*2.3.1. Transition to chaos through bifurcation cascades of stable cycles*

At the beginning let us show that the scenario of transition to chaos through the Sharkovskii subharmonic and homoclinic cascades of bifurcations of stable cycles takes place also in many-dimensional dissipative nonlinear systems of ordinary differential equations. For example consider Rikitaki system

$$
\dot{\mathbf{x}} = -\mu \mathbf{x} + y \mathbf{z},
\
\dot{y} = -\mu y + \mathbf{x}u,
\
\dot{z} = 1 - \mathbf{x}y - b\mathbf{z},
\
\dot{u} = 1 - \mathbf{x}y - cu,\tag{18}
$$

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 147

autonomous systems of ordinary differential equations takes place also in manydimensional systems. So, appearance of three-dimensional torus is not necessary condition for generation of chaotic dynamics in dissipative many-dimensional systems of differential

**Figure 13.** Projections of sections of two-dimensional invariant tori of period one (a), two (b) and three

Let us show now that the FSM scenario of transition to chaos is realized also in infinitelydimensional nonlinear autonomous dissipative systems of ordinary differential equations, namely in nonlinear ordinary differential equations with delay arguments. For instance such scenario of transition to chaos through the Sharkovskii subharmonic cascade of bifurcations of stable cycles with arbitrary period in accordance with the Sharkovskii order (3) takes

( ) () () . ( )

 is increasing, then at first a stable cycle is appearing in phase space of the equation from the stable stationary state as a result of Andronov-Hopf bifurcation. After that the period two stable cycle is appearing from this original singular cycle as a result of double period bifurcation. That is the beginning of the Feigenbaum cascade of period doubling

cascade of bifurcations of stable cycles with arbitrary period in accordance with the Sharkovskii order takes place in the Mackey-Glass equation. Projections of some main stable

Thus we can make a conclusion that universal bifurcation Feigenbaum-Sharkovskii-Magnitskii theory describes transition to dynamical chaos in all nonlinear dissipative systems of ordinary differential equations. Scenario of transition to chaos consists of subharmonic and homoclinic (heteroclinic) cascades of bifurcations of stable cycles or stable

cycles and singular attractors of the Mackey-Glass equation are presented in Fig. 14.

 

*x t x t ax t*

*n n n*

is small, then Mackey-Glass equation has unique stable stationary state. When

*x t*

 

(20)

is a bifurcation parameter. When a value of

, all subharmonic

(c) and one of the toroidal singular attractor (d) in complex Lorenz system (19).

*2.3.3. Transition to chaos in nonlinear equations with delay argument* 

place in well-known Mackey-Glass equation (Mackey & Glass, 1977).

bifurcations. Then, after further increasing of bifurcation parameter

0

In this equation the delay argument

parameter

two- or many-dimensional tori.

equations.

modelling a change in dynamics of magnetic poles of the Earth. Some main cycles of subharmonic cascade of bifurcations in the system of Eqs. (18) and some singular attractors are presented in Fig. 12.

**Figure 12.** Projections of original singular cycle (a), cycle of period two (b), Feigenbaum attractor (c) and two more complex singular attractors in the Rikitaki system (d)-(e).

### *2.3.2. Transition to chaos through bifurcation cascades of stable two-dimensional tori*

Besides described above mechanism of transition to chaos in accordance with subharmonic and homoclinic cascades of bifurcations of stable cycles, in many-dimensional dissipative nonlinear systems of ordinary differential equations there exists a scenario of transition to chaos through subharmonic and homoclinic cascades of bifurcations of stable twodimensional or many-dimensional tori along any one or several frequencies simultaneously. The mechanism of this cascade of bifurcations has the same above considered FSM nature, and presently there not discovered really any other scenarios of transition to chaos in manydimensional nonlinear systems of ordinary differential equations. Such a scenario of transition to chaos takes place in complex five-dimensional Lorenz system

$$\dot{X} = -\sigma X + \sigma Y, \quad \dot{Y} = -XZ + rX - aY, \quad \dot{Z} = -bZ + (X^\*Y + XY^\*)/2 \tag{19}$$

