Complete phase locking in modulated relaxation oscillators described by a nonsmooth circle map: Positive fractal dimension of the complementary set of phase-locked regions

As a model for modulated relaxation oscillators, an integrate-and-fire model in which the sawtooth motion of a state variable is modulated by another sawtooth oscillation is investigated. The dynamics of the system is described by a mapping function that maps successive firing times. The map is a piecewise-linear circle map having two continuous nondifferentiable points or one discontinuous point, which is equivalent to the Poincaré map investigated by Christiansen, Alstro/m, and Levinsen [Phys. Rev. A 42, 1891 (1990)]. It is proved analytically that a different type of dynamics appears in a nonchaotic region of parameter space in the present system, that is, complete phase locking (CPL) with positive fractal dimension of quasiperiodic set occurs in an entire region where the mapping function describing the system dynamics is monotonic and continuous. It is also shown that the probability of occurrence of periodic orbits with period longer than N is evaluated by a power of N, that is, by N^2(1-1/d), where d is the dimension of quasiperiodic set that is positive and less than 1. If the modulation is weak, the dimension d takes a value near 1 and the orbits with very long period appear frequently. When the modulation is enforced, a discontinuity appears in the mapping function. It has been known that a monotonic and discontinuous piecewise linear map results in CPL with zero dimension of quasiperiodicity [B. Chritiansen, P. Alstro/m, and M. T. Levinsen, Phys. Rev. A 42, 1891 (1990)]. It is identified in the present paper that the transition induced by the occurrence of discontinuity is the one within CPL such that the dimension of quasiperiodicity changes abruptly from a positive number to zero. This is the transition in which the periodic orbits with long period disappear. © 1996 The American Physical Society.