*3.3.2. Hamiltonian systems with two and a half degrees of freedom*

It is considered to be in modern literature that in case of systems with one and a half and two degrees of freedom, conservation of energy limits divergence of trajectories along all power surface, and in case of systems with two and a half and more degrees of freedom trajectories form in phase space uniform everywhere dense network named by Arnold web. Trajectories thus, as it is considered, for large enough time cover all power surface of system, approaching as much as close to its any point.

About inadequacy of the first part of this statement to the real situation all considered above examples of Hamiltonian systems with one and a half and two degrees of freedom testify. It follows from the established fact that chaotic dynamics in conservative systems is not consequence of tori resonances in nonperturbed systems, but is consequence of infinite cascades of bifurcations of births of new elliptic and hyperbolic cycles, not being cycles of nonperturbed systems. Thus the accordion of heteroclinic separatrix zigzag can be stretched on all phase space of perturbed conservative system (on all power surface), and this process is not connected in any way with tori of nonperturbed system.

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

156 Nonlinearity, Bifurcation and Chaos – Theory and Applications

(37) at reduction of values of parameter 0

dissipative system of Eqs. (38) for

freedom and with Hamiltonian

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.

is shown in Fig. 19.

**Figure 19.** Projections on the plane ( , ) *x y* of the Poincare section ( 0, 0) *r z* of solutions of

 1 and 0.029 

In conclusion of this item note that the FSM scenario of transition to chaos takes place also in

2 222 23 *H x y z r x y z r zx z* (, ,,) ( )/2 /3

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

2 2 22 2 2 *H x z xz x z* ( )/2 /2 ( )/2

It is considered to be in modern literature that in case of systems with one and a half and two degrees of freedom, conservation of energy limits divergence of trajectories along all power surface, and in case of systems with two and a half and more degrees of freedom trajectories form in phase space uniform everywhere dense network named by Arnold web. Trajectories thus, as it is considered, for large enough time cover all power surface of

About inadequacy of the first part of this statement to the real situation all considered above examples of Hamiltonian systems with one and a half and two degrees of freedom testify. It follows from the established fact that chaotic dynamics in conservative systems is not consequence of tori resonances in nonperturbed systems, but is consequence of infinite cascades of bifurcations of births of new elliptic and hyperbolic cycles, not being cycles of nonperturbed systems. Thus the accordion of heteroclinic separatrix zigzag can be stretched on all phase space of perturbed conservative system (on all power surface), and this process

.

 (a), 0.01 

(b).

 0.5 , 

*3.3.2. Hamiltonian systems with two and a half degrees of freedom* 

system, approaching as much as close to its any point.

is not connected in any way with tori of nonperturbed system.

classical Henon-Heiles system with Hamiltonian

> Let's consider the system consisting from two nonlinear oscillators with weak periodic nonlinear connection. Hamiltonian of this system looks like

$$H = \left(\dot{\mathbf{x}}^2 + \mathbf{x}^2 + \mathbf{x}^4 \,/\,\mathbf{2} + \dot{\mathbf{z}}^2 + \dot{\mathbf{z}}^2 + \mathbf{z}^4 \,/\,\mathbf{2}\right) / \mathbf{2} + \boldsymbol{\sigma} \,\mathbf{x} \,\mathbf{z} \,\text{cost}.\tag{39}$$

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 coefficients

$$
\dot{\mathbf{x}} = \mathbf{y}, \ \dot{\mathbf{y}} = -\mathbf{x} - \mathbf{x}^3 - \mathfrak{s}\ \mathbf{z}\cos t, \ \dot{\mathbf{z}} = \mathbf{r}, \ \dot{\mathbf{r}} = -\mathbf{z} - \mathbf{z}^3 - \mathfrak{s}\ \mathbf{x}\cos t. \tag{40}
$$

