and the bifurcating periodic solutions exist for *τ < τ*0. *β*<sup>2</sup> determines the stability of the bifurcating periodic solutions; bifurcating periodic solutions are stable if *β*<sup>2</sup> *<* 0, unstable if *β*<sup>2</sup> *>* 0. *T*<sup>2</sup> determines the period of the bifurcating solution; the period

In the following section, we shall give a numerical example to verify the theoretical results.

*dt* <sup>=</sup> *<sup>r</sup>*1*N*(*t*) <sup>−</sup> *�P*(*t*)*N*(*t*),

*N*(*t* − *τ*)

),

(28)

*dt* <sup>=</sup> *<sup>P</sup>*(*t*)(*r*<sup>2</sup> <sup>−</sup> *<sup>θ</sup> <sup>S</sup>*(*t*)

*dt* <sup>=</sup> *<sup>α</sup>P*(*t*) <sup>−</sup> *<sup>α</sup>S*(*t*).

*ijs* and eigenvalues *λτk*. And then we can evaluate the following values;

<sup>2</sup> <sup>−</sup> <sup>|</sup>*g*02<sup>|</sup>

� (*τk*)}

2 <sup>3</sup> ) + *<sup>g</sup>*<sup>21</sup>

2 ,

*β*<sup>2</sup> = 2*Re*{*c*1(0)}, (27)

<sup>1</sup> , *<sup>E</sup>*(2)

*dη*(*θ*)*W*11(*θ*) = −*H*11(0), (26)

*dη*(*θ*)*W*20(*θ*) = 2*iωτkW*20(0) − *H*20(0), (25)

<sup>1</sup> , *<sup>E</sup>*(3)

<sup>1</sup> )*�***R**<sup>3</sup> is a constant

Bifurcation Analysis and Its Applications 25

Thus we obtain,

$$W(A - 2i\omega \tau\_k) W\_{20}(\theta) = -H\_{20}(\theta), \qquad AW\_{11}(\theta) = -H\_{11}(\theta). \tag{22}$$

By using (19), for *θ�*[−1, 0),

$$H(z.\overline{z}, \theta) = \overline{q}^\*(0)f\_0q(\theta) - q^\*(0)f\_0(0)\overline{q}(\theta) = -\text{g}q(\theta) - \overline{\text{g}}\,\overline{q}(\theta). \tag{23}$$

Comparing the coefficients with (20), we obtain the following

$$H\_{20}(\theta) = -g\_{20}q(\theta) - \bar{g}\_{02}\bar{q}(\theta), \quad H\_{11}(\theta) = -g\_{11}q(\theta) - \bar{g}\_{11}\bar{q}(\theta). \tag{24}$$

From (22) and (24) and the definition of *A*, we get

$$
\dot{W}\_{20}(\theta) = 2i\omega \tau\_k W\_{20}(\theta) - \mathfrak{g}\_{20}q(\theta) - \mathfrak{g}\_{02}\bar{q}(\theta).
$$

Noticing *<sup>q</sup>*(*θ*) = *<sup>q</sup>*(0)*eiωτ<sup>k</sup> <sup>θ</sup>*, we evaluate *<sup>W</sup>*20(*θ*) by *<sup>E</sup>*<sup>1</sup> = (*E*(1) <sup>1</sup> , *<sup>E</sup>*(2) <sup>1</sup> , *<sup>E</sup>*(3) <sup>1</sup> )*�***R**<sup>3</sup> is a constant vector. From the definition of *A* and (22), we obtain

$$\int\_{-1}^{0} d\eta(\theta) \mathcal{W}\_{20}(\theta) = 2i\omega \tau\_k \mathcal{W}\_{20}(0) - H\_{20}(0),\tag{25}$$

and

22 Will-be-set-by-IN-TECH

<sup>20</sup> (−1)

<sup>∗</sup>(0)*f*0(*z*, *z*¯) = *D*¯ *τk*(1, *β*¯ <sup>∗</sup>

<sup>=</sup> { *AW* <sup>−</sup> <sup>2</sup>*Re*{*q*¯∗(0)*f*0*q*(*θ*)}, *θ�*[−1, 0) *AW* − 2*Re*{*q*¯∗(0)*f*0*q*(*θ*)} + *f*<sup>0</sup> *θ* = 0,

<sup>2</sup> <sup>+</sup> *<sup>H</sup>*11(*θ*)*zz*¯ <sup>+</sup> *<sup>H</sup>*02(*θ*)

(*A* − 2*iωτk*)*W*20(*θ*) = −*H*20(*θ*), *AW*11(*θ*) = −*H*11(*θ*). (22)

*H*20(*θ*) = −*g*20*q*(*θ*) − *g*¯02*q*¯(*θ*), *H*11(*θ*) = −*g*11*q*(*θ*) − *g*¯11*q*¯(*θ*). (24)

<sup>∗</sup>(0)*f*0*q*(*θ*) − *q*∗(0)*f*0(0)*q*¯(*θ*) = −*gq*(*θ*) − *g*¯ *q*¯(*θ*). (23)

