**2. Dynamical chaos in nonlinear dissipative systems of ordinary differential equations**

## **2.1. Two-dimensional systems with periodic coefficients**

Consider a smooth family of two-dimensional real nonlinear non-autonomous systems of ordinary differential equations

$$\dot{\mu} = D(t,\mu)u(t) + H(u,t,\mu), \; H(0,t,\mu) \equiv 0,\tag{1}$$

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 135

... 2 7 2 5 2 3 ... 7 5 3.

the family of systems of Eqs. (1) has only regular attractors - asymptotically

 

 ( ) 2 4:

2

, where the value

for which period doubling

Obviously, the simplest singular attractor is the Feigenbaum attractor, i.e. the first non-

bifurcations of the original cycle take place. Note that the Feigenbaum cascade of period doubling bifurcations is the beginning of the Sharkovskii subharmonic cascade. Note also, that the subharmonic cascade of bifurcations in accordance with the Sharkovskii order (3) does not exhaust the entire complexity of transition to chaos in two-dimensional nonlinear nonautonomous dissipative systems of ordinary differential equations with rotors. It can be continued at least by the Magnitskii homoclinic cascade of bifurcations of stable cycles

As an example, we consider a simplest two-dimensional nonlinear non-autonomous system

1 1 2 2 212

 

 () 2, 

2( 1 cos( )) (2sin( ) / 2) , (2sin( ) / 2) 2( 1 cos( )) . *u tu t u u u tu t u*

   

 <sup>2</sup> 

> 

(4)

are arbitrary real constants.

which is changing, then the

(3)

Sharkovskii subharmonic cascade of bifurcations of stable limit cycles is realizing in system of Eqs. (1) in accordance with the Sharkovskii order (Magnitskii, 2008; Magnitskii & Sidorov,

The ordering *n k* in (3) means that the existence of a cycle of period *k* implies the existence of a cycle of period *n* . So, if a system of Eqs. (1) has a stable limit cycle of period three then it has also all unstable cycles of all periods in accordance with the Sharkovskii order. So, the family of systems of Eqs. (1) can have irregular attractors only at infinitely many accumulation points of bifurcation values of the system parameter. Every such value is a limit of a sequence of values of some Feigenbaum subcascade of period doubling bifurcations in Sharkovskii cascade. Thus, any irregular attractor of the family of systems of Eqs. (1) with rotor type singular point is a **singular** attractor, as it is defined in (Magnitskii & Sidorov, 2006; Magnitskii, 2011). Simple singular attractor is almost stable non-periodic trajectory which is the limit of a sequence of periodic orbits of some Feigenbaum subcascade of period doubling bifurcations. Complex singular attractor exists only in bifurcation values corresponding to homoclinic or heteroclinic separatrix loops. For other values of the

23 2 2 2 1 2 2 2 ... 2 7 2 5 2 3 ...

orbitally stable periodic trajectories, even of a very large period.

periodic attractor existing in the family of systems of Eqs. (1) for

is the first limit of the sequence of bifurcation values

converging to a homoclinic loop of the rotor type singular point.

of Eqs. (1) with leading linear part (2) in which 1

 and 2 

with 2 / 

2006):

parameter


If real parts of Floquet exponents depend on parameter

*2.1.1. FSM – scenario of transition to chaos* 

with a *T*( ) -periodic matrix *D t*(, ) of the leading linear part depending on a scalar system parameter . Expansion of a function *Hut* ( ,, ) on components of vector *u* begins with members of the second order. The Floquet theory states that the fundamental matrix solution *U t*(, ) of the linear part of system of Eqs. (1) can be represented in the form ( ) (, ) (, ) *B t Ut Pt e* , where *P t*(, ) is some *T* -periodic complex matrix and *B*( ) is some constant complex matrix whose eigenvalues are named as Floquet exponents. It is important that the real linear system can have various complex but not complex-conjugate Floquet exponents 1 ( ) and 2 ( ) . Real parts Re ( ) ( ), 1 1 Re ( ) ( ), 2 2 can be different but imaginary parts 2 1 Im ( ) Im ( ) 2 *k* can be equal or differ from each other on 2 *k* . Singular point *O*(0,0) of a two-dimensional non-autonomous real system of Eqs. (1) with periodic coefficients of its leading linear part is a **rotor** if corresponding linear system has complex Floquet exponents with equal imaginary and different real parts (Magnitskii, 2008, 2011; Magnitskii & Sidorov, 2006). Canonical form of a rotor is a linear system

$$\begin{aligned} \dot{u}\_1 &= \frac{\beta\_1 + \beta\_2 + (\beta\_1 - \beta\_2)\cos\alpha t}{2} u\_1 + \frac{(\beta\_1 - \beta\_2)\sin\alpha t - \alpha}{2} u\_2\\ \dot{u}\_2 &= \frac{(\beta\_1 - \beta\_2)\sin\alpha t + \alpha}{2} u\_1 + \frac{\beta\_1 + \beta\_2 - (\beta\_1 - \beta\_2)\cos\alpha t}{2} u\_2 \end{aligned} \tag{2}$$

with 2 / - periodic coefficients. In Eqs. (2) 1 and 2 are arbitrary real constants.

