Multistable periodic and chaotic states in symmetrically
coupled period doubling systems
V.I. Ponomarenko 
, S.A. Rakitin 
, Ye.P. Seleznev
Institute of RadioEngineering and Electronics
of Russian Academy of Sciences,
Saratov Branch, Saratov, Russia;
Department of Physics, Saratov State University, Saratov, Russia
Coexistence of more than one attractors in the phase space (multistability) is typical of nonlinear dynamical systems. Attraction basins for the explored attractors can be of complicated (including fractal) structure. When control parameters are varied, the attractors evolve with undergoing different bifurcations and modifying their dimension. In this paper experimental (RL-diode circuit) and numerical (coupled quadratic maps) investigation of a system of dissipatively coupled elements that individually demonstrate transition to chaos via a period doubling cascade [1-3] are presented. The research concerns evolution of attraction basins and correlation dimension of chaotic attractors in this system.
Typical rearrangements in attraction basins have been distinguished. It has been found that when an attractor from the set becomes extinct, its attraction basin can be fractally distributed among the rest. Both experimentally and numerically it has been determined that there is a formation of a structure in the form of ``lakes'' embedded into one another [4] in attraction basins corresponding to pairs of symmetrical attractors. Variation of control parameters can lead to partition of such ``lakes''. Under critical values of parameters, degree of this partition becomes infinite (in numerical research), with fractal structure being formed. Note, despite all the transformations of attraction basins, the boundary of an individual simply connected area (both large and small) remains smooth.
This work was supported by the Russian Foundation of Fundamental Research, grant No.96-02-16753.