*3.2.1. Hyperbolic nonautonomous concervative system*

Consider nonautonomous conservative two-dimensional system of ordinary differential equations

$$
\dot{\mathbf{x}} = \mathbf{y}, \quad \dot{\mathbf{y}} = (1 + \mathbf{z}\cos t)\mathbf{x} - \mathbf{x}^3. \tag{27}
$$

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 153

there is a development and complication of heteroclinic separatrix

*t x* (30)

Let us consider alongside with system of Eqs.

(32)

the conservative system of Eqs. (31) already

the cascades of bifurcations of births of stable

0.34 two homoclinic cascades of bifurcations

 ,

0.348 .

1 when

0.38 the

 and 0.04 .

(31)

there is a merge of two tapes (separatrix manifolds) of singular attractors,

. In Fig. 17 accordions of infinitely folded heteroclinic separatrix zigzags

accompanied formation of uniform heteroclinic separatrix zigzag. At the further reduction

zigzag, accompanied a stretching of its accordion on all phase space of conservative system

Consider a standard example of a pendulum with vertically periodically oscillating point of

 

Let us write down the system of equations with Hamiltonian (30) in the form of four-

 

tends to zero. It is convenient to analyze solutions of systems of Eqs. (31)- (32)

and analyze numerically transition from solutions of dissipative system of Eqs. (32) to

possesses chaotic dynamics in sense of theory FSM. It is easy to be convinced of it if

in dissipative extended system of Eqs. (32) tends to zero. At

double period bifurcation of each of original singular stable limit cycles *C* occurs, that gives rise to cascades of Feigenbaum period doubling bifurcations. The given cascades of

cycles with the periods according to the Sharkovskii order begin. Cycles of period five, for

begin, then, as well as in other systems, there is a merge of two tapes of singular attractors (two infinitely folded heteroclinic separatrix manifolds) and then process of formation of new stable cycles proceeds on uniform infinitely folded heteroclinic separatrix surface. Development and complication of infinitely folded heteroclinic separatrix zigzag in dissipative extended system of Eqs. (32) accompanied a stretching of its accordion on the

in Poincare section of dissipative system of Eqs. (29) are shown at 0.25

*3.2.2. Standard example of a pendulum with oscillating point of fixing* 

 2 2 *Hxyt y* ( , , , ) / 2 ( cos )cos . 

<sup>2</sup> *x yy z xz rr z* , ( )sin , ,

 <sup>2</sup> *x yy z x yz rr z* , ( )sin , , 

(31) the extended dissipative system of the equations

At value of perturbation parameter 2

At further reduction of values of parameter

0.3428 . At

, 0 *z z* (0) .

solutions of conservative system of Eqs. (31) at the fixed values of parameters

bifurcations come to the end with a birth of two singular Feigenbaum attractors at

Then at 0.251 

of values of parameter

with conditions 22 2 *Hz r*

parameter

parameter

in coordinates (sin , ). *x y*

example, can be observed at

of Eqs. (28) at 0

fixing, that is a system with Hamiltonian

dimensional conservative system of the equations

Nonperturbed ( 0) system of Eqs. (27) has in the plane (,) *x y* two homoclinic separatrix loops of zero saddle singular point around singular points *O* ( 1,0) which are centers of nonperturbed system. System of Eqs. (27) is equivalent to the perturbed four-dimensional conservative autonomous system

$$
\dot{\mathbf{x}} = y, \quad \dot{y} = (1+z)\mathbf{x} - \mathbf{x}^3, \text{ } \dot{z} = \mathbf{r}, \text{ } \dot{r} = -z. \tag{28}
$$

with conditions 22 2 *Hz r* , 0 *z z* (0) . The system

$$
\dot{\mathbf{x}} = \mathbf{y}, \quad \dot{\mathbf{y}} = (1 + \mathbf{z})\mathbf{x} - \mathbf{x}^3 - \mu y, \text{ } \dot{\mathbf{z}} = \mathbf{r}, \, \dot{r} = -\mathbf{z}. \tag{29}
$$

is the extended dissipative system for conservative system of Eqs. (28). For large enough values of perturbation parameter (for example, 1.5 ) conservative system of Eqs. (28) has a chaotic dynamics, because at reduction of values of parameter in dissipative system of Eqs. (29) there are subharmonic cascades of bifurcations in full accordance with the theory FSM.

