Two-network Kuramoto-Sakaguchi model under tempered stable Lévy noise.

We examine a model of two interacting populations of phase oscillators labeled "blue" and "red." To this we apply tempered stable Lévy noise, a generalization of Gaussian noise where the heaviness of the tails parametrized by a power law exponent α can be controlled by a tempering parameter λ. This system models competitive dynamics, where each population seeks both internal phase synchronization and a phase advantage with respect to the other population, subject to exogenous stochastic shocks. We study the system from an analytic and numerical point of view to understand how the phase lag values and the shape of the noise distribution can lead to steady or noisy behavior. Comparing the analytic and numerical studies shows that the bulk behavior of the system can be effectively described by dynamics in the presence of tilted ratchet potentials. Generally, changes in α away from the Gaussian noise limit 1<α<2 disrupt the locking between blue and red, while increasing λ acts to restore it. However, we observe that with further decreases of α to small values α≪1, with λ≠0, locking between blue and red may be restored. This is seen analytically in a restoration of metastability through the ratchet mechanism, and numerically in transitions between periodic and noisy regions in a fitness landscape using a measure of noise. This nonmonotonic transition back to an ordered regime is surprising for a linear variation of a parameter such as the power law exponent and provides a mechanism for guiding the collective behavior of such a complex competitive dynamical system.

Fisher information and criticality in the Kuramoto model of nonidentical oscillators.

We use the Fisher information to provide a lens on the transition to synchronization of the Kuramoto model of nonidentical frequencies on a variety of undirected graphs. We numerically solve the equations of motion for a N=400 complete graph and N=1000 small-world, scale-free, uniform random, and random regular graphs. For large but finite graphs of small average diameter the Fisher information F as a function of coupling shows a peak closely coinciding with the critical point as determined by Kuramoto's order parameter or synchronization measure r. However, for graphs of larger average diameter the position of the peak in F differs from the critical point determined by estimates of r. On the one hand, this is a finite-size effect even at N=1000; however, we show across a range of topologies that the Fisher information peak points to a transition for smaller graphs that indicates structural changes in the numbers of locally phase-synchronized clusters, often directly from metastable to stable frequency synchronization. Solving explicitly for a two-cluster ansatz subject to Gaussian noise shows that the Fisher infomation peaks at such a transition. We discuss the implications for Fisher information as an indicator for edge-of-chaos phenomena in finite-coupled oscillator systems.

Noise-driven current reversal and stabilization in the tilted ratchet potential subject to tempered stable Lévy noise.

We consider motion of a particle in a one-dimensional tilted ratchet potential subject to two-sided tempered stable Lévy noise characterized by strength Ω, fractional index α, skew θ, and tempering λ. We derive analytic solutions to the corresponding Fokker-Planck Lévy equations for the probability density. Due to the periodicity of the potential, we carry out reduction to a compact domain and solve for the analog of steady-state solutions which we represent as wrapped probability density functions. By solving for the expected value of the current associated with the particle motion, we are able to determine thresholds for metastability of the system, namely when the particle stabilizes in a well of the potential and when the particle is in motion, for example as a consequence of the tilt of the potential. Because the noise may be asymmetric, we examine the relationship between skew of the noise and the tilt of the potential. With tempering, we find two remarkable regimes where the current may be reversed in a direction opposite to the tilt or where the particle may be stabilized in a well in circumstances where deterministically it should flow with the tilt.