4 research outputs found
Arnold maps with noise: Differentiability and non-monotonicity of the rotation number
Arnold's standard circle maps are widely used to study the quasi-periodic
route to chaos and other phenomena associated with nonlinear dynamics in the
presence of two rationally unrelated periodicities. In particular, the El
Nino-Southern Oscillation (ENSO) phenomenon is a crucial component of climate
variability on interannual time scales and it is dominated by the seasonal
cycle, on the one hand, and an intrinsic oscillatory instability with a period
of a few years, on the other. The role of meteorological phenomena on much
shorter time scales, such as westerly wind bursts, has also been recognized and
modeled as additive noise. We consider herein Arnold maps with additive,
uniformly distributed noise. When the map's nonlinear term, scaled by the
parameter , is sufficiently small, i.e. , the map is
known to be a diffeomorphism and the rotation number is a
differentiable function of the driving frequency . We concentrate on
the rotation number's behavior as the nonlinearity becomes large, and show
rigorously that is a differentiable function of ,
even for , at every point at which the noise-perturbed map is
mixing. We also provide a formula for the derivative of the rotation number.
The reasoning relies on linear-response theory and a computer-aided proof. In
the diffeomorphism case of , the rotation number
behaves monotonically with respect to . We show, using again a
computer-aided proof, that this is not the case when and the
map is not a diffeomorphism.Comment: Electronic copy of final peer-reviewed manuscript accepted for
publication in the Journal of Statistical Physic
Shared inputs, entrainment, and desynchrony in elliptic bursters: from slow passage to discontinuous circle maps
What input signals will lead to synchrony vs. desynchrony in a group of
biological oscillators? This question connects with both classical dynamical
systems analyses of entrainment and phase locking and with emerging studies of
stimulation patterns for controlling neural network activity. Here, we focus on
the response of a population of uncoupled, elliptically bursting neurons to a
common pulsatile input. We extend a phase reduction from the literature to
capture inputs of varied strength, leading to a circle map with discontinuities
of various orders. In a combined analytical and numerical approach, we apply
our results to both a normal form model for elliptic bursting and to a
biophysically-based neuron model from the basal ganglia. We find that,
depending on the period and amplitude of inputs, the response can either appear
chaotic (with provably positive Lyaponov exponent for the associated circle
maps), or periodic with a broad range of phase-locked periods. Throughout, we
discuss the critical underlying mechanisms, including slow-passage effects
through Hopf bifurcation, the role and origin of discontinuities, and the
impact of noiseComment: 17 figures, 40 page
Parallel Computation of Invariant Measures
Let S:[0,1]→[0,1] be a nonsingular transformation and let P:L1(0,1)→L1(0,1) be the corresponding Frobenius–Perron operator. In this paper we propose a parallel algorithm for computing a fixed density of P, using Ulam\u27s method and a modified Monte Carlo approach. Numerical results are also presented
The Study of Quasi Monte Carlo in the Parallel Computation of Invariant Measures
Abstract- For a non-singular multi-dimensional mapping S: X � X, the corresponding Frobenius-Perron operator is P. In the paper, the schemes for generating the quasi-random numbers are studied for the parallel computation for the fixed density of P. The numerical results for these schemes are presented