**2.2. Three-dimensional autonomous systems**

140 Nonlinearity, Bifurcation and Chaos – Theory and Applications

periodic coefficients such as Duffing-Holmes equation

<sup>3</sup>

accordance with the Sharkovskii order are presented in Fig. 4 - Fig. 6.

2 3

modified dissipative Mathieu equation

and Croquette dissipative equation

periodic coefficients.

*2.1.3. Some examples of classical two-dimensional nonautonomous systems* 

<sup>0</sup> *x kx x x f t*

 

*x x tx x*

*x x x xt*

 

 

All these equations are equivalent to two-dimensional nonlinear dissipative systems of ordinary differential equations with periodic coefficients and all of them have the same universal FSM scenario of transition to dynamical chaos (Magnitskii & Sidorov, 2006). For these equations, some important stable cycles and singular subharmonic attractors in

**Figure 4.** Original cycle (a), cycle of period two (b), Feigenbaum attractor (c), cycle of period six (d) from subharmonic cascade and more complex singular attractor (e) in the Duffing-Holmes equation (6).

**Figure 5.** Original cycle (a), cycle of period two (b), Feigenbaum attractor (c), cycle of period three (d) from subharmonic cascade and more complex singular attractor (e) in the Mathieu equation (7).

Note that double period bifurcations were found also in (Awrejcewicz, 1989; Awrejcewicz 1991) for some other nonlinear ordinary differential equations of the second order with

 

 

Consider three classical nonlinear ordinary differential equations of the second order with

cos , (6)

( cos ) 0, (7)

sin sin( ) 0. (8)

Consider a smooth family of three-dimensional nonlinear dissipative autonomous systems of ordinary differential equations

$$\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, \boldsymbol{\mu}), \qquad \mathbf{x} \in M \subset \mathbb{R}^3, \quad \boldsymbol{\mu} \in I \subset \mathbb{R}, \quad \mathbf{F} \in \mathbb{C}^{\mathrm{cr}}, \tag{9}$$

depending on a scalar system parameter .

It is shown by the author in (Magnitskii & Sidorov, 2006; Magnitskii, 2008) that if a threedimensional system of Eqs. (9) has a singular cycle of period *T* defined by complex Floquet exponents with equal imaginary parts (i.e. Moebius bands are its stable and unstable invariant manifolds), then by passing to a coordinate system rotating around the cycle, one can reduce such system to a two-dimensional nonautonomous system in coordinates, transversal to the singular cycle with zero rotor-type singular point corresponding to the cycle. So, all arguments listed in the previous section hold completely for autonomous threedimensional systems with singular cycles.

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

Therefore, three-dimensional autonomous system with singular cycle should have the same FSM scenario of transition to chaos as two-dimensional nonautonomous system with periodic coefficients and zero rotor-type singular point. As an example, consider the autonomous three-dimensional system

$$\begin{split} \dot{\mathbf{x}}\_{1} &= -\alpha \mathbf{x}\_{2} + \mathbf{x}\_{1} [((\mu - 1)\sqrt{\mathbf{x}\_{1}^{2} + \mathbf{x}\_{2}^{2}}) + \mathbf{x}\_{1})(\mathbf{x}\_{1}^{2} + \mathbf{x}\_{2}^{2} - 1) + (2\mathbf{x}\_{2} - \alpha / 2)\mathbf{x}\_{3}], \\ \dot{\mathbf{x}}\_{2} &= \alpha \mathbf{x}\_{1} + \mathbf{x}\_{2} [((\mu - 1)\sqrt{\mathbf{x}\_{1}^{2} + \mathbf{x}\_{2}^{2}}) + \mathbf{x}\_{1})(\mathbf{x}\_{1}^{2} + \mathbf{x}\_{2}^{2} - 1) + (2\mathbf{x}\_{2} - \alpha / 2)\mathbf{x}\_{3}], \\ \dot{\mathbf{x}}\_{3} &= (\mathbf{x}\_{2} + \alpha / 4)(\mathbf{x}\_{1}^{2} + \mathbf{x}\_{2}^{2} - 1) + 2(\mu - 1 - \mathbf{x}\_{1})\mathbf{x}\_{3}. \end{split} \tag{10}$$

