*3.1.3. Heteroclinic separatrix manifolds*

150 Nonlinearity, Bifurcation and Chaos – Theory and Applications

 

bifurcations in conservative system of Eqs. (21)-(22) on the boundary 0

0, 0 one can construct bifurcation diagrams of all bifurcations

.

(24)

. For each cycle of the

transition to

(25)

(26)

0.48

 1 .

existing in two-parametrical extended dissipative system of Eqs. (23) and smoothly passing to

From theoretical positions of bifurcation approach to the analysis of Hamiltonian and any

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

> sin sin( ) 0,

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

( )sin cos , , .

It is easy to check up numerically, that the two-parametrical system of Eqs. (26) with initial

are bifurcation values of the subharmonic cascade of bifurcations in conservative Croquette

(see Fig. 15).

and at reduction of values of parameter

, 0 *z z* (0) 0. Extended dissipative system for the system

of births of the given cycle. Boundary values of such curves at 0

0.497 in conservative Croquette system of the Eqs. (25) for

has the subharmonic cascade of bifurcations at each

0.45 , period two cycle (b) for

it is possible to construct monotonously increasing

 

> 

 

*x x xt*

four-dimensional conservative (not Hamiltonian) autonomous system of the equations

 *x yy r xz xz rr z* , ( )sin cos , , 

*x yy y r xz xz rr z* ,

 

*r r* (0)

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

conservative systems it follows, that at enough great values of parameter 0

with a condition 22 2 *Hz r*

conditions 0 *z z* (0) 0, <sup>0</sup>

and period four cycle (c) for

system of the Eqs. (25) for parameter 0

**Figure 15.** Projections on the plane (,) *x y* of the cycle (a) for

cascade in a plane of parameters (, )

 ( ) 

value of parameter

bifurcation curve

of Eqs. (25) will be

area of parameters

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 cascades can simultaneously coexist with tori around of them.

In Fig. 16a islands of solutions of conservative system of the Croquette Eqs. (25) are presented at 0.2 in Poincare section ( 0, ) *z r* . Around of a picture presented in Fig. 16a there is not represented in figure an area of chaotic movement around of original separatrix contour of nonperturbed system connecting the points ( ,0) . Development and complication of heteroclinic separatrix zigzag in the extended dissipative Croquette system of Eqs. (26) close to conservative system Eqs. (25) is presented in Fig. 16b,c for 0.55 . At reduction of values of dissipation parameter in system of Eqs. (26) the subharmonic cascade of bifurcations is observed. It generates the heteroclinic separatrix zigzag represented in Fig. 16b at 0.1415 . At values of parameter 0.138 the accordion of heteroclinic separatrix zigzag starts to cover all phase space of the system merging with heteroclinic separatrix manifold which is 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* of solutions of conservative Croquette system of the Eqs. (25) for 0.2 (a); development and complication of heteroclinic

separatrix zigzag in dissipative Croquette system of Eqs. (26), for 0.55 and 0.1415 (b), 0.138 (c).