## *2.1.1. FSM – scenario of transition to chaos*

134 Nonlinearity, Bifurcation and Chaos – Theory and Applications

equation and many others.

**differential equations** 

ordinary differential equations

 

 ( ) 

, where *P t*(, )

 and 2 ( ) 

but imaginary parts 2 1 Im ( ) Im ( ) 2 


with a *T*( ) 

parameter

exponents 1

2

solution *U t*(, )

( ) (, ) (, ) *B t Ut Pt e*

theory of bifurcations in such systems.

autonomous and many-dimensional autonomous nonlinear dissipative systems of ordinary differential equations, such as Duffing-Holmes, Mathieu, Croquette and Rikitaki equations. It takes place also in nonlinear partial differential equations and differential equations with delay arguments, such as Brusselyator, Ginzburg-Landau, Navier-Stokes and Mackey-Glass equations, reaction-diffusion systems and systems of differential equations describing excitable and autooscillating mediums. Moreover, the same scenario of transition to chaos takes place also in conservative and, in particularly, Hamiltonian systems such as Henon-Heiles and Yang-Mills systems, conservative Duffing-Holmes, Mathieu and Croquette

Thus, the question is about discovery and description of the uniform universal mechanism of the arranging of surrounding us infinitely complex and infinitely various nonlinear world. And this nonlinear world is arranged under uniform laws, and these laws are laws of nonlinear dynamics, qualitative theory of nonlinear systems of differential equations and

Consider a smooth family of two-dimensional real nonlinear non-autonomous systems of

 

members of the second order. The Floquet theory states that the fundamental matrix

constant complex matrix whose eigenvalues are named as Floquet exponents. It is important that the real linear system can have various complex but not complex-conjugate Floquet

> 1 1

 . Real parts Re ( ) ( ), 

2008, 2011; Magnitskii & Sidorov, 2006). Canonical form of a rotor is a linear system

12 12 1 2 1 12

1 2 12 12 21 2

*uu u*

 

*u uu*

 

 

 *k* . Singular point *O*(0,0) of a two-dimensional non-autonomous real system of Eqs. (1) with periodic coefficients of its leading linear part is a **rotor** if corresponding linear system has complex Floquet exponents with equal imaginary and different real parts (Magnitskii,

> ( )cos ( )sin 2 2 ( )sin ( )cos 2 2

*t t*

 

*t t*

 

 

is some *T* -periodic complex matrix and *B*( )

 Re ( ) ( ), 2 2 

> 

> >

*k* can be equal or differ from each other on

of the linear part of system of Eqs. (1) can be represented in the form

of the leading linear part depending on a scalar system

on components of vector *u* begins with

(1)

can be different

(2)

is some

**2. Dynamical chaos in nonlinear dissipative systems of ordinary** 

**2.1. Two-dimensional systems with periodic coefficients** 

 *u Dt ut Hut H t* ( , ) ( ) ( , , ), (0, , ) 0, 

. Expansion of a function *Hut* ( ,, )

 

 

> 

If real parts of Floquet exponents depend on parameter which is changing, then the Sharkovskii subharmonic cascade of bifurcations of stable limit cycles is realizing in system of Eqs. (1) in accordance with the Sharkovskii order (Magnitskii, 2008; Magnitskii & Sidorov, 2006):

$$\begin{aligned} 1 \lhd 2 \lhd 2^2 \lhd 2^3 \lhd \dots \lhd 2^2 \cdot 7 \lhd 2^2 \cdot 5 \lhd 2^2 \cdot 3 \lhd \dots \\ \dots \lhd 2 \cdot 7 \lhd 2 \cdot 5 \lhd 2 \cdot 3 \lhd \dots \lhd 7 \lhd 5 \lhd 3. \end{aligned} \tag{3}$$

The ordering *n k* in (3) means that the existence of a cycle of period *k* implies the existence of a cycle of period *n* . So, if a system of Eqs. (1) has a stable limit cycle of period three then it has also all unstable cycles of all periods in accordance with the Sharkovskii order. So, the family of systems of Eqs. (1) can have irregular attractors only at infinitely many accumulation points of bifurcation values of the system parameter. Every such value is a limit of a sequence of values of some Feigenbaum subcascade of period doubling bifurcations in Sharkovskii cascade. Thus, any irregular attractor of the family of systems of Eqs. (1) with rotor type singular point is a **singular** attractor, as it is defined in (Magnitskii & Sidorov, 2006; Magnitskii, 2011). Simple singular attractor is almost stable non-periodic trajectory which is the limit of a sequence of periodic orbits of some Feigenbaum subcascade of period doubling bifurcations. Complex singular attractor exists only in bifurcation values corresponding to homoclinic or heteroclinic separatrix loops. For other values of the parameter the family of systems of Eqs. (1) has only regular attractors - asymptotically orbitally stable periodic trajectories, even of a very large period.