To determine *g*21, we need to compute *W*20(*θ*) and *W*11(*θ*). By (15) and (18), we have

*z*2

<sup>11</sup> (0)*zz*¯ <sup>+</sup> *<sup>W</sup>*(1)

<sup>11</sup> (0)*zz*¯ <sup>+</sup> *<sup>W</sup>*(2)

<sup>11</sup> (0)*zz*¯ <sup>+</sup> *<sup>W</sup>*(3)

*z*2 <sup>2</sup> <sup>+</sup> *<sup>W</sup>*(1)

<sup>02</sup> (0) *z*¯ 2

> <sup>02</sup> (0) *z*¯ 2

<sup>02</sup> (0) *z*¯ 2

, *γ*¯ ∗)

⎡ ⎣ *f* 0 11 *f* 0 12 *f* 0 13

*zq* (19)

*z*¯ 2

*W*˙ = *Wzz*˙ + *Wz*¯*z*¯. (21)

<sup>11</sup> (−1)*zz*¯ <sup>+</sup> *<sup>W</sup>*(1)

⎤ ⎦

<sup>2</sup> <sup>+</sup> *<sup>O</sup>*(|(*z*, *<sup>z</sup>*¯)<sup>|</sup>

<sup>2</sup> <sup>+</sup> *<sup>O</sup>*(|(*z*, *<sup>z</sup>*¯)<sup>|</sup>

<sup>2</sup> <sup>+</sup> *<sup>O</sup>*(|(*z*, *<sup>z</sup>*¯)<sup>|</sup>

3 ),

<sup>02</sup> (−1)

<sup>2</sup> <sup>+</sup> .... (20)

3 ),

> 3 ),

> > *z*¯2 2

By using (17), we have *xt*(*x*1*t*(*θ*), *x*2*t*(*θ*), *x*3*t*(*θ*)) = *W*(*t*, *θ*) + *zq*(*θ*) + *zq*(*θ*),

<sup>20</sup> (0) *z*2 <sup>2</sup> <sup>+</sup> *<sup>W</sup>*(2)

<sup>20</sup> (0) *z*2 <sup>2</sup> <sup>+</sup> *<sup>W</sup>*(3)

<sup>20</sup> (0) *z*2 <sup>2</sup> <sup>+</sup> *<sup>W</sup>*(1)

*q*(*θ*)=(1, *β*, *γ*)*Teiωτ<sup>k</sup> <sup>θ</sup>*, and

*<sup>x</sup>*1*t*(0) = *<sup>z</sup>* <sup>+</sup> *<sup>z</sup>*¯ <sup>+</sup> *<sup>W</sup>*(1)

*<sup>x</sup>*2*t*(0) = *<sup>β</sup><sup>z</sup>* <sup>+</sup> *<sup>β</sup>*¯ *<sup>z</sup>* <sup>+</sup> *<sup>W</sup>*(2)

*<sup>x</sup>*3*t*(0) = *<sup>γ</sup><sup>z</sup>* <sup>+</sup> *γα* <sup>+</sup> *<sup>W</sup>*(3)

From the definition of *F*(*μ*, *xt*), we have

and we evaluate *g*(*z*, *z*¯).

where

Thus we obtain,

By using (19), for *θ�*[−1, 0),

*<sup>x</sup>*1*t*(−1) = *ze*−*iωτ<sup>k</sup> <sup>θ</sup>* <sup>+</sup> *ze*¯ *<sup>i</sup>ωτk<sup>θ</sup>* <sup>+</sup> *<sup>W</sup>*(1)

*g*(*z*, *z*¯) = *q*¯

*<sup>W</sup>*˙ <sup>=</sup> *<sup>x</sup>*˙*<sup>t</sup>* <sup>−</sup> *zq*˙ <sup>+</sup> .

*H*(*z*.*z*¯, *θ*) = *H*20(*θ*)

Comparing the coefficients with (20), we obtain the following

Note that on the center manifold *C*<sup>0</sup> near to the origin,

*H*(*z*.*z*¯, *θ*) = *q*¯

= *AW* + *H*(*z*.*z*¯, *θ*),

*de f*

3 ).

+*O*(|(*z*, *z*¯)|

$$\int\_{-1}^{0} d\eta(\theta) W\_{11}(\theta) = -H\_{11}(0),\tag{26}$$

where *dη*(*θ*) = *η*(*θ*, 0). Next we compute *W*20(*θ*) and *W*11(*θ*) from (25) and (26) and determine the following values to investigate the qualities of bifurcating periodic solution in the center manifold at the critical value *τk*. For this purpose, we express the direction of Hopf bifurcation in terms of *g*� *ijs* and eigenvalues *λτk*. And then we can evaluate the following values;

$$c\_1(0) = \frac{i}{2\omega\tau\_k} (\lg\_2 \varrho\_{11} - 2|\varrho\_{11}|^2 - \frac{|\varrho\_{02}|^2}{3}) + \frac{\varrho\_{21}}{2},$$

$$\mu\_2 = -\frac{\text{Re}\{c\_1(0)\}}{\text{Re}\{\lambda'(\tau\_k)\}},$$

$$\beta\_2 = 2\text{Re}\{c\_1(0)\},\tag{27}$$

$$T\_2 = -\frac{\text{Im}\{c\_1(0)\} + \mu\_2 \text{Im}\{\lambda'(\tau\_k)\}}{\omega\tau\_k}$$

and we state this as in the following theorem.

