**3.1. Bifurcation approach to analysis of Hamiltonian and conservative systems**

The fact that the dynamics of any conservative system is a limit case of a dynamics of an extended dissipative system with weak dissipation as the dissipation parameter tends to zero was proved by author in (Magnitskii, 2008b; Magnitskii, 2011) and illustrated by numerous examples of Hamiltonian systems with one and a half, two and three degrees of 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 systems with strong dissipation.

### *3.1.1. Theoretical basis of bifurcation approach*

148 Nonlinearity, Bifurcation and Chaos – Theory and Applications

on value of perturbation.

zero.

**Figure 14.** Projections of period one (a), two (b) and three (d) cycles, Feigenbaum attractor (c) and one

The modern classical theory of Hamiltonian systems reduces a problem of the analysis of dynamics of such system to the problem of its integralability, i.e. to a problem of construction of the canonical transformation reducing system to variables "action - angle" in which, as it is considered to be, movement occurs on a surface of *n* -dimensional torus and is periodic or quasiperiodic. Any nonintegrable nonlinear Hamiltonian system is considered as perturbation of integrable system, and the analysis of its dynamics is reduced to finding-out of a question on destruction or nondestruction some tori of nonperturbed system depending

In the present Section absolutely other bifurcation approach is considered for analysis of chaotic dynamics not only Hamiltonian, but also any conservative system of nonlinear differential equations. The method consists in consideration of approximating extended two-parametrical dissipative system of the equations, stable solutions (attractors) of which are as much as exact aproximations to solutions of original Hamiltonian (conservative) system. Attractors (stable cycles, tori and singular attractors) of extended dissipative system one can search by numerical methods with use the results of universal FSM (Feigenbaum-Sharkovskii-Magnitskii) theory, developed initially for nonlinear dissipative systems of ordinary differential equations and considered in detail in the previous Section of the chapter. It becomes clear what chaos is in Hamiltonian and simply conservative systems. And this chaos is not a result of destruction of some tori of nonperturbed system as it is considered to be in the modern literature, but, on the contrary, it is a result of bifurcation cascades of a birth of regular (cycles and tori) and singular attractors in extended dissipative system in accordance with the universal FSM theory when dissipation parameter tends to

**3.1. Bifurcation approach to analysis of Hamiltonian and conservative systems** 

The fact that the dynamics of any conservative system is a limit case of a dynamics of an extended dissipative system with weak dissipation as the dissipation parameter tends to zero was proved by author in (Magnitskii, 2008b; Magnitskii, 2011) and illustrated by numerous examples of Hamiltonian systems with one and a half, two and three degrees of

of more complex singular attractor (e) in the Mackey-Glass equation (20).

**3. Chaos in Hamiltonian and conservative systems** 

Let's consider generally nonlinear conservative system of ordinary differential equations with a smooth right part

$$
\dot{\mathbf{x}} = f(\mathbf{x}), \mathbf{x} \in \mathbb{R}^n, \ d\upsilon \, f(\mathbf{x}) = 0 \tag{21}
$$

which variables are connected by some equation

$$H(\mathbf{x}\_1, \dots, \mathbf{x}\_n) = \mathbf{z}.\tag{22}$$

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

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

$$
\dot{\mathbf{x}} = g(\mathbf{x}, \boldsymbol{\varepsilon}, \boldsymbol{\mu}), \ \mathbf{x} \in \boldsymbol{\mathcal{R}}^n,\tag{23}
$$

possesses following properties: 1) the only solutions of system of Eqs. (21)-(22) are solutions of system of Eqs. (23) with initial conditions 10 0 ( ,..., ) *Hx xn* at 0 ; 2) at all 0 the 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 small 0 are as much as exact approximations of solutions of conservative system of Eqs. (21)- (22) (see proof in (Magnitskii, 2008; Magnitskii, 2011)).

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 properties 1) and 2). Then for everyone 0 one should to find numerically all stable solutions and their cascades of bifurcations according to the FSM scenario in extended dissipative system of Eqs. (23) when tends to zero, starting from the various initial conditions, satisfying the equality (22). Areas of stability of the found simple regular solutions (simple cycles) will generate at 0 regular solutions (tori) of original conservative system of 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 area of parameters 0, 0 one can construct bifurcation diagrams of all bifurcations existing in two-parametrical extended dissipative system of Eqs. (23) and smoothly passing to bifurcations in conservative system of Eqs. (21)-(22) on the boundary 0 .

