CHAOS IN THE MODEL OF REPETITIVE PHASE-TRANSITIONS WITH HYSTERESIS - APPLICATION TO THE SELF-SUSTAINED POTENTIAL OSCILLATIONS OF LIPID-BILAYER MEMBRANES INDUCED BY GEL LIQUID-CRYSTAL PHASE-TRANSITIONS

To clarify the mechanism of chaos generation and the routes to chaos i n the self-sustained oscillation of the electric potential difference between two solutions divided by a lipid-bilayer membrane, a simple mo del of the system, the model of repetitive phase transitions with hyst eresis, is presented in which oscillation is driven by repetitive gel- liquid-crystal phase transitions with hysteresis occurring in the lipi d membrane and at the same time by a periodic external current. The dy namical property of the system is completely described by the nature o f the function mapping the times at which the phase transition occurs successively. There exist various kinds of routes to chaos in the mode l of repetitive phase transitions with hysteresis (RPTH model) such as the period-doubling cascades, the intermittency, the quasiperiodic-ch aotic transition, and the transition to chaos from complete phase lock ing. When the values of the system parameters satisfy certain conditio ns, the RPTH model becomes equivalent to the integrate-and-fire model and similar to the driven-relaxation-oscillator model. The model also generates structurally stable chaotic attractors which are never destr oyed by a slight change in the values of system parameters. The attrac tors appear only for the regions of parameter values where the mapping function has at least one discontinuous point. This model contains th e essential features of evolution behavior in various kinds of systems which generate iterative phase transitions with hysteresis.