**Theorem :** *μ*<sup>2</sup> determines the direction of Hopf bifurcation; if *μ*<sup>2</sup> *>* 0, then the Hopf bifurcation is supercritical and the bifurcating periodic solutions exist for *τ > τ*0, if *μ*<sup>2</sup> *<* 0, then the Hopf bifurcation is subcritical and the bifurcating periodic solutions exist for *τ < τ*0. *β*<sup>2</sup> determines the stability of the bifurcating periodic solutions; bifurcating periodic solutions are stable if *β*<sup>2</sup> *<* 0, unstable if *β*<sup>2</sup> *>* 0. *T*<sup>2</sup> determines the period of the bifurcating solution; the period increases if *T*<sup>2</sup> *>* 0, it decreases if *T*<sup>2</sup> *<* 0.

In the following section, we shall give a numerical example to verify the theoretical results.

#### *4.2.1 Numerical example of Center Manifold reduction*

Consider the following system with discrete time delay *τ*;

$$\begin{aligned} \frac{dN(t)}{dt} &= r\_1 N(t) - \epsilon P(t) N(t),\\ \frac{dP(t)}{dt} &= P(t) (r\_2 - \theta \frac{S(t)}{N(t-\tau)}),\\ \frac{dS(t)}{dt} &= \alpha P(t) - \alpha S(t). \end{aligned} \tag{28}$$

#### 24 Will-be-set-by-IN-TECH 26 Numerical Simulation – From Theory to Industry Bifurcation Analysis and Its Applications <sup>25</sup>

Now we present some numerical simulations by using MATLAB(7.6.0) programming (Çelik-3). We simulate the predator-prey system (28) by choosing the parameters *r*<sup>1</sup> = 0.45, *r*<sup>2</sup> = 0.1, *θ* = 0.05, *�* = 0.03, and *α* = 1, i.e., we consider the following system

$$\begin{aligned} \frac{dN(t)}{dt} &= 0.45N(t) - 0.03P(t)N(t),\\ \frac{dP(t)}{dt} &= P(t)(0.1 - 0.05\frac{S(t)}{N(t-\tau)}),\\ \frac{dS(t)}{dt} &= P(t) - S(t), \end{aligned} \tag{29}$$

0 50 100 150 200 250

Bifurcation Analysis and Its Applications 27

0 50 100 150 200 250

Time(t)

**Figure 10.** The trajectory of predator density versus time with the initial condition *P*<sup>0</sup> = 25. The graph of solutions of the model (29) when *τ* = 1.5 *< τ*0, where the equilibrium point *E*<sup>∗</sup> is asymptotically stable.

Let *g*(*y*) be a positive function. Deduce the stability of *y*<sup>0</sup> as an equilibrium solution of the

Time(t)

**Figure 9.** The trajectory of prey density versus time with the initial condition *N*<sup>0</sup> = 50. The graph of solutions of the model (29) when *τ* = 1.5 *< τ*0, where the equilibrium point *E*<sup>∗</sup> is asymptotically stable.

5

equation

10

15

20

P(t)

25

30

35

N(t)

which has only one positive equilibrium *E*∗ = (*N*∗ <sup>0</sup> , *P*<sup>∗</sup> <sup>0</sup> , *S*<sup>∗</sup> <sup>0</sup> )=(7.5, 15, 15). By algorithms in the previous sections, we obtain *τ*<sup>0</sup> = 1.5663, *ω*<sup>1</sup> = 0.0045, *z*<sup>1</sup> = *ω*<sup>2</sup> <sup>1</sup> and *g*� (*z*1) = 1.1944 <sup>×</sup> <sup>10</sup>−<sup>4</sup> *<sup>&</sup>gt;* 0 which leads to *dReλ*(*τ*<sup>0</sup> ) *<sup>d</sup><sup>τ</sup>* = 5.8982 *>* 0. So by the theorem above, the equilibrium point *E*∗ is asymptotically stable when *τ�* [0, *τ*0)=[0, 1.5663) and unstable when *τ >* 1.5663 and also Hopf bifurcation occurs at *τ* = *τ*<sup>0</sup> = 1.5663 as it is illustrated by computer simulations.

By the theory of Hassard et al.,[10], as it is discussed in previous section, we also determine the direction of Hopf bifurcation and the other properties of bifurcating periodic solutions. From the formulae in Section 5.2 we evaluate the values of *μ*2, *β*<sup>2</sup> and *T*<sup>2</sup> as

$$\mu\_2 = 0.0981 > 0, \; \beta\_2 = -0.5785 < 0, \; T\_2 = 7.1165 > 0,$$

from which we conclude that Hopf bifurcation of system (29) occurring at *τ*<sup>0</sup> = 1.5663 is supercritical and the bifurcating periodic solution exists when *τ* crosses *τ*<sup>0</sup> to the right, and also the bifurcating periodic solution is stable.

