Bifurcation mechanisms, properties and structure
of the choatic attractors in the laser model with saturable absorber
S.A. Tatarkova, V.V. Tuchin
Saratov State University,
Saratov, Russia
At the last time many papers were dedicated by the special role of homoclinic structures in the dissipative system dynamics. So, being the certain limit solution of the nonlinear equations, the homoclinic trajectories influence essentially on the properties and behavior of the dynamical system. Investigations have shown that the some universality of behavior exist for many dynamical systems having homoclinic trajectories.
In [1] it has been demonstrated that the specific spiral-like
movement of the phase trajectory near the unstable nonzero
steady-state appears if we have the additional complex conjugated
eigenvalues with negative real part ( 
).
Moreover, it will that only if the real part of the complex pairs
satisfies by the relation 
. Indeed it is the condition of
the Shil'nikov theorem estimating the presence of the specific
dynamics if the repulsion from the steady-state is stronger then
the attraction.
In the generalized multistability region near Andronov-Hopf
bifurcation point the relation 
is satisfied. When in
corresponding of the Shil'nikov theorem, the continues subset of
the periodic trajectories and the period increase until infinity
when the parameter approaches to the homoclinic bifurcation of
the saddle-focus steady-state and submit by the relation:
, where 
- limit cycle period with number of the
peaks equal n, 
- imaginary part of the complex-conjugated pair
with negative real part and correspond to the parameters when
homoclinic bifurcation happens. At this case the chaotic
attractor mapping has branch-like structure.