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.

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.

Wave Generation in Unidirectional Chains of Idealized Neural Oscillators.

We investigate the dynamics of unidirectional semi-infinite chains of type-I oscillators that are periodically forced at their root node, as an archetype of wave generation in neural networks. In previous studies, numerical simulations based on uniform forcing have revealed that trajectories approach a traveling wave in the far-downstream, large time limit. While this phenomenon seems typical, it is hardly anticipated because the system does not exhibit any of the crucial properties employed in available proofs of existence of traveling waves in lattice dynamical systems.

Typical trajectories of coupled degrade-and-fire oscillators: from dispersed populations to massive clustering.

We consider the dynamics of a piecewise affine system of degrade-and-fire oscillators with global repressive interaction, inspired by experiments on synchronization in colonies of bacteria-embedded genetic circuits. Due to global coupling, if any two oscillators happen to be in the same state at some time, they remain in sync at all subsequent times; thus clusters of synchronized oscillators cannot shrink as a result of the dynamics. Assuming that the system is initiated from random initial configurations of fully dispersed populations (no clusters), we estimate asymptotic cluster sizes as a function of the coupling strength.

Modeling temporal networks using random itineraries.

We propose a procedure to generate dynamical networks with bursty, possibly repetitive and correlated temporal behaviors. Regarding any weighted directed graph as being composed of the accumulation of paths between its nodes, our construction uses random walks of variable length to produce time-extended structures with adjustable features. The procedure is first described in a general framework.

Epigenetic aspects of lymphocyte antigen receptor gene rearrangement or 'when stochasticity completes randomness'.

To perform their specific functional role, B and T lymphocytes, cells of the adaptive immune system of jawed vertebrates, need to express one (and, preferably, only one) form of antigen receptor, i.e. the immunoglobulin or T-cell receptor (TCR), respectively.

Corepressive interaction and clustering of degrade-and-fire oscillators.

Strongly nonlinear degrade-and-fire (DF) oscillations may emerge in genetic circuits having a delayed negative feedback loop as their core element. Here we study the synchronization of DF oscillators coupled through a common repressor field. For weak coupling, initially distinct oscillators remain desynchronized.

Clustering in cell cycle dynamics with general response/signaling feedback.

Motivated by experimental and theoretical work on autonomous oscillations in yeast, we analyze ordinary differential equations models of large populations of cells with cell-cycle dependent feedback. We assume a particular type of feedback that we call responsive/signaling (RS), but do not specify a functional form of the feedback. We study the dynamics and emergent behavior of solutions, particularly temporal clustering and stability of clustered solutions.

TCR beta allelic exclusion in dynamical models of V(D)J recombination based on allele independence.

Allelic exclusion represents a major aspect of TCRbeta gene assembly by V(D)J recombination in developing T lymphocytes. Despite recent progress, its comprehension remains problematic when confronted with experimental data. Existing models fall short in terms of incorporating into a unique distribution all the cell subsets emerging from the TCRbeta assembly process.

Spatial chaos of traveling waves has a given velocity.

We study the complexity of stable waves in unidirectional bistable coupled map lattices as a test tube to spatial chaos of traveling patterns in open flows. Numerical calculations reveal that, grouping patterns into sets according to their velocity, at most one set of waves has positive topological entropy for fixed parameters. By using symbolic dynamics and shadowing, we analytically determine velocity-dependent parameter domains of existence of pattern families with positive entropy.

Athermal dynamics of strongly coupled stochastic three-state oscillators.

We study the behavior of globally coupled ensembles of cyclic stochastic three-state units with transition rates from i-1 to i proportional to the number of units in state i. Contrary to mean-field theory predictions, numerical simulations show significant stochastic oscillations for sufficiently large coupling strength. The order parameter characterizing units synchrony increases monotonically with coupling while the coherence of oscillations has a maximum at a certain coupling strength.

On the global orbits in a bistable CML.

In an infinite one-dimensional coupled map lattice (CML) for which the local map is piecewise affine and bistable, we study the global orbits using a spatiotemporal coding introduced in a previous work. The set of all the fixed points is first considered. It is shown that, under some restrictions on the parameters, the latter is a Cantor set, and we introduce an order to study the fixed points' existence.

Kink dynamics in one-dimensional coupled map lattices.

We examine the problem of the dynamics of interfaces in a one-dimensional space-time discrete dynamical system. Two different regimes are studied: the non-propagating and the propagating one. In the first case, after proving the existence of such solutions, we show how they can be described using Taylor expansions.