Avalanches in a short-memory excitable network

We study propagation of avalanches in a certain excitable network. The model is a particular case of the one introduced by Larremore et al. (Phys. Rev. E, 2012) and is mathematically equivalent to an endemic variation of the Reed–Frost epidemic model introduced by Longini (Math. Biosci., 1980). Two types of heuristic approximation are frequently used for models of this type in applications: a branching process for avalanches of a small size at the beginning of the process and a deterministic dynamical system once the avalanche spreads to a significant fraction of a large network. In this paper we prove several results concerning the exact relation between the avalanche model and these limits, including rates of convergence and rigorous bounds for common characteristics of the model.

Non-autonomous invariant sets and attractors: Random dynamical system

In this paper the concepts of pullback attractor ,pullback absorbing family in (deterministic) dynamical system are defined in (random) dynamical systems. Also some main result such as (existence) of pullback attractors ,upper semi-continuous of pullback attractors and uniform and global attractors are proved in random dynamical system .

Strange Attractors Generated by Multiple-Valued Static Memory Cell with Polynomial Approximation of Resonant Tunneling Diodes

This paper brings analysis of the multiple-valued memory system (MVMS) composed by a pair of the resonant tunneling diodes (RTD). Ampere-voltage characteristic (AVC) of both diodes is approximated in operational voltage range as common in practice: by polynomial scalar function. Mathematical model of MVMS represents autonomous deterministic dynamical system with three degrees of freedom and smooth vector field. Based on the very recent results achieved for piecewise-linear MVMS numerical values of the parameters are calculated such that funnel and double spiral chaotic attractor is observed. Existence of such types of strange attractors is proved both numerically by using concept of the largest Lyapunov exponents (LLE) and experimentally by computer-aided simulation of designed lumped circuit using only commercially available active elements.

Singular perturbations in noisy dynamical systems

Consider a deterministic dynamical system in a domain containing a stable equilibrium, e.g., a particle in a potential well. The particle, independent of initial conditions, eventually reaches the bottom of the well. If however, the particle is subjected to white noise, due, e.g., to collisions with a population of smaller, lighter particles comprising the medium through which the particle travels, a dramatic difference in the behaviour of the Brownian particle occurs. The particle will exit the well. The natural questions then are how long will it take for it to exit and from where on the boundary of the domain of attraction of the deterministic equilibrium (the rim of the well) will it exit. We compute the mean first passage time to the boundary and the mean probabilities of the exit positions. When the noise is small each quantity satisfies a singularly perturbed deterministic boundary value problem. We treat the problem by the method of matched asymptotic expansions (MAE) and generalizations thereof. MAE has been used successfully to solve problems in many applications. However, there exist problems for which MAE does not suffice. Among these are problems exhibiting boundary layer resonance, i.e., the problem of ‘spurious solutions’, which led some to conclude that this was ‘the failure of MAE’. We present a physical argument and four mathematical arguments to modify or augment MAE to make it successful. Finally, we discuss applications of the theory.

Predictability of Extreme Waves in the Lorenz-96 Model Near Intermittency and Quasi-Periodicity

We introduce a method for quantifying the predictability of the event that the evolution of a deterministic dynamical system enters a specific subset of state space at a given lead time. The main idea is to study the distribution of finite-time growth rates of errors in initial conditions along the attractor of the system. The predictability of an event is measured by comparing error growth rates for initial conditions leading to that event with all possible growth rates. We illustrate the method by studying the predictability of extreme amplitudes of traveling waves in the Lorenz-96 model. Our numerical experiments show that the predictability of extremes is affected by several routes to chaos in a different way. In a scenario involving intermittency due to a periodic attractor disappearing through a saddle-node bifurcation we find that extremes become better predictable as the intensity of the event increases. However, in a similar intermittency scenario involving the disappearance of a 2-torus attractor we find that extremes are just as predictable as nonextremes. Finally, we study a scenario which involves a 3-torus attractor in which case the predictability of extremes depends nonmonotonically on the prediction lead time.

White noise perturbation of locally stable dynamical systems

The influence of small random perturbations on a deterministic dynamical system with a locally stable equilibrium is considered. The perturbed system is described by the Itô stochastic differential equation. It is assumed that the noise does not vanish at the equilibrium. In this case the trajectories of the stochastic system may leave any bounded domain with probability one. We analyze the stochastic stability of the equilibrium under a persistent perturbation by white noise on an asymptotically long time interval [Formula: see text], where [Formula: see text] is a perturbation parameter.

Metastability in a class of hyperbolic dynamical systems perturbed by heavy-tailed Lévy type noise

We consider a finite dimensional deterministic dynamical system with finitely many local attractors Kι, each of which supports a unique ergodic probability measure Pι, perturbed by a multiplicative non-Gaussian heavy-tailed Lévy noise of small intensity ε > 0. We show that the random system exhibits a metastable behavior: there exists a unique ε-dependent time scale on which the system reminds of a continuous time Markov chain on the set of the invariant measures Pι. In particular our approach covers the case of dynamical systems of Morse–Smale type, whose attractors consist of points and limit cycles, perturbed by multiplicative α-stable Lévy noise in the Itô, Stratonovich and Marcus sense. As examples we consider α-stable Lévy perturbations of the Duffing equation and Pareto perturbations of a biochemical birhythmic system with two nested limit cycles.

Stability of Multi-Dimensional Birth-and-Death Processes with State-Dependent 0-Homogeneous Jumps

We study the conditions for positive recurrence and transience of multi-dimensional birth-and-death processes describing the evolution of a large class of stochastic systems, a typical example being the randomly varying number of flow-level transfers in a telecommunication wire-line or wireless network. First, using an associated deterministic dynamical system, we provide a generic method to construct a Lyapunov function when the drift is a smooth function on ℝN. This approach gives an elementary and direct proof of ergodicity. We also provide instability conditions. Our main contribution consists of showing how discontinuous drifts change the nature of the stability conditions and of providing generic sufficient stability conditions having a simple geometric interpretation. These conditions turn out to be necessary (outside a negligible set of the parameter space) for piecewise constant drifts in dimension two.

Stability of Multi-Dimensional Birth-and-Death Processes with State-Dependent 0-Homogeneous Jumps

We study the conditions for positive recurrence and transience of multi-dimensional birth-and-death processes describing the evolution of a large class of stochastic systems, a typical example being the randomly varying number of flow-level transfers in a telecommunication wire-line or wireless network. First, using an associated deterministic dynamical system, we provide a generic method to construct a Lyapunov function when the drift is a smooth function on ℝN. This approach gives an elementary and direct proof of ergodicity. We also provide instability conditions. Our main contribution consists of showing how discontinuous drifts change the nature of the stability conditions and of providing generic sufficient stability conditions having a simple geometric interpretation. These conditions turn out to be necessary (outside a negligible set of the parameter space) for piecewise constant drifts in dimension two.