**Figure 17.** Projections on the plane ( , ) *x y* of Poincare section ( 0, 0) *r z* of solutions of dissipative system of Eqs. (29) for 1.5 and 0.25 (a), 0.04 (b).

Then at 0.251 there is a merge of two tapes (separatrix manifolds) of singular attractors, accompanied formation of uniform heteroclinic separatrix zigzag. At the further reduction of values of parameter there is a development and complication of heteroclinic separatrix zigzag, accompanied a stretching of its accordion on all phase space of conservative system of Eqs. (28) at 0 . In Fig. 17 accordions of infinitely folded heteroclinic separatrix zigzags in Poincare section of dissipative system of Eqs. (29) are shown at 0.25 and 0.04 .

### *3.2.2. Standard example of a pendulum with oscillating point of fixing*

152 Nonlinearity, Bifurcation and Chaos – Theory and Applications

0.138 (c).

equations

Nonperturbed ( 0)

system of Eqs. (29) for

 1.5 and 0.25 

conservative autonomous system

with conditions 22 2 *Hz r*

separatrix zigzag in dissipative Croquette system of Eqs. (26), for

example of such a system. Let us analyze some other examples.

*3.2.1. Hyperbolic nonautonomous concervative system* 

<sup>3</sup> *x y y tx x* , (1 cos ) .

chaotic dynamics, because at reduction of values of parameter

values of perturbation parameter (for example, 1.5

<sup>3</sup> *x y y zx x y z rr z* , (1 ) , , .

, 0 *z z* (0) .

**3.2. Hamiltonian systems with one and a half degrees of freedom** 

In modern scientific literature Hamiltonian systems with one and a half degrees of freedom refer to as nonautonomous conservative two-dimensional systems of ordinary differential equations with time-dependent Hamiltonian. Considered above Croquette system is an

Consider nonautonomous conservative two-dimensional system of ordinary differential

loops of zero saddle singular point around singular points *O* ( 1,0) which are centers of nonperturbed system. System of Eqs. (27) is equivalent to the perturbed four-dimensional

<sup>3</sup> *x y y zx x z rr z* , (1 ) , , . (28)

is the extended dissipative system for conservative system of Eqs. (28). For large enough

(29) there are subharmonic cascades of bifurcations in full accordance with the theory FSM.

**Figure 17.** Projections on the plane ( , ) *x y* of Poincare section ( 0, 0) *r z* of solutions of dissipative

(b).

 (a), 0.04 

system of Eqs. (27) has in the plane (,) *x y* two homoclinic separatrix

The system

 0.55 and 0.1415 

(b),

(27)

(29)

) conservative system of Eqs. (28) has a

in dissipative system of Eqs.

Consider a standard example of a pendulum with vertically periodically oscillating point of fixing, that is a system with Hamiltonian

$$H(\mathbf{x}, y, t, \varepsilon) = y^2 \,/\, \mathcal{D} + (\alpha^2 + \varepsilon \cos t) \cos \mathbf{x}.\tag{30}$$