In computer simulations, the initial conditions are taken as (*N*0, *P*0, *S*0)=(50, 25, 25) and MATLAB DDE (Delay Differential Equations) solver is used to simulate the system (29). We first take *τ* = 1.5 *< τ*<sup>0</sup> and plot the density functions *N*(*t*), *P*(*t*) and *S*(*t*) in Figs.9,10,11 respectively which shows the positive equilibrium is asymptotically stable for *τ < τ*0.. Moreover in Fig.12, we illustrate the asymptotic stability in three dimension.

However in Figs.13,14,15 and 16 below, we take *τ* = 2 *> τ*<sup>0</sup> sufficiently close to *τ*<sup>0</sup> which illustrates the existence of bifurcating periodic solutions from the equilibrium point *E*∗.

#### **4.3 Exercises**

**1.** Consider the Van der Pol equation

$$
\ddot{\mathbf{x}} - (1 - x^2)\dot{\mathbf{x}} + \mathbf{x} = \mathbf{0}
$$

and convert this into a system and check the stability of fixed points.

**2.** Let *y*<sup>0</sup> be an equilibrium point of the equation

$$\frac{dy}{dt} = f(y).$$

24 Will-be-set-by-IN-TECH

Now we present some numerical simulations by using MATLAB(7.6.0) programming (Çelik-3). We simulate the predator-prey system (28) by choosing the parameters *r*<sup>1</sup> = 0.45,

*dt* <sup>=</sup> 0.45*N*(*t*) <sup>−</sup> 0.03*P*(*t*)*N*(*t*),

*N*(*t* − *τ*)

<sup>0</sup> , *P*<sup>∗</sup> <sup>0</sup> , *S*<sup>∗</sup> ),

*<sup>d</sup><sup>τ</sup>* = 5.8982 *>* 0. So by the theorem above, the

<sup>0</sup> )=(7.5, 15, 15). By algorithms

<sup>1</sup> and *g*�

(29)

(*z*1) =

*dt* <sup>=</sup> *<sup>P</sup>*(*t*)(0.1 <sup>−</sup> 0.05 *<sup>S</sup>*(*t*)

equilibrium point *E*∗ is asymptotically stable when *τ�* [0, *τ*0)=[0, 1.5663) and unstable when *τ >* 1.5663 and also Hopf bifurcation occurs at *τ* = *τ*<sup>0</sup> = 1.5663 as it is illustrated by computer

By the theory of Hassard et al.,[10], as it is discussed in previous section, we also determine the direction of Hopf bifurcation and the other properties of bifurcating periodic solutions.

*μ*<sup>2</sup> = 0.0981 *>* 0, *β*<sup>2</sup> = −0.5785 *<* 0, *T*<sup>2</sup> = 7.1165 *>* 0,

from which we conclude that Hopf bifurcation of system (29) occurring at *τ*<sup>0</sup> = 1.5663 is supercritical and the bifurcating periodic solution exists when *τ* crosses *τ*<sup>0</sup> to the right, and

In computer simulations, the initial conditions are taken as (*N*0, *P*0, *S*0)=(50, 25, 25) and MATLAB DDE (Delay Differential Equations) solver is used to simulate the system (29). We first take *τ* = 1.5 *< τ*<sup>0</sup> and plot the density functions *N*(*t*), *P*(*t*) and *S*(*t*) in Figs.9,10,11 respectively which shows the positive equilibrium is asymptotically stable for *τ < τ*0..

However in Figs.13,14,15 and 16 below, we take *τ* = 2 *> τ*<sup>0</sup> sufficiently close to *τ*<sup>0</sup> which illustrates the existence of bifurcating periodic solutions from the equilibrium point *E*∗.

*<sup>x</sup>*¨ <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>x</sup>*2)*x*˙ <sup>+</sup> *<sup>x</sup>* <sup>=</sup> <sup>0</sup>

*dt* <sup>=</sup> *<sup>f</sup>*(*y*).

*dy*

*r*<sup>2</sup> = 0.1, *θ* = 0.05, *�* = 0.03, and *α* = 1, i.e., we consider the following system

*dt* <sup>=</sup> *<sup>P</sup>*(*t*) <sup>−</sup> *<sup>S</sup>*(*t*),

in the previous sections, we obtain *τ*<sup>0</sup> = 1.5663, *ω*<sup>1</sup> = 0.0045, *z*<sup>1</sup> = *ω*<sup>2</sup>

From the formulae in Section 5.2 we evaluate the values of *μ*2, *β*<sup>2</sup> and *T*<sup>2</sup> as

Moreover in Fig.12, we illustrate the asymptotic stability in three dimension.

and convert this into a system and check the stability of fixed points.

*dN*(*t*)

*dP*(*t*)

*dS*(*t*)

which has only one positive equilibrium *E*∗ = (*N*∗

1.1944 <sup>×</sup> <sup>10</sup>−<sup>4</sup> *<sup>&</sup>gt;* 0 which leads to *dReλ*(*τ*<sup>0</sup> )

also the bifurcating periodic solution is stable.