Having designated cos *t u* we shall receive from the system of Eqs. (40) the conservative six-dimensional autonomous system of ordinary differential equations

$$
\dot{\mathbf{x}} = \mathbf{y}, \quad \dot{\mathbf{y}} = -\mathbf{x} - \mathbf{x}^3 - \mathbf{z} \,\mathbf{u}, \quad \dot{\mathbf{z}} = \mathbf{r}, \quad \dot{\mathbf{r}} = -\mathbf{z} - \mathbf{z}^3 - \mathbf{x} \,\mathbf{u}, \quad \dot{\mathbf{u}} = \mathbf{v}, \quad \dot{\mathbf{v}} = -\mathbf{u} \tag{41}
$$

with the condition <sup>222</sup> *Hu v* , *u v* (0) , (0) 0 . In this case extended dissipative system can have a kind of

$$\dot{\mathbf{x}} = \mathbf{y}, \quad \dot{\mathbf{y}} = -\mathbf{x} - \mathbf{x}^3 - \mathbf{z}\,\mathbf{u} - \mu\,\mathbf{y}, \quad \dot{\mathbf{z}} = \mathbf{r}, \quad \dot{\mathbf{r}} = -\mathbf{z} - \mathbf{z}^3 - \mathbf{x}\,\mathbf{u} - \mu\,\mathbf{r}, \quad \dot{\mathbf{u}} = \mathbf{v}, \quad \dot{\mathbf{v}} = -\mathbf{u}.\tag{42}$$

It is easy to see, that solutions of conservative system of Eqs. (41) with initial conditions 0 00 0 *z xr y* , are solutions of four-dimensional conservative system

$$
\dot{\mathbf{x}} = \mathbf{y}\_{\prime} \quad \dot{\mathbf{y}} = -\mathbf{x} - \mathbf{x}^{3} - \mathbf{x} \, u\_{\prime} \quad \dot{\mathbf{u}} = \mathbf{v}\_{\prime} \quad \dot{\mathbf{v}} = -\mathbf{u} \tag{43}
$$

The right part of last system coincides with the right part of the considered above conservative generalized Mathieu system of Eqs. (36) with 1 . At large enough values of parameter (for example, 1.8 ) conservative system of Eqs. (41) possesses chaotic 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 such large values of perturbation parameter (see in Fig. 20b).

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 159

. All

tends to zero. Corresponding

, 0 00 0 *u xv y* , and

The system of Eqs. (44) is interesting to those, that character of its dynamics contradicts practically to all propositions of the modern classical theory of Hamiltonian systems. In system of Eqs. (44) there exist simultaneously areas of regular movement on twodimensional tori around of basic cycles of the system, areas of regular movement on threedimensional tori around of mentioned above two-dimensional tori, areas of local chaotic behavior of trajectories of the system in four-dimensional subspace of a five-dimensional power surface and areas of global chaotic behavior of trajectories of the system in other

part of a power surface even at enough large value of the perturbation parameter

system of Eqs. (45) when dissipation parameter

 and 0.005 

(Magnitskii , 2009b; Magnitskii , 2011).

3 , 0 00 0 *u xv y* , and

(45) for 

 0.125 , 0.095 

Magnitskii , 2011)).

solutions of extended dissipative system of Eqs. (45) at 3

tori of the system are not tori of so-called nonperturbed system, but they are born as a result of various bifurcations. Global chaos in the system is not consequence of destruction of any mythical tori of nonperturbed system as this phenomenon is treated by the modern classical Hamiltonian mechanics and KAM theory. It is extreme consequence of complication of infinitely folded heteroclinic separatrix manifold of extended dissipative

heteroclinic separatrix zigzags in projections to the plane (,) *x y* of the section 0 *r* of

Thus we can make a conclusion that universal bifurcation Feigenbaum-Sharkovskii-Magnitskii theory describes also transition to dynamical chaos in nonlinear conservative and, in particular, Hamiltonian systems of ordinary differential equations at large enough values of perturbation parameter. Note that for small values of perturbation parameter the key role in complication of dynamics of any conservative system is played by nonlocal effect of duplication of hyperbolic and elliptic cycles and tori in a neighborhood of separatrix contour (or surface) of nonperturbed system opened and analyzed by the author in

**Figure 21.** Development and complication of heteroclinic separatrix zigzag in dissipative system of Eqs.

0.095 (b) and

0.005 (c).