of two complex variables *X x ix* 1 2 and *Y y iy* 1 2 and one real variable *Z* . If values of parameters *a b*, , and Re*r* are fixed and the value of parameter Im*r* is decreasing, then at first a stable invariant torus is appearing from the stable cycle as a result of Andronov-Hopf bifurcation. After that the period two invariant torus is appearing from this original singular saddle torus as a result of double period bifurcation (Fig. 13). That is the beginning of Feigenbaum cascade of period doubling bifurcations. Then, after further decreasing of bifurcation parameter Im*r* , all subharmonic cascade of bifurcations of stable twodimensional tori with arbitrary period in accordance with the Sharkovskii order (3) takes place in the complex Lorenz system. Projections of sections of period one, two and three two-dimensional invariant tori and one of the toroidal singular attractor are presented in Fig. 13.

This example shows that the FSM (Feigenbaum-Sharkovskii-Magnitskii) scenario of transition to dynamical chaos in two-dimensional nonautonomous and three-dimensional autonomous systems of ordinary differential equations takes place also in manydimensional systems. So, appearance of three-dimensional torus is not necessary condition for generation of chaotic dynamics in dissipative many-dimensional systems of differential equations.

**Figure 13.** Projections of sections of two-dimensional invariant tori of period one (a), two (b) and three (c) and one of the toroidal singular attractor (d) in complex Lorenz system (19).

### *2.3.3. Transition to chaos in nonlinear equations with delay argument*

146 Nonlinearity, Bifurcation and Chaos – Theory and Applications

are presented in Fig. 12.

*x x yz y y xu z xy bz u xy cu*

modelling a change in dynamics of magnetic poles of the Earth. Some main cycles of subharmonic cascade of bifurcations in the system of Eqs. (18) and some singular attractors

**Figure 12.** Projections of original singular cycle (a), cycle of period two (b), Feigenbaum attractor (c)

*2.3.2. Transition to chaos through bifurcation cascades of stable two-dimensional tori* 

transition to chaos takes place in complex five-dimensional Lorenz system

\* \* *X X Y Y XZ rX aY Z bZ X Y XY*

 

parameters *a b*, ,

Fig. 13.

Besides described above mechanism of transition to chaos in accordance with subharmonic and homoclinic cascades of bifurcations of stable cycles, in many-dimensional dissipative nonlinear systems of ordinary differential equations there exists a scenario of transition to chaos through subharmonic and homoclinic cascades of bifurcations of stable twodimensional or many-dimensional tori along any one or several frequencies simultaneously. The mechanism of this cascade of bifurcations has the same above considered FSM nature, and presently there not discovered really any other scenarios of transition to chaos in manydimensional nonlinear systems of ordinary differential equations. Such a scenario of

of two complex variables *X x ix* 1 2 and *Y y iy* 1 2 and one real variable *Z* . If values of

first a stable invariant torus is appearing from the stable cycle as a result of Andronov-Hopf bifurcation. After that the period two invariant torus is appearing from this original singular saddle torus as a result of double period bifurcation (Fig. 13). That is the beginning of Feigenbaum cascade of period doubling bifurcations. Then, after further decreasing of bifurcation parameter Im*r* , all subharmonic cascade of bifurcations of stable twodimensional tori with arbitrary period in accordance with the Sharkovskii order (3) takes place in the complex Lorenz system. Projections of sections of period one, two and three two-dimensional invariant tori and one of the toroidal singular attractor are presented in

This example shows that the FSM (Feigenbaum-Sharkovskii-Magnitskii) scenario of transition to dynamical chaos in two-dimensional nonautonomous and three-dimensional

, , ( )/2 (19)

and Re*r* are fixed and the value of parameter Im*r* is decreasing, then at

and two more complex singular attractors in the Rikitaki system (d)-(e).