For 1 system of Eqs.(10) has the singular point (0,0, / 8( 1)) and limit cycle <sup>0</sup> , (cos ,sin ,0)*<sup>T</sup> xt t t* with period 2 / *T* in the plane of variables 1 2 (,) *x x* . By changing the variables <sup>0</sup> 1 2 ( , ) , ( , )(0, ( ), ( ))*<sup>T</sup> xt x t Qt u t u t* with 2 / -periodic matrix ( ) ( ), ( ),(0,0,1) , 0 0 *<sup>T</sup> Qt x t x t* one can reduce the system of Eqs. (10) to two-dimensional nonautonomous system with 2 / -periodic coefficients and zero rotor-type singular point

$$\begin{aligned} \dot{u}\_1 &= 2(\mu - 1 + \cos \alpha t)u\_1 + (2 \sin \alpha t - \alpha / 2)u\_2 + h\_1 \{u\_1, u\_2, t, \alpha, \mu\}, \\ \dot{u}\_2 &= (2 \sin \alpha t + \alpha / 2)u\_1 + 2(\mu - 1 - \cos \alpha t)u\_2 + h\_2 \{u\_1, u\_2, t, \alpha, \mu\}, \end{aligned} \tag{11}$$

where

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

Leading linear part of system of Eqs. (11) coincides with the linear part of system of Eqs. (4) with rotor. So, for 0 zero solution of system of Eqs. (11) and singular cycle <sup>0</sup> *x t*, of system of Eqs. (10) are stable. For 0 all cascades of bifurcations in accordance with the theory FSM take place in both systems. Some cycles and singular attractors from these cascades are presented in Fig. 7, rotor and singular cycle separatrix loops are presented in Fig. 8. Thus, if parameter is changing, then the Sharkovskii subharmonic and Magnitskii

**Figure 7.** Projections of period four and six cycles and singular attractors of system of Eqs. (10) (above) and corresponding to them period two and three cycles and singular attractors of system of Eqs. (11) (below).

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 143

 

 

and does not influence the generation of the

 

 

(13)

 , which

. By linearizing system of


(12)

/ 2 (( ) )(1 ),

2( / 4) ( ) (1 )

 

2( ) ( / 4)(1 ).

Eqs. (12) on the cycle with respect to deviations 123 *y* ,*y y*, from the cycle and by performing

0 0 ( ) ( ( ), ( ),(0,0,1) ) *<sup>T</sup> Qt x t x t* , one can obtain the following system of equations in the rotating

32 3

System of Eqs. (13) coincides with the linear part of system of Eqs. (4) considered in previous Section, and in addition, the coordinate tangent to the cycle has the form

dynamics of solutions in a neighborhood of the cycle. Consequently, the heteroclinic separatrix manifold generated around the cycle of system of Eqs. (12) as the bifurcation

a rotor-type singular point and should be similar to a heteroclinic separatrix zigzag in the Poincare section for small dissipation parameter *b* . The projection of Poincare section

 

 

(2( ) 2 cos ) (2 sin / 2) , (2 sin / 2) (2( ) 2 cos ) . *z b b tz b t z z b t z b b tz*

 

grows has the same structure as that of the heteroclinic separatrix manifold of

 

*x y xz b x b x y y x b y z by x y z b bx z by x y*

System of Eqs. (12) has the periodic solution (the cycle) <sup>0</sup> , (cos ,sin ,0)*<sup>T</sup> xt t t*

 

 

homoclinic cascades of bifurcations of stable limit cycles are realizing in any system of Eqs. (9) in accordance with the Sharkovskii and homoclinic orders. Cycle of period three is the last cycle in the Sharkovskii order. Therefore, to verify an existence of subharmonic cascade of bifurcations in any system one should to find a stable cycle of period three in this system

The three-dimensional phase space of three-dimensional autonomous system containing the original singular cycle of period *T* is diffeomorphic to three-dimensional manifold of an autonomous four-dimensional system of the form of Eqs. (5), the first two equations of which have linear part of the form of Eqs. (2), and the remaining two equations with some condition define a motion on a plane along a simple cycle of period*T* . Therefore, the separatrix heteroclinic manifold constructed in previous Section for system of Eqs. (5) in the section ( 0, 0) *q p* (in the section of the singular cycle corresponding to the rotor) should be completely similar to the separatrix heteroclinic manifold of a three-dimensional

or any stable homoclinic cycle.

*2.2.2. Topological structure of singular attractors* 

autonomous system in the section of the original singular cycle.