Let us write down the system of equations with Hamiltonian (30) in the form of fourdimensional conservative system of the equations

$$
\dot{\mathbf{x}} = \mathbf{y}\_{\prime} \quad \dot{\mathbf{y}} = (\alpha^2 + \mathbf{z})\sin\mathbf{x}\_{\prime} \quad \dot{\mathbf{z}} = \mathbf{r}\_{\prime} \quad \dot{\mathbf{r}} = -\mathbf{z} \tag{31}
$$

with conditions 22 2 *Hz r* , 0 *z z* (0) . Let us consider alongside with system of Eqs. (31) the extended dissipative system of the equations

$$
\dot{\mathbf{x}} = \mathbf{y}\_t \quad \dot{\mathbf{y}} = (\alpha \mathbf{o}^2 + \mathbf{z})\sin\mathbf{x} - \mu y\_t \quad \dot{\mathbf{z}} = \mathbf{r}\_t \quad \dot{\mathbf{r}} = -\mathbf{z} \tag{32}
$$

and analyze numerically transition from solutions of dissipative system of Eqs. (32) to solutions of conservative system of Eqs. (31) at the fixed values of parameters , 1 when parameter tends to zero. It is convenient to analyze solutions of systems of Eqs. (31)- (32) in coordinates (sin , ). *x y*

At value of perturbation parameter 2 the conservative system of Eqs. (31) already possesses chaotic dynamics in sense of theory FSM. It is easy to be convinced of it if parameter in dissipative extended system of Eqs. (32) tends to zero. At 0.38 the double period bifurcation of each of original singular stable limit cycles *C* occurs, that gives rise to cascades of Feigenbaum period doubling bifurcations. The given cascades of bifurcations come to the end with a birth of two singular Feigenbaum attractors at 0.348 .

At further reduction of values of parameter the cascades of bifurcations of births of stable cycles with the periods according to the Sharkovskii order begin. Cycles of period five, for example, can be observed at 0.3428 . At 0.34 two homoclinic cascades of bifurcations begin, then, as well as in other systems, there is a merge of two tapes of singular attractors (two infinitely folded heteroclinic separatrix manifolds) and then process of formation of new stable cycles proceeds on uniform infinitely folded heteroclinic separatrix surface. Development and complication of infinitely folded heteroclinic separatrix zigzag in dissipative extended system of Eqs. (32) accompanied a stretching of its accordion on the

most part of phase space of conservative system of Eqs. (31) at reduction of values of parameter is shown in Fig. 18.

**Figure 18.** Projections on the plane (sin , ) *x y* of Poincare section ( 0, 0) *r z* of solutions of dissipative system of Eqs. (32) for 2 and 0.337 (a), 0.33 (b) and 0.29 (c).

### *3.2.3. Conservative Duffing-Holmes equation*

Rewrite conservative Duffing-Holmes equation in the form of two-dimensional nonautonomous conservative system of the equations

$$
\dot{\mathbf{x}} = \mathbf{y}\_{\prime} \cdot \dot{\mathbf{y}} = \delta \, \mathbf{x} - \mathbf{x}^{3} + \mathbf{z} \cos \alpha \, t. \tag{33}
$$

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 155

conservative system of Eqs. (34) has also homoclinic cascade of

(37)

( 0) *x y* of Hamiltonian

period doubling

(36)

0

(38)

Note in conclusion of this item that the FSM scenario of transition to chaos takes place also in many other nonautonomous two-dimensional nonlinear conservative systems and, in

In modern scientific literature Hamiltonian systems with two degrees of freedom refer to as autonomous Hamiltonian four-dimensional systems of ordinary differential equations, Hamiltonian systems with two and a half degrees of freedom refer to as nonautonomous conservative four-dimensional systems of ordinary differential equations with timedependent Hamiltonian and Hamiltonian systems with three degrees of freedom refer to as autonomous Hamiltonian six-dimensional systems of ordinary differential equations. We consider examples of such systems and show that all such conservative systems satisfy the

perturbation parameter

(Magnitskii , 2008b; Magnitskii , 2011)).

universal FSM theory of transition to chaos.

extended dissipative system can have a kind of

Let's consider a case 0.5

Hamiltonian system of Eqs. (37) when

system of Eqs. (37) is an elliptic cycle at enough small

wth Hamiltonian

*3.3.1. Hamiltonian systems with two degrees of freedom* 

bifurcations in full accordance with the FSM theory.

