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Rate processes are simple and analytically tractable models for many dynamical systems which switch stochastically between a discrete set of quasi stationary states but they may also approximate continuous processes by coarse grained,…
We consider synchronization of chaotic systems coupled indirectly through a common environmnet where the environment has an intrinsic dynmics of its own modulated via feedback from the systems. We find that a rich vareity of synchronization…
For non-equilibrium systems described by finite Markov processes, we consider the number of times that a system traverses a cyclic sequence of states (a cycle). The joint distribution of the number of forward and backward instances of any…
We provide complete characterizations, on Banach spaces with cotype 2, of those linear operators which happen to be weakly mixing or strongly mixing transformations with respect to some nondegenerate Gaussian measure. These…
In the context of the long-standing issue of mixing in infinite ergodic theory, we introduce the idea of mixing for observables possessing an infinite-volume average. The idea is borrowed from statistical mechanics and appears to be…
In this article we give several characterizations for various transitivity properties for linear operators. We define a general form of `Hypercyclicity Criterion' using a Furstenberg family $\mathcal{F}$ to characterize…
An operator algebra implementation of Markov chain Monte Carlo algorithms for simulating Markov random fields is proposed. It allows the dynamics of networks whose nodes have discrete state spaces to be specified by the action of an update…
The Master Stability Function is a robust and useful tool for determining the conditions of synchronization stability in a network of coupled systems. While a comprehensive classification exists in the case in which the nodes are chaotic…
We consider general Markov chains with discrete time in an arbitrary measurable (phase) space and homogeneous in time. Markov chains are defined by the classical transition function which within the framework of the operator treatment…
For an attracting periodic orbit (limit cycle) of a deterministic dynamical system, one defines the isochron for each point of the orbit as the cross-section with fixed return time under the flow. Equivalently, isochrons can be…
Despite being a foundational concept of modern systems theory, there have been few studies on observability of non-linear stochastic systems under partial observations. In this paper, we introduce a definition of observability for…
We explore the concept of metastability in random dynamical systems, focussing on connections between random Perron-Frobenius operator cocycles and escape rates of random maps, and on topological entropy of random shifts of finite type. The…
This paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability. We consider a general class of hidden Markov models in continuous time with compact signal state space, and call such a…
The convergence rate of a Markov chain to its stationary distribution is typically assessed using the concept of total variation mixing time. However, this worst-case measure often yields pessimistic estimates and is challenging to infer…
We study a class of dynamical systems generated by random substitutions, which contains both intrinsically ergodic systems and instances with several measures of maximal entropy. In this class, we show that the measures of maximal entropy…
The concept of random dynamical system is a comparatively recent development combining ideas and methods from the well developed areas of probability theory and dynamical systems. Due to our inaccurate knowledge of the particular physical…
Say that a finite group $G$ is mixable if a product of random elements, each chosen independently from two options, can distribute uniformly on $G$. We present conditions and obstructions to mixability. We show that $2$-groups, the…
We study the existence and uniqueness of (locally) absolutely continuous trajectories of a dynamical system governed by a nonexpansive operator. The weak convergence of the orbits to a fixed point of the operator is investigated by relying…
We consider stochastic processes arising from dynamical systems simply by evaluating an observable function along the orbits of the system and study marked point processes associated to extremal observations of such time series…
We consider irreversible Markov chains on finite commutative rings randomly generated using both addition and multiplication. We restrict ourselves to the case where the addition is uniformly random and multiplication is arbitrary. We first…