**1.** Consider the Van der Pol equation

**2.** Let *y*<sup>0</sup> be an equilibrium point of the equation

simulations.

**4.3 Exercises**

**Figure 9.** The trajectory of prey density versus time with the initial condition *N*<sup>0</sup> = 50. The graph of solutions of the model (29) when *τ* = 1.5 *< τ*0, where the equilibrium point *E*<sup>∗</sup> is asymptotically stable.

**Figure 10.** The trajectory of predator density versus time with the initial condition *P*<sup>0</sup> = 25. The graph of solutions of the model (29) when *τ* = 1.5 *< τ*0, where the equilibrium point *E*<sup>∗</sup> is asymptotically stable.

Let *g*(*y*) be a positive function. Deduce the stability of *y*<sup>0</sup> as an equilibrium solution of the equation

**Figure 11.** The trajectories of *S*(*t*) versus time with the initial condition *S*<sup>0</sup> = 25. The graph of solutions of the model (29) when *τ* = 1.5 *< τ*0, where the equilibrium point *E*<sup>∗</sup> is asymptotically stable.

0 50 100 150 200 250

Time(t)

*dt* <sup>=</sup> *<sup>f</sup>*(*y*)*g*(*y*)

*<sup>x</sup>*� <sup>=</sup> <sup>−</sup>*x*<sup>3</sup> <sup>−</sup> *<sup>y</sup>*,

*<sup>y</sup>*� <sup>=</sup> *<sup>x</sup>* <sup>−</sup> *<sup>y</sup>*3,

*<sup>x</sup>*� <sup>=</sup> *<sup>y</sup>* <sup>−</sup> *xy*2,

*<sup>y</sup>*� <sup>=</sup> <sup>−</sup>*<sup>x</sup>* <sup>+</sup> *<sup>x</sup>*2*y*,

**5.** Analyze the bifurcation properties of the following problems choosing *r* as bifurcation

*dt* = *f*(*y*). Repeat the same

Bifurcation Analysis and Its Applications 29

*dy*

from its stability as an equilibrium solution of the equation *dy*

0

**Figure 13.** The trajectory of *N*(*t*) when *τ* = 2 *> τ*0.

question if *g*(*y*)is a negative function. **3.** For the following nonlinear system

determine the stability of fixed points. **4.** For the following nonlinear system

classify the fixed points.

parameter,

20

40

60

80

N(t)

100

120

140

160

**Figure 12.** The trajectory of *N*, *P*, *S* in three dimension with the initial condition (50, 25, 25) when *τ* = 1.5 *< τ*<sup>0</sup> for which the equilibrium point *E*<sup>∗</sup> is asymptotically stable.

**Figure 13.** The trajectory of *N*(*t*) when *τ* = 2 *> τ*0.

26 Will-be-set-by-IN-TECH

0 50 100 150 200 250

Time(t)

**Figure 11.** The trajectories of *S*(*t*) versus time with the initial condition *S*<sup>0</sup> = 25. The graph of solutions

0

N(t) P(t)

0

**Figure 12.** The trajectory of *N*, *P*, *S* in three dimension with the initial condition (50, 25, 25) when

10

20

*τ* = 1.5 *< τ*<sup>0</sup> for which the equilibrium point *E*<sup>∗</sup> is asymptotically stable.

30

S(t)

10

20

30

40

50

of the model (29) when *τ* = 1.5 *< τ*0, where the equilibrium point *E*<sup>∗</sup> is asymptotically stable.

5

10

15

20

S(t)

25

30

35

$$\frac{dy}{dt} = f(y)g(y)$$

from its stability as an equilibrium solution of the equation *dy dt* = *f*(*y*). Repeat the same question if *g*(*y*)is a negative function.

**3.** For the following nonlinear system

$$\mathbf{x}' = -\mathbf{x}^3 - \mathbf{y}\_{\prime\prime}$$

$$\mathbf{y}' = \mathbf{x} - \mathbf{y}^3\_{\prime\prime}$$

determine the stability of fixed points.

**4.** For the following nonlinear system

$$\mathbf{x}' = \mathbf{y} - \mathbf{x}\mathbf{y}^2$$

$$\mathbf{y}' = -\mathbf{x} + \mathbf{x}^2\mathbf{y}\_r$$

classify the fixed points.

**5.** Analyze the bifurcation properties of the following problems choosing *r* as bifurcation parameter,

0 50 100 150 200 250

Bifurcation Analysis and Its Applications 31

Time(t)

*dt* <sup>=</sup> *<sup>x</sup>* <sup>+</sup> *<sup>y</sup>* <sup>−</sup> *<sup>x</sup>*<sup>2</sup> <sup>+</sup> *<sup>y</sup>*2,

*dt* <sup>=</sup> <sup>−</sup>2*<sup>x</sup>* <sup>−</sup> *<sup>y</sup>* <sup>+</sup> *xy*,

*dt* <sup>=</sup> <sup>2</sup>*<sup>x</sup>* <sup>+</sup> *<sup>y</sup>* <sup>−</sup> *<sup>x</sup>*(*x*<sup>2</sup> <sup>+</sup> *<sup>y</sup>*2),

*dt* <sup>=</sup> <sup>−</sup>*<sup>x</sup>* <sup>+</sup> <sup>2</sup>*<sup>y</sup>* <sup>−</sup> *<sup>y</sup>*(*x*<sup>2</sup> <sup>+</sup> *<sup>y</sup>*2).