Obviously, the simplest singular attractor is the Feigenbaum attractor, i.e. the first nonperiodic attractor existing in the family of systems of Eqs. (1) for , where the value is the first limit of the sequence of bifurcation values for which period doubling bifurcations of the original cycle take place. Note that the Feigenbaum cascade of period doubling bifurcations is the beginning of the Sharkovskii subharmonic cascade. Note also, that the subharmonic cascade of bifurcations in accordance with the Sharkovskii order (3) does not exhaust the entire complexity of transition to chaos in two-dimensional nonlinear nonautonomous dissipative systems of ordinary differential equations with rotors. It can be continued at least by the Magnitskii homoclinic cascade of bifurcations of stable cycles converging to a homoclinic loop of the rotor type singular point.

As an example, we consider a simplest two-dimensional nonlinear non-autonomous system of Eqs. (1) with leading linear part (2) in which 1 () 2, <sup>2</sup> ( ) 2 4:

$$\begin{aligned} \dot{u}\_1 &= 2(\mu - 1 + \cos(\alpha t))u\_1 + (2\sin(\alpha t) - \alpha/2)u\_2 - u\_2^2, \\ \dot{u}\_2 &= (2\sin(\alpha t) + \alpha/2)u\_1 + 2(\mu - 1 - \cos(\alpha t))u\_2. \end{aligned} \tag{4}$$

For 4 and for growth of the parameter stable cycles are generated in the system of Eqs. (4) in accordance with the Sharkovskii order (3) and then in accordance with the Magnitskii homoclinic order. These cycles of period two and three, one of the singular attractors and homoclinic cycles of periods four and five are presented in Fig.1 (Magnitskii & Sidorov, 2006). Thus, in this system full bifurcation FSM (Feigenbaum-Sharkovskii-Magnitskii) scenario of transition to dynamical chaos is realized.

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 137

 

(5)

, that is for the system of Eqs. (4). In this case, as the parameter

0.12 lying between cycles of period 5 and 3 in the Sharkovskii

 

in system of Eqs. (5) is a

it is strongly dissipative.

Rewrite the system of Eqs. (4) in the form of autonomous 4d-system

with 2 *b uv* 1, and with the cycle 2 2 *p q* <sup>1</sup> . The parameter

*b* the system of Eqs. (5) is conservative.

<sup>2</sup> (2( ) 2 ) (2 / 2) , (2 / 2) (2( ) 2 ) ,

the system is weakly dissipative, for large *b* and small

*u b bp u bq v v p q v bq u b bp v q p*

bifurcation parameter, and the parameter *b* is responsible for dissipation. For small *b* and

As a rule, all known dissipative systems of nonlinear differential equations are strongly dissipative, which has for many decades prevented one from studying the structure of their irregular attractors even with the use of most advanced computers. Last circumstance stimulated the development of numerous definitions of irregular attractors, ostensibly distinguished in their topological structure (strange, chaotic, stochastic, etc.). We illustrate this circumstance by the example of system of Eqs. (5) with strong

 0 increases, system of Eqs. (5) has not only a complete subharmonic cascade of bifurcations in accordance with the Sharkovskii order, but also it has complete homoclinic cascade of bifurcations of cycles converging to the rotor homoclinic loop. The cause is clarified in Fig. 2a in which the Poincare section ( 0, 0) *q p* of the singular attractor of

order is shown. The graph of the section almost coincides with the graph of onedimensional unimodal mapping of a segment into itself, which has the above-listed cascades of bifurcations (Feigenbaum, 1978; Sharkovskii, 1964; Magnitskii & Sidorov, 2006). The projection of the manifold of the singular attractor onto the plane (,) *p u*

**Figure 2.** The projection of the Poincare section ( 0, 0) *q p* of solution of system of Eqs. (5) for

(a) and the projection of the manifold of the singular attractor onto the plane (,) *p u*

 

small 

Besides at

dissipation for *b* 1, 4

system of Eqs. (5) for

*b* 1, 4, 0.12 

corresponding to the section (b).

corresponding to the section is shown in Fig. 2b.

**Figure 1.** Stable cycles of period two (a) and period three (b), singular attractor (c) and homoclinic cycles of periods four (d) and five (e) in the system of Eqs. (4).
