Multistability and Transient Processes in Coupled Period Doubling Systems

Basins of attraction and transient processes are experimentally and numerically investigated for the coupled period doubling systems. The influence of bifurcattion of the unstable point on the transient phenomenas is shown. The results of experimental and numerical investigations quolitatively coincide. 1.Introduction. Coexistence of two or more attractors with their basins of attraction in phase space (multistability) is typical for nonlinear dynamical systems. When control parameters are varied, the attractors and unstable points evolve with undergoing different bifurcation. The bifurcations of attractors lead to transformations of their basins of attraction, so that their structure may be very complicated or even fractal [1]. The bifurcations of unstable points do not influence on the basins of attraction, but modify the transient processes. In this paper the evolution of basins of attraction and transient processes in a system of two dissipatively coupled elements, each demonstrating period doubling rout to chaos [2], are investigated both experimentally and numerically. 2.Investigated systems. The experimental system (Fig.1) consists of two resistively coupled RL-diode circuits driven in-phase by external harmonic force. Elecrtonic keys K1 and K2 connect the diodes D1 and D2 to direct current sources X0 and Y0, which are used as sources of initial conditions. Then they are inphase connected to inductors L1 and L2 , respectively. Control parameters are the amplitude of external force V and the conductivity of the coupling resistor 1/R. The external frequency f and dissipation were constant. Experimental system makes it possible to observe the time serieses by the duration from 2 to 2 periods (1/f) of external force. The dynamics of each circuit in the limited region of its parameters is qualitatively modeled by the quadratic map. Two dissipatively [3,4] coupled quadratic maps can be used as an appropriate model for the above experimental system: xn+1=λ-xn+k(xn-yn) yn+1=λ-yn+k(yn-xn) where xn and yn are the dynamical variables, n denoted the discret time, λ and k are the parameters of nonlinearity and coupling , respectively. In the experimental system , parameter λ corresponds to the amplitude of external force, and k corresponds to the conductivity 1/R of the coupling resistor. Fig.1. The scheme of the experimental system. 3. Basins and transient processes. As parameter λ increases, each subsystem demonstrates transition to chaos via universal cascade of period doubling befurcations. In the case of coupled system it is possible to classify the dynamical states as following: in the limit of vanishingly small coupling k→0, any regime of period N=1,2,3,... (in term of discret time) can be realized by N possible ways [4]. They differ from each other by the value of phase shift m=0,1,2,3,... . Further the possible variations of dynamical period N regime of will be called the oscillation types and denoted by indices Nm. At λ=0.8 in each subsystem there are cycles of period 2. At k =0 in coupled system there are two cycles: in-phase 20 and out-of-phase 21. Fig.2 illustrates basins of attraction (on the right) and dependence of transient process time on initial conditions (on the left) for period 2 cycles when λ=0.8 and k decrease from 0.5 to 0. At strong coupling (k=0.5) the stable cycle 20 and the unstable cycle 1 exist only. On the plane of initial conditions (X0 ,Y0) there is single basin of round shape (Fig.2b). The dependence of transient time from initial conditions has the shape of concentric waves, which wavelength decreases near the border of basin (Fig.2a). As coupling decreases the basin is out of round, and the ′′ waves′′ are not concentric (Fig.2cd). The points on the top of the ′′ waves′′ correspond to transient process after which the system arrives into unstable period 1 cycle. As k decreases this unstable cycle undergo the period doubling bifurcation, and the unstable out-of-phase cycle 21 is occured in its neighbourhood. This bifurcation does not effect on the basin, but on transient time dependence the platos are formed (Fig.2ef). The initial points on these platos iterate to the unstable cycle 21 and after that system arrives to the stable cycle 20. Fig3. illustrates first two hundred iterations, unstable and stable cycles. As k decreases the unstable cycle undergoes a next bifurcation and it becomes stable. Fig.2ef and Fig.2gh illustrate basins and transient time dependences before and after bifurcation, respectively. Near the bifurcation the platos occur, because the system is on the unstable cycle for a long time.After bifurcation the cycle 21 becomes stable, and the single basin