*dt* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>y</sup>*<sup>2</sup> <sup>−</sup> *<sup>x</sup>*2,

0

**8.**

**Figure 15.** The trajectory of *S*(*t*) when *τ* = 2 *> τ*0.

for the above system, classify the fixed points.

**7.** Show that the following system is structurally unstable,

*dx*

*dy*

and the following system is structural stable. Explain our reason.

*dx*

*dy*

*dx dt* <sup>=</sup> *xy*,

*dy*

5

10

15

20

S(t)

25

30

35

40

**Figure 14.** The trajectory of *P*(*t*) when *τ* = 2 *> τ*0.

$$a) \mathbf{x'} = -\mathbf{x} + \beta \tanh \mathbf{x},$$

$$b) \mathbf{x'} = r\mathbf{x} - 4\mathbf{x}^3,$$

$$c) \mathbf{x'} = r\mathbf{x} - \sin(\mathbf{x}),$$

$$d) \mathbf{x'} = r\mathbf{x} + 4\mathbf{x}^3,$$

$$e) \mathbf{x'} = r\mathbf{x} - \sin h(\mathbf{x}),$$

$$c) \mathbf{x'} = \mathbf{x} + \frac{r\mathbf{x}}{1 + \mathbf{x}^2}.$$

**6.** Find the equilibrium points and identify the bifurcation in the following system, and sketch the appropriate bifurcation diagram and phase portraits:

$$\frac{dx}{dt} = (1+\pi)y - x - 2x^3,$$

$$\frac{dy}{dt} = x - y - y^3.$$

Then compute the extended center manifold near the bifurcation point by choosing *τ* as bifurcation parameter.

**Figure 15.** The trajectory of *S*(*t*) when *τ* = 2 *> τ*0.

**7.** Show that the following system is structurally unstable,

$$\frac{dx}{dt} = x + y - x^2 + y^2y$$

$$\frac{dy}{dt} = -2x - y + xy\_t$$

and the following system is structural stable. Explain our reason.

$$\frac{d\mathbf{x}}{dt} = 2\mathbf{x} + \mathbf{y} - \mathbf{x}(\mathbf{x}^2 + \mathbf{y}^2),$$

$$\frac{d\mathbf{y}}{dt} = -\mathbf{x} + 2\mathbf{y} - \mathbf{y}(\mathbf{x}^2 + \mathbf{y}^2).$$

**8.**

28 Will-be-set-by-IN-TECH

0 50 100 150 200 250

Time(t)

*a*)*x*� = −*x* + *β* tanh *x*,

*<sup>b</sup>*)*x*� <sup>=</sup> *rx* <sup>−</sup> <sup>4</sup>*x*3,

*d*)*x*� = *rx* + 4*x*3,

*c*)*x*� = *x* +

the appropriate bifurcation diagram and phase portraits:

*dx*

*dy*

*c*)*x*� = *rx* − *sin*(*x*),

*e*)*x*� = *rx* − sin *h*(*x*),

**6.** Find the equilibrium points and identify the bifurcation in the following system, and sketch

*dt* = (<sup>1</sup> <sup>+</sup> *<sup>τ</sup>*)*<sup>y</sup>* <sup>−</sup> *<sup>x</sup>* <sup>−</sup> <sup>2</sup>*x*3,

Then compute the extended center manifold near the bifurcation point by choosing *τ* as

*dt* <sup>=</sup> *<sup>x</sup>* <sup>−</sup> *<sup>y</sup>* <sup>−</sup> *<sup>y</sup>*3.

*rx* <sup>1</sup> <sup>+</sup> *<sup>x</sup>*<sup>2</sup> .

0

bifurcation parameter.

**Figure 14.** The trajectory of *P*(*t*) when *τ* = 2 *> τ*0.

5

10

15

20

P(t)

25

30

35

40

$$\begin{aligned} \frac{dx}{dt} &= xy\_\prime\\ \frac{dy}{dt} &= 1 - y^2 - x^2 \end{aligned}$$

for the above system, classify the fixed points.

in the (*x*, *y*) plane.

the trajectories?

them.

or not.

**Author details**

**5. References**

Canan Çelik Karaaslanlı *Bahçe¸sehir University, Turkey*

(a) Determine all fixed points of the system.

Which fixed point does the trajectory through (2, 0) approach?

in the earth's atmosphere) in three dimension as follows;

**12.** For the Holling-Tanner type Predator-Prey model

**13.** Consider the Delayed Predator-Prey model

*x*�

*y*�

[1] Allen L.J.S., An Introduction to Mathematical Biology, 2007.

