**1. Introduction**

Continuous dynamical systems that involve differential equations mostly contain parameters. It can happen that a slight variation in a parameter can have significant impact on the solution. The main questions of interest in this chapter are: How to continue equilibria and periodic orbits of dynamical systems with respect to a parameter? How to compute stability boundaries of equilibria and limit cycles in the parameter space? How to predict qualitative changes in system's behavior (bifurcations) occurring at these equilibrium points? This chapter will also cover the classification of bifurcations in terms of equilibria and periodic orbits. Especially it will present the specific bifurcation called "Hopf bifurcation" which refers to the development of periodic orbits from stable equilibrium point, as a bifurcation parameter crosses a critical value. Since the theory of bifurcation from equilibria based on center manifold reduction and Poincaré-Normal forms, the direction of bifurcations for the mathematical models will also be explained using this theory. Finally, by introducing several software packages and numerical methods this chapter will also cover the techniques to determine and continue in some control parameters all local bifurcations of periodic orbits of dynamical systems and relevant normal form computations combined with the center manifold theorem, including periodic normal forms for periodic orbits.

In general, in a dynamical system, a parameter is allowed to vary, then the differential system may change. An equilibrium can become unstable and a periodic solution may appear or a new stable equilibrium may appear making the previous equilibrium unstable. The value of parameter at which these changes occur is known as "bifurcation value" and the parameter that is varied is known as the "bifurcation parameter".

In this chapter, we also discuss several types of bifurcations, saddle node, transcritical, pitchfork and Hopf bifurcation. Among these types, we especially focus on Hopf bifurcation. The first three types of bifurcation occur in scalar and in systems of differential equations. The fourth type called Hopf bifurcation does not occur in scalar differential equations because this type of bifurcation involves a change to a periodic solution. Scalar autonomous

©2012 Çelik,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.

#### 2 Will-be-set-by-IN-TECH 4 Numerical Simulation – From Theory to Industry Bifurcation Analysis and Its Applications <sup>3</sup>

differential equations can not have periodic solutions. Hopf bifurcation occurs in systems of differential equations consisting of two or more equations. This type is also referred to as a "Poincare-Andronov-Hopf bifurcation".

For a given system of differential equations first we shall consider the stability and the local Hopf bifurcation. By using the Hopf bifurcation theorem we prove the occurrence of the Hopf bifurcation. And then, based on the normal form method and the center manifold reduction introduced by Hassard et al.,[10], we derive the formulae determining the direction, stability and the period of the bifurcating periodic solution at the critical value of the bifurcation parameter. To verify the theoretical analysis, numerical simulations for bifurcation analysis are given in this chapter. For references see [1]-[22].

We also introduce the Hopf bifurcation for continuous dynamical systems and state the Hopf bifurcation theorem for these models. As it is well known, Hopf bifurcations occur when a conjugated complex pair of eigenvalues crosses the boundary of stability. In the time-continuous case, a limit cycle bifurcates. It has an angular frequency which is given by the imaginary part of the crossing pair. In the discrete case, the bifurcating orbit is generally quasi-periodic, except that the argument of the crossing pair times an integer gives just 2*π*. If we consider an ordinary differential equation (ODE) that depends on one or more parameters *α*

$$\mathbf{x}' = f(\mathbf{x}, \mathbf{a}), \tag{1}$$

The attracting or repelling can occur via straight orbits (a node) or via spiralling orbits (a spiral or focus). Note that, it is not possible to have a saddle focus in two dimensions. It is possible

To prove the existence of Hopf bifurcation, we first obtain the Hopf bifurcation theorem hypothesis, i.e., the existence of purely imaginary eigenvalues of the corresponding characteristic equation with respect to the parameter *α* and also we prove the transversality

at the critical value *α*<sup>0</sup> where Hopf bifurcation occurs. Then based on the normal form approach and the center manifold theory introduced by Hassard et. al,[10], we derive the

Finally in this chapter, to support these theoretical results, we illustrate them by numerical simulations. In numerical analysis, generally MATLAB solver packages are used to analyze the dynamics of nonlinear models. In these solvers, differential equation systems are simulated by difference equations. In this chapter, we also give some examples from biology (such as well known predator-prey models with time delay) with numerical simulations and by graphing the solutions in two or three dimensions, we illustrate the occurrence of periodic

