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

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 universal FSM theory of transition to chaos.

## *3.3.1. Hamiltonian systems with two degrees of freedom*

Consider generalized Hamiltonian-Mathieu system with two degrees of freedom

$$\dot{\mathbf{x}} = \mathbf{y}, \quad \dot{\mathbf{y}} = -(\boldsymbol{\delta} + \mathbf{z})\,\mathbf{x} - \mathbf{x}^3, \quad \dot{\mathbf{z}} = \mathbf{r}, \quad \dot{\mathbf{r}} = -\mathbf{z} - \mathbf{x}^2/2 \tag{37}$$

wth Hamiltonian

154 Nonlinearity, Bifurcation and Chaos – Theory and Applications

is shown in Fig. 18.

*3.2.3. Conservative Duffing-Holmes equation* 

dimensional conservative autonomous system

 2 and 0.337 

nonautonomous conservative system of the equations

<sup>3</sup> *x yy x x t* ,

loops of zero saddle singular point around singular points 1/2 *O* ( ,0)

, 0 *z z* (0) .

<sup>3</sup> *x y y x x zz rr z* , ,,

 <sup>3</sup> *x y y x x z yz rr z* , 

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

 

 

parameter

system of Eqs. (32) for

Nonperturbed ( 0)

of nonperturbed system.

with conditions 22 2 *Hz r*

(35) in space of parameters (, )

some values of parameters ( , 0)

most part of phase space of conservative system of Eqs. (31) at reduction of values of

**Figure 18.** Projections on the plane (sin , ) *x y* of Poincare section ( 0, 0) *r z* of solutions of dissipative

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

As other above considered systems, system of Eqs. (33) is equivalent to the perturbed four-

In the work (Dubrovsky, 2010) the two-parametrical bifurcation diagram of system of Eqs.

bifurcations up to the cycle of period three, stable in dissipative system of Eqs. (35) at the

 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

 (b) and 0.29 

> 

system of Eqs. (33) has in the plane (,) *x y* two homoclinic separatrix

The system

 

> 

is constructed. All cycles of the subharmonic cascade of

, are continued in a plane of parameters up to the value

, , (35)

(c).

cos . (33)

which are centers

(34)

 (a), 0.33 

$$H(\mathbf{x}, \mathbf{y}, \mathbf{z}, r) = (\delta \mathbf{x}^2 + \mathbf{y}^2 + \mathbf{z}^2 + r^2) / 2 + \mathbf{z} \mathbf{x}^2 / 2 + \mathbf{x}^4 / 4 = \varepsilon.$$

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 extended dissipative system can have a kind of

$$\dot{\mathbf{x}} = y\_r \cdot \dot{\mathbf{y}} = -(\boldsymbol{\delta} + \mathbf{z})\mathbf{x} - \mathbf{x}^3 - \mu y\_r \cdot \dot{\mathbf{z}} = \mathbf{r}\_r \cdot \dot{\mathbf{r}} = -\mathbf{z} - \mathbf{x}^2 / 2 + (\mathbf{z} - H(\mathbf{x}, y, z, r))\mathbf{r} . \tag{38}$$

Let's consider a case 0.5 at which the cycle 22 2 *z r* ( 0) *x y* of Hamiltonian system of Eqs. (37) is an elliptic cycle at enough small . At 0.185 period doubling 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 Hamiltonian system of Eqs. (37) when 0 . Development and complication of infinitely

folded heteroclinic separatrix zigzag in dissipative extended system of Eqs. (38) at 1 accompanied a stretching of its accordion on all phase space of conservative system of Eqs. (37) at reduction of values of parameter 0 is shown in Fig. 19.

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 157

 cos , , cos . (40)

. In this case extended dissipative

. At large enough values of

 , , ,, . (42)

) conservative system of Eqs. (41) possesses chaotic

*t u* we shall receive from the system of Eqs. (40) the conservative

(39)

Let's show now that the second part of the above mentioned statement does not correspond also to the real situation, and that in Hamiltonian systems with two and a half degrees of freedom trajectories are not obliged to cover all power surface even at the large perturbations. Thus, areas with regular, local chaotic and global chaotic dynamics can exist simultaneously

Let's consider the system consisting from two nonlinear oscillators with weak periodic

Hamiltonian (39) generates so called Hamiltonian system with two and a half degrees of freedom, i.e. four-dimensional system of ordinary differential equations with periodic

3 3 *x y y x x zu z r r z z xu u v v u* , ,, , , (41)

 , *u v* (0) , (0) 0 

3 3 *x y y x x zu y z r r z z xu r u v v u* ,

It is easy to see, that solutions of conservative system of Eqs. (41) with initial conditions

<sup>3</sup> *x y y x x xu u v v u* , ,, (43)

The right part of last system coincides with the right part of the considered above

dynamics even on solutions of system of Eqs. (43), as at reduction of values of parameter

the subharmonic cascade of bifurcations of stable cycles exists in dissipative system of Eqs. (42) giving rise complex heteroclinic separatrix manifolds in four-dimensional subspace of solutions of conservative system of Eqs. (41) being solutions of system of Eqs. (43). However, chaotic dynamics of solutions of system of Eqs. (41) is local even inside this fourdimensional subspace of solutions and is limited by area of regular movements on twodimensional tori (see in Fig. 20a). At the same time for solutions, not satisfying conditions 0 00 0 *z xr y* , or 0 00 0 *z xr y* , conservative system of Eqs. (41) has areas of complex global chaotic dynamics and areas of regular movement on three-dimensional tori even at

on power surface of such systems even at large values of perturbation parameter.

224 224 *H x x x z z z xz t* ( /2 / 2) / 2 cos .

 3 3 *x yy x x z t z r r z z x t* , 

six-dimensional autonomous system of ordinary differential equations

0 00 0 *z xr y* , are solutions of four-dimensional conservative system

conservative generalized Mathieu system of Eqs. (36) with 1

such large values of perturbation parameter (see in Fig. 20b).

(for example, 1.8

nonlinear connection. Hamiltonian of this system looks like

coefficients

parameter

Having designated cos

system can have a kind of

with the condition <sup>222</sup> *Hu v*

**Figure 19.** Projections on the plane ( , ) *x y* of the Poincare section ( 0, 0) *r z* of solutions of dissipative system of Eqs. (38) for 0.5 , 1 and 0.029 (a), 0.01 (b).

In conclusion of this item note that the FSM scenario of transition to chaos takes place also in classical Henon-Heiles system with Hamiltonian

$$H(\mathbf{x}, \mathbf{y}, \mathbf{z}, r) = \left(\mathbf{x}^2 + \mathbf{y}^2 + \mathbf{z}^2 + r^2\right) / 2 + z\mathbf{x}^2 - \mathbf{z}^3 / 3$$

and in Yang-Mills-Higgs system (Magnitskii, 2008b; Magnitskii , 2009) with two degrees of freedom and with Hamiltonian

$$H = \left(\dot{\mathbf{x}}^2 + \dot{\mathbf{z}}^2\right) / \text{ } \mathbf{2} + \mathbf{x}^2 \mathbf{z}^2 \text{ } \text{ } \mathbf{2} + \nu(\mathbf{x}^2 + \mathbf{z}^2) / \text{ } \mathbf{2} \dots$$