, ,1 , 1 , (18)

 

> Let us show now that the FSM scenario of transition to chaos is realized also in infinitelydimensional nonlinear autonomous dissipative systems of ordinary differential equations, namely in nonlinear ordinary differential equations with delay arguments. For instance such scenario of transition to chaos through the Sharkovskii subharmonic cascade of bifurcations of stable cycles with arbitrary period in accordance with the Sharkovskii order (3) takes place in well-known Mackey-Glass equation (Mackey & Glass, 1977).

$$\dot{\mathbf{x}}(t) = -a\mathbf{x}(t) + \beta\_0 \frac{\theta^n \mathbf{x}(t-\tau)}{\theta^n + \mathbf{x}^n(t-\tau)}.\tag{20}$$

In this equation the delay argument is a bifurcation parameter. When a value of parameter is small, then Mackey-Glass equation has unique stable stationary state. When is increasing, then at first a stable cycle is appearing in phase space of the equation from the stable stationary state as a result of Andronov-Hopf bifurcation. After that the period two stable cycle is appearing from this original singular cycle as a result of double period bifurcation. That is the beginning of the Feigenbaum cascade of period doubling bifurcations. Then, after further increasing of bifurcation parameter , all subharmonic cascade of bifurcations of stable cycles with arbitrary period in accordance with the Sharkovskii order takes place in the Mackey-Glass equation. Projections of some main stable cycles and singular attractors of the Mackey-Glass equation are presented in Fig. 14.

Thus we can make a conclusion that universal bifurcation Feigenbaum-Sharkovskii-Magnitskii theory describes transition to dynamical chaos in all nonlinear dissipative systems of ordinary differential equations. Scenario of transition to chaos consists of subharmonic and homoclinic (heteroclinic) cascades of bifurcations of stable cycles or stable two- or many-dimensional tori.

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 149

freedom and by examples of simply conservative but not Hamiltonian systems. The stability domains of cycles of such a system with zero dissipation become tori of a conservative (Hamiltonian) system around its elliptic cycles into which the stable cycles themselves go. Complicated separatrix heteroclinic manifolds spanned by unstable singular cycles of the dissipative system become (for zero dissipation) even more complicated separatrix manifolds of the conservative (Hamiltonian) system along which the motion of a trajectory is treated as chaotic dynamics. Thus, it becomes clear why the order of the tori alternation in conservative (Hamiltonian) systems can differ from the Sharkovskii order existing in

Let's consider generally nonlinear conservative system of ordinary differential equations

( ), , ( ) 0 *<sup>n</sup> x f x x R div f x* (21)

Any Hamiltonian system is a special case of system of Eqs. (21)-(22) at even value of dimension *n* and at the given integral of movement (22) generating system of Eqs. (21). Movement in

> 

possesses following properties: 1) the only solutions of system of Eqs. (21)-(22) are solutions

system of Eqs. (23) is dissipative system on its solutions laying in neighborhoods of solutions of system of Eqs. (21)-(22). Then attractors of dissipative system of Eqs. (23) at

So, for application of the offered approach to the analysis of conservative and, in particular, Hamiltonian systems it is necessary to construct an extended dissipative system, satisfying the

solutions and their cascades of bifurcations according to the FSM scenario in extended

conditions, satisfying the equality (22). Areas of stability of the found simple regular solutions

Eqs. (21)-(22), and areas of stability of complex cycles and singular attractors and also heteroclinic separatrix manifolds will generate chaotic solutions. By the same method in the

are as much as exact approximations of solutions of conservative system of Eqs.

system of Eqs. (21) occurs in 1 *n* -dimensional subspace, set by the equation (22).

**Theorem.** Let two-parametrical system of ordinary differential equations

(22)

(23)

; 2) at all 0

the

 at 0 

one should to find numerically all stable

tends to zero, starting from the various initial

0 regular solutions (tori) of original conservative system of

systems with strong dissipation.

with a smooth right part

small 0 

*3.1.1. Theoretical basis of bifurcation approach* 

which variables are connected by some equation

1 ( ,..., ) . *Hx xn*

( , , ), , *<sup>n</sup> x gx x R*

of system of Eqs. (23) with initial conditions 10 0 ( ,..., ) *Hx xn*

(21)- (22) (see proof in (Magnitskii, 2008; Magnitskii, 2011)).

properties 1) and 2). Then for everyone 0

dissipative system of Eqs. (23) when

(simple cycles) will generate at

**Figure 14.** Projections of period one (a), two (b) and three (d) cycles, Feigenbaum attractor (c) and one of more complex singular attractor (e) in the Mackey-Glass equation (20).