0.125 (a),

are presented in Fig. 21 (see (Magnitskii , 2008b;

**Figure 20.** Projections of the section *v* 0 of system of Eqs. (41) on the plane (,) *x y* for 1.8 , 0 00 0 *z xr y* , (a) and 0 00 0 *z xr y* , (b).}

So, in conservative system of Eqs. (41) even at enough large values of parameter there exist simultaneously areas of regular movement on two-dimensional tori around of basic cycles of the system, areas of regular movement on three-dimensional tori around of mentioned above two-dimensional tori, areas of local chaotic behaviour of trajectories of the system in four-dimensional subspace of five-dimensional phase space and areas of global chaotic behavior of trajectories of the system in the other part of phase space. All tori of the system are not tori of nonperturbed system, and are born as a result of various bifurcations in accordance with FSM theory. Global chaos in the system is not consequence of destruction of any mythical tori of nonperturbed system as this phenomenon is treated by the modern classical Hamiltonian mechanics and KAM (Kolmogorov-Arnold-Mozer) theory, and it is extreme consequence of complication of infinitely folded heteroclinic separatrix manifold of extended dissipative system of Eqs. (42) when dissipation parameter tends to zero (Magnitskii , 2011).

### *3.3.3. Hamiltonian system with three degrees of freedom*

Let's consider a complex Hamiltonian system with three degrees of freedom

$$\dot{\mathbf{x}} = \mathbf{y}, \ \dot{\mathbf{y}} = - (\mathcal{S} + \mathbf{z}) \mathbf{x} - \mathbf{x}^3, \ \dot{\mathbf{z}} = \mathbf{r}, \ \dot{\mathbf{r}} = -\mathbf{z} - \mathbf{x}^2 \ / \ \mathcal{D} - \mathbf{u}^2 \ / \ \mathcal{D}, \ \dot{\mathbf{u}} = \mathbf{v}, \ \dot{\mathbf{v}} = -(\mathbf{y} + \mathbf{z}) \mathbf{u} - \mathbf{u}^3 \tag{44}$$

with Hamiltonian

$$H(\mathbf{x}, \mathbf{y}, \mathbf{z}, r, u, v) = \left(\mathcal{S}\mathbf{x}^2 + \mathbf{y}^2 + \mathbf{z}^2 + \mathbf{y}\,\mathbf{u}^2 + \mathbf{v}^2\right)/2 + \mathbf{z}(\mathbf{x}^2 + \mathbf{u}^2)/2 + (\mathbf{x}^4 + \mathbf{u}^4)/4 = \varepsilon.$$

Extened dissipative two-parametrical system in this case can look like

$$\begin{aligned} \dot{\mathbf{x}} = \mathbf{y}\_{\prime\prime} \ \dot{\mathbf{y}} = -(\boldsymbol{\delta} + \mathbf{z})\mathbf{x} - \mathbf{x}^3 - \mu \, \mathbf{y}\_{\prime\prime} \ \dot{\mathbf{z}} = \mathbf{r}\_{\prime\prime} \ \dot{\mathbf{r}} = -\mathbf{z} - \mathbf{x}^2 \ / \ 2 - \mathbf{u}^2 \ / \ 2 \\ + (\boldsymbol{\varepsilon} - \boldsymbol{H})\mathbf{r}\_{\prime\prime} \ \dot{\mathbf{u}} = \mathbf{v}\_{\prime\prime} \ \dot{\mathbf{v}} = -(\boldsymbol{\gamma} + \mathbf{z})\boldsymbol{u} - \boldsymbol{u}^3 - \mu \, \mathbf{v} . \end{aligned} \tag{45}$$