Grundlehren Math. Wiss.,(250), Springer, 1983.

Dynamical Systems on a Plane, Israel Program Sci. Transl., 1971.

(b) Let *r*<sup>2</sup> = *x*<sup>2</sup> + *y*2. Show that *r*� = 0 Considering the result of (c), what does this imply for

Bifurcation Analysis and Its Applications 33

(c) Sketch the phase portrait in the (*x*, *y*) plane, including trajectories through (1, 0) and (2, 0).

**11.** Consider the Lorenz system (the model of heat convection by Rayleigh-Benard occurring

*x*� = *σ*(*y* − *x*), *y*� = *rx* + −*xz* − *y*, *z*� = *xy* − *bz*,

*<sup>x</sup>*� <sup>=</sup> *<sup>x</sup>*(<sup>1</sup> <sup>−</sup> *<sup>x</sup>*) <sup>−</sup> *<sup>x</sup>*

*x* ),

where *x*(0) *>* 0, *y*(0) *>* 0 and *a*, *δ*, *β* are positive constants. Find the equilibria and classify

(*t*) = *rx*(*t*) − *bx*(*t*)*x*(*t* − *τ*) − *αx*(*t*)*y*(*t*),

with time delay *τ* and positive constants *r*, *b*, *α*, *c* and *β*. By choosing *τ* as bifurcation parameter, check whether bifurcating periodic solutions occur around the equilibrium points

[2] Andronov A.A., Leontovich E.A., Gordon I.I., Maier A.G., Theory of Bifurcations of

[3] Arnold V.I., Geometrical Methods in the Theory of Ordinary Differential Equation,

*<sup>y</sup>*� <sup>=</sup> *<sup>y</sup>*(*<sup>δ</sup>* <sup>−</sup> *<sup>β</sup> <sup>y</sup>*

(*t*) = −*cy*(*t*) + *βx*(*t*)*y*(*t*),

*a* + *x y*,

where *σ*,*r*, *b* are constants. Perfom the stability analysis of this nonlinear system.

**Figure 16.** When *τ* = 2 *> τ*<sup>0</sup> a stable periodic orbit bifurcates from the equilibrium point *E*∗.

**9.** For *r* ∈ *R*, consider the differential equation

$$\mathbf{x}' = r\mathbf{x} - 2\mathbf{x}^2 + \mathbf{x}^3$$

on the real line.

(a) Show that *x*∗ = 0 is a fixed point for any value of the parameter r, and determine its stability. Hence identify a bifurcation point *r*1.

(b) Show that for certain values of the parameter *r* there are additional fixed points. For which values of *r* do these fixed points exist? Determine their stability and identify a further bifurcation points *r*2.

(c)Using a Taylor expansion of the differential equation above, determine the normal form of the bifurcation at *r*1. What type of bifurcation takes place.

(d) Similarly, determine the normal from of the bifurcation at *r*2. What type bifurcation takes place?

(e) Sketch the bifurcation diagram for all values of *r* and *x*∗. (Use a full line to denote a curve of stable fixed points, and a dashed line for a curve of unstable fixed points).

**10.** Consider the system of differential equations

$$\begin{aligned} x' &= y - xy^2 \\ y' &= -x + yx^2 \end{aligned}$$

in the (*x*, *y*) plane.

30 Will-be-set-by-IN-TECH

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup>

0

P(t) N(t)

**Figure 16.** When *τ* = 2 *> τ*<sup>0</sup> a stable periodic orbit bifurcates from the equilibrium point *E*∗.

*<sup>x</sup>*� <sup>=</sup> *rx* <sup>−</sup> <sup>2</sup>*x*<sup>2</sup> <sup>+</sup> *<sup>x</sup>*<sup>3</sup>

(a) Show that *x*∗ = 0 is a fixed point for any value of the parameter r, and determine its

(b) Show that for certain values of the parameter *r* there are additional fixed points. For which values of *r* do these fixed points exist? Determine their stability and identify a further

(c)Using a Taylor expansion of the differential equation above, determine the normal form of

(d) Similarly, determine the normal from of the bifurcation at *r*2. What type bifurcation takes

(e) Sketch the bifurcation diagram for all values of *r* and *x*∗. (Use a full line to denote a curve

*<sup>x</sup>*� <sup>=</sup> *<sup>y</sup>* <sup>−</sup> *xy*2, *<sup>y</sup>*� <sup>=</sup> <sup>−</sup>*<sup>x</sup>* <sup>+</sup> *yx*2,

of stable fixed points, and a dashed line for a curve of unstable fixed points).

200

**9.** For *r* ∈ *R*, consider the differential equation

stability. Hence identify a bifurcation point *r*1.

the bifurcation at *r*1. What type of bifurcation takes place.

**10.** Consider the system of differential equations

400

on the real line.

bifurcation points *r*2.

place?

0

5

10

15

20

S(t)

25

30

35

40

(a) Determine all fixed points of the system.

(b) Let *r*<sup>2</sup> = *x*<sup>2</sup> + *y*2. Show that *r*� = 0 Considering the result of (c), what does this imply for the trajectories?

