Nonlocal techniques for chaotic behavior and synchronization of a
dynamical system
Vladimir N. Belykh, Igor V. Belykh
University of Nizhny Novgorod, Nizhny Novgorod, Russia
Many modern problems arising from the real system especially the systems from radiophysics modeled by multy-dimensional, strongly nonlinear, nonintegrable differential or difference equations. The study of such systems has spawn numerous new concepts and new results; e.g. Smale horseshoe, hetero-homoclinic orbits, hyberbolic sets, quasiattractors, strange attractors, etc. In this talk we will present a development of nonlocal 2-d comparison systems method exploited as a main tool of the theoretical and the computational analysis. This approach is related to the coupled system
where 
and h(t) are scalar functions,
are nonnegative parameters, L is a differential operator
where the coefficients 
may be dependent on 
.
We present three following assertions concerning system (
).
1. In the case of 
with zero-flux boundary conditions at
system () is globally synchronized (solution 
is globally stable) for large 
and the system
being stable.
2. In the case of 
the stationary system is
hyperbolic for nonmonotonous coninuous function and some K;SPMgt;1
such that 
.
3. In the case of some nonmonotonous functions there exists a
bifurcation set corresponding to the hetero-homoclinic orbits of the system
().
We will present also some geometric models of ODE having strange and quasistrange attractors in order to explain the complex structures of limiting sets for "simple" dynamical systems.
Finally, the computational complexity in the case of infinite set of
bifurcations will be discussed.
This work was supported by grant INTAS-93-0570.
