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 b y another sawtooth oscillation is investigated. The dynamics of the sy stem is described by a mapping function that maps successive firing ti mes. The map is a piecewise-linear circle map having two continuous no ndifferentiable points or one discontinuous point, which is equivalent to the Poincare map investigated by Christiansen, Alstrom, and Levins en [Phys. Rev. A 42, 1891 (1990)]. It is proved analytically that a di fferent type of dynamics appears in a nonchaotic region of parameter s pace in the present system, that is, complete phase locking (CPL) with positive fractal dimension of quasiperiodic set occurs in an entire r egion when the mapping function describing the system dynamics is mono tonic and continuous. It is also shown that the probability of occurre nce of periodic orbits with period longer than N is evaluated by a pow er of N, that is, by N-2(1-1/d), where d is the dimension of quasiperi odic 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 per iod appear frequently, When the modulation is enforced, a discontinuit y appears in the mapping function. It has been known that a monotonic and discontinuous piecewise Linear map results in CPL with zero dimens ion of quasiperiodicity [B. Chritiansen, P. Alstrom, and M. T. Levinse n, Phys. Rev. A 42, 1891 (1990)]. It is identified in the present pape r that the transition induced by the occurrence of discontinuity is th e one within CPL such that the dimension of quasiperiodicity changes a bruptly from a positive number to zero. This is the transition in whic h the periodic orbits with long period disappear.