The system of Eqs. (44) is interesting to those, that character of its dynamics contradicts practically to all propositions of the modern classical theory of Hamiltonian systems. In system of Eqs. (44) there exist simultaneously areas of regular movement on twodimensional tori around of basic cycles of the system, areas of regular movement on threedimensional tori around of mentioned above two-dimensional tori, areas of local chaotic behavior of trajectories of the system in four-dimensional subspace of a five-dimensional power surface and areas of global chaotic behavior of trajectories of the system in other part of a power surface even at enough large value of the perturbation parameter . All tori of the system are not tori of so-called nonperturbed system, but they are born as a result of various bifurcations. Global chaos in the system is not consequence of destruction of any mythical tori of nonperturbed system as this phenomenon is treated by the modern classical Hamiltonian mechanics and KAM theory. It is extreme consequence of complication of infinitely folded heteroclinic separatrix manifold of extended dissipative system of Eqs. (45) when dissipation parameter tends to zero. Corresponding heteroclinic separatrix zigzags in projections to the plane (,) *x y* of the section 0 *r* of solutions of extended dissipative system of Eqs. (45) at 3 , 0 00 0 *u xv y* , and 0.125 , 0.095 and 0.005 are presented in Fig. 21 (see (Magnitskii , 2008b; Magnitskii , 2011)).

158 Nonlinearity, Bifurcation and Chaos – Theory and Applications

0 00 0 *z xr y* , (a) and 0 00 0 *z xr y* , (b).}

tends to zero (Magnitskii , 2011).

*3.3.3. Hamiltonian system with three degrees of freedom* 

Let's consider a complex Hamiltonian system with three degrees of freedom

<sup>3</sup> 2 2 <sup>3</sup> *x y y zx x z r r z x u u v v zu u* , () , ,

222 22 22 44 *Hxyzruv x y z u v zx u x u* (, ,,,,) (

*x y y zx x y z r r z x u*

 

 

Extened dissipative two-parametrical system in this case can look like

with Hamiltonian

**Figure 20.** Projections of the section *v* 0 of system of Eqs. (41) on the plane (,) *x y* for

So, in conservative system of Eqs. (41) even at enough large values of parameter

exist simultaneously areas of regular movement on two-dimensional tori around of basic cycles of the system, areas of regular movement on three-dimensional tori around of mentioned above two-dimensional tori, areas of local chaotic behaviour of trajectories of the system in four-dimensional subspace of five-dimensional phase space and areas of global chaotic behavior of trajectories of the system in the other part of phase space. All tori of the system are not tori of nonperturbed system, and are born as a result of various bifurcations in accordance with FSM theory. Global chaos in the system is not consequence of destruction of any mythical tori of nonperturbed system as this phenomenon is treated by the modern classical Hamiltonian mechanics and KAM (Kolmogorov-Arnold-Mozer) theory, and it is extreme consequence of complication of infinitely folded heteroclinic separatrix manifold of extended dissipative system of Eqs. (42) when dissipation parameter

1.8 ,

 

> 

/ 2 / 2, , ( ) (44)

)/2 ( )/2 ( )/4 .

(45)

3 2 2

, ( ) , , /2 /2 ( ), , ( ) .

*Hr u v v zu u v*

3

there

> Thus we can make a conclusion that universal bifurcation Feigenbaum-Sharkovskii-Magnitskii theory describes also transition to dynamical chaos in nonlinear conservative and, in particular, Hamiltonian systems of ordinary differential equations at large enough values of perturbation parameter. Note that for small values of perturbation parameter the key role in complication of dynamics of any conservative system is played by nonlocal effect of duplication of hyperbolic and elliptic cycles and tori in a neighborhood of separatrix contour (or surface) of nonperturbed system opened and analyzed by the author in (Magnitskii , 2009b; Magnitskii , 2011).

**Figure 21.** Development and complication of heteroclinic separatrix zigzag in dissipative system of Eqs. (45) for 3 , 0 00 0 *u xv y* , and 0.125 (a), 0.095 (b) and 0.005 (c).