(c) Sketch the phase portrait in the (*x*, *y*) plane, including trajectories through (1, 0) and (2, 0). Which fixed point does the trajectory through (2, 0) approach?

**11.** Consider the Lorenz system (the model of heat convection by Rayleigh-Benard occurring in the earth's atmosphere) in three dimension as follows;

$$\begin{aligned} x' &= \sigma(y - x), \\ y' &= rx + -xz - y, \\ z' &= xy - bz, \end{aligned}$$

where *σ*,*r*, *b* are constants. Perfom the stability analysis of this nonlinear system.

**12.** For the Holling-Tanner type Predator-Prey model

$$\begin{aligned} x' &= x(1-x) - \frac{x}{a+x}y\_{\prime\prime} \\ y' &= y(\delta - \beta \frac{y}{x})\_{\prime} \end{aligned}$$

where *x*(0) *>* 0, *y*(0) *>* 0 and *a*, *δ*, *β* are positive constants. Find the equilibria and classify them.

**13.** Consider the Delayed Predator-Prey model

$$\begin{aligned} x'(t) &= rx(t) - bx(t)x(t-\tau) - ax(t)y(t), \\ y'(t) &= -cy(t) + \beta x(t)y(t), \end{aligned}$$

with time delay *τ* and positive constants *r*, *b*, *α*, *c* and *β*. By choosing *τ* as bifurcation parameter, check whether bifurcating periodic solutions occur around the equilibrium points or not.

## **Author details**

Canan Çelik Karaaslanlı *Bahçe¸sehir University, Turkey*

#### **5. References**

	- [4] Çelik C., The stability and Hopf bifurcation for a predator-prey system with time delay, Chaos, Solitons & Fractals, 2008, (37), 87–99.
	- [5] Çelik C., Hopf bifurcation of a ratio-dependent predator-prey system with time delay, Chaos, Solitons & Fractals, 2009, (42), 1474–1484.
	- [6] Çelik C., Dynamical Behavior of a Ratio Dependent Predator-Prey System with Distributed Delay, Discrete and Continuous Dynamical Systems, Series B, 2011, (16), 719-738.
	- [7] Guckenheimer J., Holmes P., Nonlinear Oscilllations, Dynamical Systems and Bifurcations of Vector Fields, Springer, 1983.
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	- [10] B.D. Hassard B.D., Kazarinoff N. D., Wan Y-H., Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.
	- [11] Hale J., Koçak H., Dynamics and Bifurcations. Texts in Applied Mathematics, New York: Springer-Verlag, 1991.
	- [12] Iooss G., Bifurcations of Maps and Applications, North-Holland, 1979.
	- [13] Kahoui M.E., Weber A., Deciding Hopf bifurcations by quantifier elimination in a software component architecture. Journal of Symbolic Computation, 2000, 30 (2), 161-179.
	- [14] Kelley A., The stable, center stable, center, center unstable and unstable manifolds, J. Diff. Eq., 1967, (3), 546-570.
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	- [16] Kuang Y., Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993.
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	- [22] Wilhelm T., Heinrich R., Smallest chemical reaction system with Hopf bifurcation, Journal of Mathematical Chemistry, 1995, (17,1), 1-14. doi:10.1007/BF01165134.

© 2012 Lee and Nataraj, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

and reproduction in any medium, provided the original work is properly cited.

**Model-Based Adaptive Tracking Control of** 

DongBin Lee and C. Nataraj

http://dx.doi.org/10.5772/51625

**1. Introduction** 

uncertainty [4, 5].

Additional information is available at the end of the chapter

**Linear Time-Varying System with Uncertainties** 

This primary purpose of this research is concerned with adaptive tracking control of a nonlinear system [6, 9]. Particularly, time-varying control approach has been designed for tracking of the system with application to a nonlinear dynamic model [1]. Furthermore, the time-varying system is further complicated by parametric uncertainty or disturbances such as external forces, continuous or discrete noise where the parameters are unknown. Over the past several years, trajectory tracking issue as a high-level control of a nonlinear system has been received a wide attention from control community. Hence, the discussion here is principally devoted to model-based adaptive trajectory tracking control algorithm of linear time-varying (LTV) systems in the presence of

A system undergoing slow time variation in comparison to its time constants can usually be considered to be linear time invariant (LTI) and thus, slow time-variation is often ignored in dealing with systems in practice. An example of this is the aging and wearing of electronic and mechanical components, which happens on a scale of years, and thus does not result in any behavior qualitatively different from that observed in a time invariant system on a day-to-day basis. There are many well developed techniques for dealing with the response of linear time invariant systems such as Laplace and Fourier transforms, but not applicable to linear time varying or nonlinear systems, nor feasible to implement for complicated real-world systems. In addition, time-varying system may be difficult to satisfy global controllability or to show whether the time-varying system is even stable or not, due to difficulties in computing or finding solution. Unlike LTI systems, linear time varying systems may behave more like nonlinear systems [1, 2, 3]. In general all systems are time-varying in principle and a large number of systems arising in practice are time-varying. Time variation is a result of system
