The variation of invariant graphs in forced systems

In skew-product systems with contractive factors, all orbits asymptotically approach the graph of the so-called sync function; hence, the corresponding regularity properties primarily matter. In the literature, sync function Lipschitz continuity and differentiability have been proved to hold depending on the derivative of the base reciprocal, if not on its Lyapunov exponent. However, forcing topological features can also impact the sync function regularity. Here, we estimate the total variation of sync functions generated by one-dimensional Markov maps. A sharp condition for bounded variation is obtained depending on parameters, that involves the Markov map topological entropy. The results are illustrated with examples

The mathematics of asymptotic stability in the Kuramoto model

Now a standard in Nonlinear Sciences, the Kuramoto model is the perfect example of the transition to synchrony in heterogeneous systems of coupled oscillators. While its basic phenomenology has been sketched in early works, the corresponding rigorous validation has long remained problematic and was achieved only recently. This paper reviews the mathematical results on asymptotic stability of stationary solutions in the continuum limit of the Kuramoto model, and provides insights into the principal arguments of proofs. This review is complemented with additional original results, various examples, and possible extensions to some variations of the model in the literature.Comment: 20 page

Population Dynamics of Globally Coupled Degrade-and-Fire Oscillators

This paper reports the analysis of the dynamics of a model of pulse-coupled oscillators with global inhibitory coupling. The model is inspired by experiments on colonies of bacteria-embedded synthetic genetic circuits. The total population can be either of finite (arbitrary) size or infinite, and is represented by a one-dimensional profile. Profiles can be discontinuous, possibly with infinitely many jumps. Their time evolution is governed by a singular differential equation. We address the corresponding initial value problem and characterize the dynamics' main features. In particular, we prove that trajectory behaviors are asymptotically periodic, with period only depending on the profile (and on the model parameters). A criterion is obtained for the existence of the corresponding periodic orbits, which reveals the existence of a sharp transition as the coupling parameter is increased. The transition separates a regime where any profile can be obtained in the limit of large times, to a situation where only trajectories with sufficiently large groups of synchronized oscillators perdure

Breaking of Ergodicity in Expanding Systems of Globally Coupled Piecewise Affine Circle Maps

International audienceTo identify and to explain coupling-induced phase transitions in Coupled Map Lattices (CML) has been a lingering enigma for about two decades. In numerical simulations, this phenomenon has always been observed preceded by a lowering of the Lyapunov dimension, suggesting that the transition might require changes of linear stability. Yet, recent proofs of co-existence of several phases in specially designed models work in the expanding regime where all Lyapunov exponents remain positive. In this paper, we consider a family of CML composed by piecewise expanding individual map, global interaction and finite number N of sites, in the weak coupling regime where the CML is uniformly expanding. We show, mathematically for N=3 and numerically for N> 2, that a transition in the asymptotic dynamics occurs as the coupling strength increases. The transition breaks the (Milnor) attractor into several chaotic pieces of positive Lebesgue measure, with distinct empiric averages. It goes along with various symmetry breaking, quantified by means of magnetization-type characteristics. Despite that it only addresses finite-dimensional systems, to some extend, this result reconciles the previous ones as it shows that loss of ergodicity/symmetry breaking can occur in basic CML, independently of any decay in the Lyapunov dimension