Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. III. Natural hysteresis memory effects

The nonlinear reflection curve due to a van der Pol type boundary condition at the right end becomes a multivalued relation when one of the parameters (alpha) exceeds the characteristic impedance value (alpha = 1). From stability and continuity considerations, we prescribe kinematic admissibility and define hysteresis iterations with memory effects, whose dynamical behavior is herein investigated. Assume first that the left end boundary condition is fixed. We show that asymptotically there are two types of stable periodic solutions: (i) a single period-2k orbit, or (ii) coexistence of a period-2k and a period-2(k + 1) orbits, where as the parameter alpha increases, k will also increase and assume all positive integral values. Even though unstable periodic solutions do appear, there is obviously no chaos. When the left end boundary condition is energy-injecting, however, we show that for a certain parameter range a shift sequence of subintervals of an invariant interval can be constructed and, therefore, chaos appears. Numerical simulations of chaotic and nonchaotic phenomena are also illustrated.