As it is stated above, in dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden "qualitative" or topological change in its behavior. Generally, at a bifurcation, the local stability properties of equilibria, periodic orbits or other invariant sets changes. It has two

*Local bifurcations*, which can be analyzed entirely through changes in the local stability properties of equilibria, periodic orbits or other invariant sets as parameters cross through

*Global bifurcations,* which often occur when larger invariant sets of the system "collide" with each other, or with equilibria of the system. They cannot be detected purely by a stability

In dynamical systems, only the solutions of linear systems may be found explicitly. The problem is that in general real life problems may only be modeled by nonlinear systems. The main idea is to approximate a nonlinear system by a linear one (around the equilibrium point). Of course, we do hope that the behavior of the solutions of the linear system will be the same as the nonlinear one. But this is not always true. Before the linear stability analysis, we give

analysis of the equilibria (fixed or equilibrium points, see the next section).

*<sup>d</sup><sup>α</sup>* (*<sup>α</sup>* <sup>=</sup> *<sup>α</sup>*0) <sup>=</sup> <sup>0</sup> (3)

Bifurcation Analysis and Its Applications 5

*dλ*

formula for determining the properties of Hopf bifurcation of the model.

solutions. For some examples see references by Çelik, [4], [5] and [6].

**2. Basic concepts of bifurcation analysis**

though in three of higher dimensional systems.

condition

types;

critical thresholds; and

**2.1 Equilibrium points**

some basic definitions below.

where, for simplicity, we assume *α* to be the only parameter. There is the possibility that under variation of *α* nothing interesting happens to Equation (1). There is only a quantitatively different behavior. Let us define Equation (1) to be structurally stable in the case there are no qualitative changes occurring. However, the ODE (Ordinary Differential Equation) might change qualitatively. At that point, bifurcations will have occurred.

Many of the basic principles for one dimensional systems apply also for two-dimensional systems. Let us define a two-dimensional system

$$\begin{aligned} \mathbf{x}' &= f(\mathbf{x}, \mathbf{y}, \mathbf{a}), \\ \mathbf{y}' &= \mathbf{g}(\mathbf{x}, \mathbf{y}, \mathbf{a}), \end{aligned} \tag{2}$$

where biologically we mostly interpret x as prey or resource and y as predator or consumer. Equilibria can be found by taking the equations equal to zero, i.e.,

$$\begin{aligned} f(x, y, \mathfrak{a}) &= 0, \\ g(x, y, \mathfrak{a}) &= 0. \end{aligned}$$

We have three possibilities for the stability of an equilibrium. Next to the stable and unstable equilibrium, there is the saddle equilibrium. A two-dimensional stable equilibrium is attracting in two directions, while a two-dimensional unstable equilibrium is repelling in two directions. A saddle point is attracting in one direction and repelling in the other direction. In the less formal literature saddles are often considered just unstable equilibria. A second remark is that also the dynamics of the system around the equilibria can differ.

The attracting or repelling can occur via straight orbits (a node) or via spiralling orbits (a spiral or focus). Note that, it is not possible to have a saddle focus in two dimensions. It is possible though in three of higher dimensional systems.

To prove the existence of Hopf bifurcation, we first obtain the Hopf bifurcation theorem hypothesis, i.e., the existence of purely imaginary eigenvalues of the corresponding characteristic equation with respect to the parameter *α* and also we prove the transversality condition

$$\frac{d\lambda}{d\mathfrak{a}}(\mathfrak{a} = \mathfrak{a}\_0) \neq 0 \tag{3}$$

at the critical value *α*<sup>0</sup> where Hopf bifurcation occurs. Then based on the normal form approach and the center manifold theory introduced by Hassard et. al,[10], we derive the formula for determining the properties of Hopf bifurcation of the model.

Finally in this chapter, to support these theoretical results, we illustrate them by numerical simulations. In numerical analysis, generally MATLAB solver packages are used to analyze the dynamics of nonlinear models. In these solvers, differential equation systems are simulated by difference equations. In this chapter, we also give some examples from biology (such as well known predator-prey models with time delay) with numerical simulations and by graphing the solutions in two or three dimensions, we illustrate the occurrence of periodic solutions. For some examples see references by Çelik, [4], [5] and [6].
