3,173 research outputs found
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
Synchronization of stochastic hybrid oscillators driven by a common switching environment
Many systems in biology, physics and chemistry can be modeled through
ordinary differential equations, which are piecewise smooth, but switch between
different states according to a Markov jump process. In the fast switching
limit, the dynamics converges to a deterministic ODE. In this paper we suppose
that this limit ODE supports a stable limit cycle. We demonstrate that a set of
such oscillators can synchronize when they are uncoupled, but they share the
same switching Markov jump process. The latter is taken to represent the effect
of a common randomly switching environment. We determine the leading order of
the Lyapunov coefficient governing the rate of decay of the phase difference in
the fast switching limit. The analysis bears some similarities to the classical
analysis of synchronization of stochastic oscillators subject to common white
noise. However the discrete nature of the Markov jump process raises some
difficulties: in fact we find that the Lyapunov coefficient from the
quasi-steady-state approximation differs from the Lyapunov coefficient one
obtains from a second order perturbation expansion in the waiting time between
jumps. Finally, we demonstrate synchronization numerically in the radial
isochron clock model and show that the latter Lyapinov exponent is more
accurate
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