COMPLETE PHASE-LOCKING IN MODULATED RELAXATION-OSCILLATORS DESCRIBED BY A NONSMOOTH CIRCLE MAP - POSITIVE FRACTAL DIMENSION OF THE COMPLEMENTARY SET OF PHASE-LOCKED REGIONS
K. Yagisawa et al., COMPLETE PHASE-LOCKING IN MODULATED RELAXATION-OSCILLATORS DESCRIBED BY A NONSMOOTH CIRCLE MAP - POSITIVE FRACTAL DIMENSION OF THE COMPLEMENTARY SET OF PHASE-LOCKED REGIONS, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 54(3), 1996, pp. 2392-2403
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.