As an example, consider the autonomous three-dimensional system:

lies in the plane of the variables (,) *x y* and has the period 2 /

 

1 23 *z b tz b tz* (( 2 / )sin ) ((2 / )cos ) ,

the change of variables *y*() ()() *t Qtzt* with 2 /

2 23

( 0.1, 0) *y x* of solution of system of Eqs. (12) is presented in Fig. 9a.

 

variables transversal to the cycle:

parameter

**Figure 8.** Rotor separatrix loop of system of Eqs. (11) (a) and corresponding to it separatrix loop of singular cycle of system of Eqs. (10) (b).

homoclinic cascades of bifurcations of stable limit cycles are realizing in any system of Eqs. (9) in accordance with the Sharkovskii and homoclinic orders. Cycle of period three is the last cycle in the Sharkovskii order. Therefore, to verify an existence of subharmonic cascade of bifurcations in any system one should to find a stable cycle of period three in this system or any stable homoclinic cycle.

## *2.2.2. Topological structure of singular attractors*

142 Nonlinearity, Bifurcation and Chaos – Theory and Applications

with rotor. So, for 0

Fig. 8. Thus, if parameter

system of Eqs. (10) are stable. For

where

(below).

singular cycle of system of Eqs. (10) (b).

> 

<sup>1</sup> <sup>1</sup> 2 112

2 1 2 212

22 2 1 11 1 1 1 2 2

*h t u u u u t uu*

Leading linear part of system of Eqs. (11) coincides with the linear part of system of Eqs. (4)

theory FSM take place in both systems. Some cycles and singular attractors from these cascades are presented in Fig. 7, rotor and singular cycle separatrix loops are presented in

**Figure 7.** Projections of period four and six cycles and singular attractors of system of Eqs. (10) (above) and corresponding to them period two and three cycles and singular attractors of system of Eqs. (11)

**Figure 8.** Rotor separatrix loop of system of Eqs. (11) (a) and corresponding to it separatrix loop of

(2sin ( 1)sin / 4) 2 cos .

*h t u t u uu t*

 

*u tu t u h u u t u tu tu h u u t*

 

2 1 1 12

2( 1 cos ) (2sin / 2) ( , , , , ), (2sin / 2) 2( 1 cos ) ( , , , , ),

( 1 cos )((2 ) ) ((2 4)sin / 2) ,

(11)

zero solution of system of Eqs. (11) and singular cycle <sup>0</sup> *x t*,

 

 

> of

 

0 all cascades of bifurcations in accordance with the

is changing, then the Sharkovskii subharmonic and Magnitskii

 

> The three-dimensional phase space of three-dimensional autonomous system containing the original singular cycle of period *T* is diffeomorphic to three-dimensional manifold of an autonomous four-dimensional system of the form of Eqs. (5), the first two equations of which have linear part of the form of Eqs. (2), and the remaining two equations with some condition define a motion on a plane along a simple cycle of period*T* . Therefore, the separatrix heteroclinic manifold constructed in previous Section for system of Eqs. (5) in the section ( 0, 0) *q p* (in the section of the singular cycle corresponding to the rotor) should be completely similar to the separatrix heteroclinic manifold of a three-dimensional autonomous system in the section of the original singular cycle.

As an example, consider the autonomous three-dimensional system:

$$\begin{aligned} \dot{\mathbf{x}} &= -a\boldsymbol{\alpha}\boldsymbol{y} - a\boldsymbol{\alpha}\boldsymbol{x}\boldsymbol{z} / 2 - ( (\boldsymbol{\mu} - b)\boldsymbol{x} + b)(1 - \mathbf{x}^2 - \mathbf{y}^2) \boldsymbol{\lambda} \\ \dot{\mathbf{y}} &= a\boldsymbol{\alpha}\boldsymbol{x} + 2(b - a\boldsymbol{y} \, \boldsymbol{y} \, / 4)\boldsymbol{z} - (\boldsymbol{\mu} - b)\boldsymbol{y}(1 - \mathbf{x}^2 - \mathbf{y}^2) \\ \dot{\mathbf{z}} &= 2(\boldsymbol{\mu} - b - b\boldsymbol{\alpha})\boldsymbol{z} - (b\boldsymbol{y} + a\boldsymbol{\alpha} \, / 4)(1 - \mathbf{x}^2 - \mathbf{y}^2) .\end{aligned} \tag{12}$$