## *3.1.2. Subharmonic cascade of bifurcations in Hamiltonian and conservative systems*

From theoretical positions of bifurcation approach to the analysis of Hamiltonian and any conservative systems it follows, that at enough great values of parameter 0 transition to chaotic dynamics in system of Eqs. (21)-(22) occurs according to universal FSM scenario and that bifurcation diagram of this scenario can be received by limiting transition at 0 from similar bifurcation diagram of two-parametrical extended dissipative system of Eqs. (23). Let's illustrate this position by the example of classical conservative Croquette equation

$$
\ddot{\mathbf{x}} + \alpha \sin \mathbf{x} + \beta \sin \mathbf{(x} - \alpha \mathbf{t}) = \mathbf{0},
\tag{24}
$$

Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations 151

. Around of a picture presented in Fig. 16a there is

(a); development and complication of heteroclinic

in system of Eqs. (26) the subharmonic cascade of bifurcations is

the accordion of heteroclinic separatrix zigzag starts to

. Development and complication of

. At reduction of values

of solutions of conservative

0.1415 .

Cascade of saddle-node bifurcations in extended dissipative system, consisting in a simultaneous birth of stable and saddle cycles, leads to formation in conservative (Hamiltonian) system of family of complex multiturnaround tori around of elliptic cycles and heteroclinic separatrix manifold which is tense on complex multiturnaround hyperbolic cycles of the system. In Poincare section it looks like a family of the hyperbolic singular points connected by separatrix contours. This picture at any shift in initial conditions passes into a family of so-called islands (points in Poincare section forming closed curves around of points of elliptic cycle).

At the same time, as follows from the theory, at enough great values of perturbation parameter

 0 in extended dissipative system there are cascades of bifurcations in accordance with scenario FSM. These cascades of bifurcations generate considered in the previous Section of the chapter infinitely folded heteroclinic separatrix manifolds having in Poincare section a kind of heteroclinic separatrix zigzag. These manifolds are tense on unstable singular cycles of FSM-cascade of dissipative system and they pass at zero dissipation in even more complex separatrix manifolds of conservative (Hamiltonian) system, movement of trajectories on which looks like as chaotic dynamics. Thus there is a stretching of an accordion of infinitely folded heteroclinic separatrix zigzag on some area of phase space of the conservative system. In the remained part of phase space elliptic cycles from the right part of subharmonic and homoclinic

In Fig. 16a islands of solutions of conservative system of the Croquette Eqs. (25) are presented

not represented in figure an area of chaotic movement around of original separatrix contour

heteroclinic separatrix zigzag in the extended dissipative Croquette system of Eqs. (26) close

cover all phase space of the system merging with heteroclinic separatrix manifold which is

observed. It generates the heteroclinic separatrix zigzag represented in Fig. 16b at

*3.1.3. Heteroclinic separatrix manifolds* 

cascades can simultaneously coexist with tori around of them.

of nonperturbed system connecting the points ( ,0)

to conservative system Eqs. (25) is presented in Fig. 16b,c for 0.55

tense on hyperbolic cycles from the cascade a saddle-node bifurcations.

**Figure 16.** Projections on the plane (,) *x y* of Poincare section ( 0, ) *z r*

in Poincare section ( 0, ) *z r*

at 0.2 

of dissipation parameter

At values of parameter 0.138

Croquette system of the Eqs. (25) for 0.2

modeling a magnet rotary fluctuations in an external magnetic field in absence of friction. It is easy to see, that the equation (24) is equivalent to two-dimensional conservative system with periodic coefficients (Hamiltonian system with one and a half degrees of freedom) and also to four-dimensional conservative (not Hamiltonian) autonomous system of the equations

$$\dot{\mathbf{x}} = y, \quad \dot{y} = -(\alpha + r)\sin\mathbf{x} + z\cos\mathbf{x}, \quad \dot{z} = \alpha r, \ \dot{r} = -\alpha z \tag{25}$$

with a condition 22 2 *Hz r* , 0 *z z* (0) 0. Extended dissipative system for the system of Eqs. (25) will be

$$
\dot{\mathbf{x}} = y, \quad \dot{y} = -\mu \, y - (\alpha + r) \sin \mathbf{x} + z \cos \mathbf{x}, \quad \dot{z} = \alpha r, \quad \dot{r} = -\alpha z. \tag{26}
$$

It is easy to check up numerically, that the two-parametrical system of Eqs. (26) with initial conditions 0 *z z* (0) 0, <sup>0</sup> *r r* (0) has the subharmonic cascade of bifurcations at each value of parameter and at reduction of values of parameter . For each cycle of the cascade in a plane of parameters (, ) it is possible to construct monotonously increasing bifurcation curve ( ) of births of the given cycle. Boundary values of such curves at 0 are bifurcation values of the subharmonic cascade of bifurcations in conservative Croquette system of the Eqs. (25) for parameter 0 (see Fig. 15).

**Figure 15.** Projections on the plane (,) *x y* of the cycle (a) for 0.45 , period two cycle (b) for 0.48 and period four cycle (c) for 0.497 in conservative Croquette system of the Eqs. (25) for 1 .