particular, in classical generalized conservative Mathieu system

<sup>3</sup> *x y y zx x z rr z* , () , ,

**3.3. More complex Hamiltonian and conservative systems** 

which is equivalent to conservative generalized Mathieu equation (7) with

Consider generalized Hamiltonian-Mathieu system with two degrees of freedom

2 222 2 4 *H x y z r x y z r zx x* (, ,,) (

3 2 *x y y zx x y z r r z x Hxyzr r* , () , ,

at which the cycle 22 2 *z r*

bifurcation of the elliptic cycle occurs giving rise to various cascades of period doubling bifurcations and subharmonic cascades of bifurcations, generating infinitely folded heteroclinic separatrix manifolds both in extended dissipative system of Eqs. (38) and in

 

The system of Eqs. (37) contains additional composed <sup>2</sup> *x* / 2 in the fourth equation of the conservative four-dimensional generalized Mathieu system of Eqs. (36). In this case

)/2 /2 /4 .

/ 2 ( ( , , , )) .

 . At 0.185 

0 . Development and complication of infinitely

 3 2 *x y y zx x z r r z x* , ( ) , , /2 

Nonperturbed ( 0) system of Eqs. (33) has in the plane (,) *x y* two homoclinic separatrix loops of zero saddle singular point around singular points 1/2 *O* ( ,0) which are centers of nonperturbed system.

As other above considered systems, system of Eqs. (33) is equivalent to the perturbed fourdimensional conservative autonomous system

$$
\dot{\mathbf{x}} = \mathbf{y}\_r \cdot \dot{\mathbf{y}} = \delta \, \mathbf{x} - \mathbf{x}^3 + \mathbf{z}\_r \, \dot{\mathbf{z}} = \alpha \mathbf{r}\_r \, \dot{r} = -\alpha \mathbf{z} \tag{34}
$$

with conditions 22 2 *Hz r* , 0 *z z* (0) . The system

$$
\dot{\mathbf{x}} = \mathbf{y}, \ \dot{\mathbf{y}} = \boldsymbol{\delta} \times -\mathbf{x}^3 + \boldsymbol{z} - \mu \boldsymbol{y}, \ \dot{\mathbf{z}} = \alpha \mathbf{r}, \ \dot{r} = -\alpha \mathbf{z} \tag{35}
$$

is the extended dissipative system for conservative system of Eqs. (34).

In the work (Dubrovsky, 2010) the two-parametrical bifurcation diagram of system of Eqs. (35) in space of parameters (, ) is constructed. All cycles of the subharmonic cascade of bifurcations up to the cycle of period three, stable in dissipative system of Eqs. (35) at the some values of parameters ( , 0) , are continued in a plane of parameters up to the value 0 (when the system becomes conservative) by the modified Magnitskii method of stabilization (Magnitskii & Sidorov, 2006). Thus, it is proved an existence of full subharmonic cascade of bifurcations of cycles of any period according to Sharkovskii order in conservative system of Duffing-Holmes equations (34). For large enough values of perturbation parameter conservative system of Eqs. (34) has also homoclinic cascade of bifurcations in full accordance with the FSM theory.

Note in conclusion of this item that the FSM scenario of transition to chaos takes place also in many other nonautonomous two-dimensional nonlinear conservative systems and, in particular, in classical generalized conservative Mathieu system

$$
\dot{\mathbf{x}} = \mathbf{y}, \text{ } \dot{\mathbf{y}} = - (\delta + \mathbf{z})\mathbf{x} - \alpha \mathbf{x}^3,
\dot{\mathbf{z}} = \alpha \mathbf{r}, \dot{r} = -\alpha \mathbf{z} \tag{36}
$$

which is equivalent to conservative generalized Mathieu equation (7) with 0 (Magnitskii , 2008b; Magnitskii , 2011)).