System of Eqs. (12) has the periodic solution (the cycle) <sup>0</sup> , (cos ,sin ,0)*<sup>T</sup> xt t t* , which lies in the plane of the variables (,) *x y* and has the period 2 / . By linearizing system of Eqs. (12) on the cycle with respect to deviations 123 *y* ,*y y*, from the cycle and by performing the change of variables *y*() ()() *t Qtzt* with 2 / - periodic matrix 0 0 ( ) ( ( ), ( ),(0,0,1) ) *<sup>T</sup> Qt x t x t* , one can obtain the following system of equations in the rotating variables transversal to the cycle:

$$\begin{aligned} \dot{\mathbf{z}}\_2 &= (2(\mu - b) + 2b \cos \alpha t) \mathbf{z}\_2 + (2b \sin \alpha t - \alpha / 2) \mathbf{z}\_{3^\prime} \\ \dot{\mathbf{z}}\_3 &= (2b \sin \alpha t + \alpha / 2) \mathbf{z}\_2 + (2(\mu - b) - 2b \cos \alpha t) \mathbf{z}\_3. \end{aligned} \tag{13}$$

System of Eqs. (13) coincides with the linear part of system of Eqs. (4) considered in previous Section, and in addition, the coordinate tangent to the cycle has the form 1 23 *z b tz b tz* (( 2 / )sin ) ((2 / )cos ) , and does not influence the generation of the dynamics of solutions in a neighborhood of the cycle. Consequently, the heteroclinic separatrix manifold generated around the cycle of system of Eqs. (12) as the bifurcation parameter grows has the same structure as that of the heteroclinic separatrix manifold of a rotor-type singular point and should be similar to a heteroclinic separatrix zigzag in the Poincare section for small dissipation parameter *b* . The projection of Poincare section ( 0.1, 0) *y x* of solution of system of Eqs. (12) is presented in Fig. 9a.

**Figure 9.** Projection of the Poincare section ( 0.1, 0) *y x* of solution of system of Eqs. (12) for *b* 0.08, 1, 0.051 (a) and the projection of the manifold of the singular attractor onto the plane (,) *x z* corresponding to the section (b).

### *2.2.3. Some examples of classical tree-dimensional autonomous nonlinear systems*

For instance let consider four classical tree-dimensional chaotic systems of nonlinear ordinary differential equations describing different natural and social processes:

the Lorenz hydrodynamic system

$$
\dot{\mathbf{x}} = \sigma(y - \mathbf{x}), \quad \dot{y} = \mathbf{x}(r - z) - y, \quad \dot{z} = \mathbf{x}y - bz,\tag{14}
$$

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 145

**Figure 10.** Cycles of period three in Lorenz (14) (a), Ressler (15) (b), Chua (16) (c) and Magnitskii (17)

**Figure 11.** Homoclinic cascade of bifurcations of stable cycles (a)-(c) and heteroclinic butterfly

All these classical systems have also arbitrary stable cycles from the Sharkovskii subharmonic cascade of bifurcations and all singular attractors from this cascade. Moreover, these systems have also more complex complete or incomplete homoclinic or heteroclinic cascades of bifurcations which take place after Sharkovskii cascade and infinitely many homoclinic or heteroclinic singular attractors (Magnitskii & Sidorov, 2006; Magnitskii , 2008;

In conclusion of this Section note that also very many other nonlinear three-dimensional autonomous systems of ordinary differential equations considered in the scientific literature have the same universal scenario of transition to dynamical chaos in accordance with the Feigenbaum-Sharkovskii-Magnitskii (FSM) theory. Among them there are systems of: Vallis, Anishchenko-Astakhov, Rabinovich-Fabricant, Pikovskii-Rabinovich-Trakhtengertz, Sviregev, Volterra-Gause, Sprott, Chen, Rucklidge, Genezio-Tesi, Wiedlich-Trubetskov and

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

separatrix contour (d) in the Lorenz system (14).

many others (Magnitskii , 2011; Magnitsky , 2007).

example consider Rikitaki system

**2.3. Many- and infinitely- dimensional autonomous systems** 

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

(d) systems.

