Oscillation types of dissipatively coupled
period-doubling systems at large coupling
M.D. Prokhorov
Institute of RadioEngineering and Electronics
of Russian Academy of Sciences,
Saratov Branch, Saratov, Russia
A system of two symmetrically coupled identical objects demonstrating a transition to chaos via a sequence of period-doubling bifurcations is one of the basic objects in nonlinear dynamics. This system was investigated for various types of coupling between its elements. The type of coupling, named as a dissipative [1], was studied theoretically and experimentally in [2, 3].
In general, the system of two dissipatively coupled subsystems is described by the following equations:
where x, y - are the dynamical variables, 
and 
- are the functions describing the behavior of the isolated system,
k - is the coupling coefficient. Such systems can demonstrate a
variety of periodic, quasiperiodic and chaotic oscillations. However,
the study of the dynamics of the model and experimental systems was
conducted for the small coupling between subsystems (k;SPMlt;0.5). In the
present work the behavior of the system of two symmetrically coupled
logistic maps ( 
, 
)
was considered for the case of large coupling ( 
).
On the plane 
the regions of existence of the system
possible oscillation states were constructed. It was shown, that
in-phase regimes are invariant to the change of k to 1-k. Period-2
out-of-phase regimes existing at small k values are absent at large
coupling. Instead of them in the region, symmetrical to the region of
period-2 out-of-phase oscillations about k=0.5, exist period-1
out-of-phase regimes (it is a case when both subsystems possess
period-1 oscillations, but the amplitude of these oscillations is
different in the subsystems). Out-of-phase regimes with higher-order
period exist at large k values, but their phase portraits differ
qualitatively from those at small k.
The obtained results can be extended to the systems of different
nature, which can be described by the system (1). Besides, the
obtained results are also true for any form of the identical functions
and demonstrating period-doubling route to chaos.
This work was supported by the Russian Foundation of Fundamental Research, Grant No 96-02-16753 and by a fellowship of INTAS Grant 93-2492 and was carried out within the research program of International Center for Fundamental Physics in Moscow.