Critical phenomena and fractality of attraction basin
boundaries in unidirectionally coupled
period doubling systems
B.P. Bezruchko 
, V.I. Ponomarenko 
, Ye.P. Seleznev
Department of Physics, Saratov State University, Saratov, Russia;
Institute of RadioEngineering and Electronics
of Russian Academy of Sciences,
Saratov Branch, Saratov, Russia
Two unidirectionally coupled LR-diode circuits driven in-phase by an external sinusoidal source are experimentally investigated. The subsystems are coupled by the amplifier in a manner that the first circuit acts upon the second one, but the second circuit does not act upon the first. For the nonlinearity parameter values, when the dynamics of each resonator corresponds to that of Feigenbaum [1], we use a two-dimensional map as a numerical model describing behavior of the experimental system:
where 
are the nonlinearity parameters of
the two subsystems, Š is the coupling coefficient, n = 1,2,3,...
We investigated the attraction basins of various multistable states experimentally and numerically and demonstrated fractalization of the attraction basin boundaries in accordance to the known scenario [2]. We demonstrated experimentally for the real system that there exist critical behavior kinds differing from the Feigenbaum types of critical phenomena, and namely bicritical and tricritical [3,4]. These types of behavior are observed in the regions of evolution of some dynamical states of the system. Bicritical and tricritical lines or a bi-tricritical point corresponds to them in the phase space of the system.
These results expand our knowledge about unidirectionally coupled nonlinear period doubling systems. This type of coupling is widely presented for systems of different nature and often used for modeling open flow systems [5,6].
This work was supported by the Russian Foundation of Fundamental Research, grant No.96-02-16753.