Magnitskii , 2011).

the Ressler chemical system

$$
\dot{\mathbf{x}} = -(y+z), \; \dot{y} = \mathbf{x} + ay, \; \dot{z} = b + z(\mathbf{x} - \mu), \tag{15}
$$

the Chua electro technical system

$$
\dot{\mathbf{x}} = \mu \mathbf{l} \,\mathrm{J}\,\mathrm{y} - h(\mathbf{x}) \mathbf{l}, \text{ } \dot{\mathbf{y}} = \mathbf{x} - \mathbf{y} + \mathbf{z}, \text{ } \dot{\mathbf{z}} = -\beta \mathbf{y}, \tag{16}
$$

where *h x*( ) is a piecewise linear function; and the Magnitskii macroeconomic system

$$\dot{\mathbf{x}} = b\mathbf{x}((1-\sigma)\mathbf{z} - \delta\mathbf{y}),\\\dot{\mathbf{y}} = \mathbf{x}(1 - (1-\delta)\mathbf{y} + \sigma\mathbf{z}),\\\dot{\mathbf{z}} = a(\mathbf{y} - d\mathbf{x}).\tag{17}$$

To demonstrate that the transition to chaos under variation of a system parameter in all these classical chaotic systems occurs in accordance with the described above unique FSM scenario, let show that all these systems have period three stable cycles in accordance with the Sharkovskii order (3). This stable period three cycles are presented in Fig. 10.

In Fig. 11 it is presented homoclinic cascade of bifurcations of stable cycles in the Lorenz system and the most complex separatrix contour in this system named as **heteroclinic butterfly** which is the limit of the heneroclinic cascade of bifurcations of stable heteroclinic cycles (Magnitskii & Sidorov, 2006; Magnitskii , 2008; Magnitskii , 2011).

*b* 0.08, 1, 0.051 

(,) *x z* corresponding to the section (b).

the Lorenz hydrodynamic system

the Chua electro technical system

the Ressler chemical system

in Fig. 10.

**Figure 9.** Projection of the Poincare section ( 0.1, 0) *y x* of solution of system of Eqs. (12) for

*2.2.3. Some examples of classical tree-dimensional autonomous nonlinear systems* 

ordinary differential equations describing different natural and social processes:

 *x y x y x r z y z xy bz* 

*x y z y x ay z b z x* ( ), , ( ),

 *x y hx y x y z z y* 

For instance let consider four classical tree-dimensional chaotic systems of nonlinear

[ ( )], , ,

 

To demonstrate that the transition to chaos under variation of a system parameter in all these classical chaotic systems occurs in accordance with the described above unique FSM scenario, let show that all these systems have period three stable cycles in accordance with the Sharkovskii order (3). This stable period three cycles are presented

In Fig. 11 it is presented homoclinic cascade of bifurcations of stable cycles in the Lorenz system and the most complex separatrix contour in this system named as **heteroclinic butterfly** which is the limit of the heneroclinic cascade of bifurcations of stable heteroclinic

where *h x*( ) is a piecewise linear function; and the Magnitskii macroeconomic system

*x bx z y y x y z z a y dx* ((1 ) ), (1 (1 ) ), ( ).

cycles (Magnitskii & Sidorov, 2006; Magnitskii , 2008; Magnitskii , 2011).

(a) and the projection of the manifold of the singular attractor onto the plane

( ), ( ) , , (14)

(15)

(16)

(17)

**Figure 10.** Cycles of period three in Lorenz (14) (a), Ressler (15) (b), Chua (16) (c) and Magnitskii (17) (d) systems.

**Figure 11.** Homoclinic cascade of bifurcations of stable cycles (a)-(c) and heteroclinic butterfly separatrix contour (d) in the Lorenz system (14).

All these classical systems have also arbitrary stable cycles from the Sharkovskii subharmonic cascade of bifurcations and all singular attractors from this cascade. Moreover, these systems have also more complex complete or incomplete homoclinic or heteroclinic cascades of bifurcations which take place after Sharkovskii cascade and infinitely many homoclinic or heteroclinic singular attractors (Magnitskii & Sidorov, 2006; Magnitskii , 2008; Magnitskii , 2011).

In conclusion of this Section note that also very many other nonlinear three-dimensional autonomous systems of ordinary differential equations considered in the scientific literature have the same universal scenario of transition to dynamical chaos in accordance with the Feigenbaum-Sharkovskii-Magnitskii (FSM) theory. Among them there are systems of: Vallis, Anishchenko-Astakhov, Rabinovich-Fabricant, Pikovskii-Rabinovich-Trakhtengertz, Sviregev, Volterra-Gause, Sprott, Chen, Rucklidge, Genezio-Tesi, Wiedlich-Trubetskov and many others (Magnitskii , 2011; Magnitsky , 2007).
