![]() Through his adventure, the astronaut meets many alien creatures. Unfortunately, he wakes up in an unknown location, far away from the Earth. Onboard of his ship, the Nomad, the astronaut sleeps in cryonics until reaching his destiny. An astronaut is sent to Ganymede, a moon of Jupiter, in order to find viable resources to supply Earth. Earth is exhausted of resources and no longer a viable planet for its human civilization, whose unsustainable growth made it impossible for developing society any further. Ser.The story begins in the 22nd century. Ye, F.X.-F., Wang, Y., Qian, H.: Stochastic dynamics: Markov chains and random transformations. Ye, F.X.-F., Qian, H.: Stochastic dynamics II: Finite random dynamical systems, linear representation, and entropy production. Walters, P.: An Introduction to Ergodic Theory. Viana, M.: (Dis)continuity of Lyapunov exponents. Shmulevich, I., Dougherty, E.R.: Probabilistic Boolean Networks: The Modeling and Control of Gene Regulatory Networks. Oseledec, V.I.: A multiplicative ergodic theorem. ![]() ![]() Newman, J.: Necessary and sufficient conditions for stable synchronization in random dynamical systems. Ma, Y., Qian, H., Ye, F.X.-F.: Stochastic dynamics: Models for intrinsic and extrinsic noises and their applications (in Chinese). Jarret, A.: Desynchronization of random dynamical system under perturbation by an intrinsic noise. Huang, W., Yi, Y.: On Lyapunov exponents of continuous Schrödinger cocycles over irrational rotations. Huang, W., Qian, H., Wang, S., Ye, F.X.-F., Yi, Y.: Synchronization in discrete-time, discrete-state random dynamical systems. 13(3), 265–280 (1998)Īrnold, L., Gundlach, V., Demetrius, L.: Evolutionary formalism for products of positive random matrices. Springer, Berlin (1998)Īrnold, L., Chueshov, I.: Order-preserving random dynamical systems: equilibria, attractors, applications. An example of a smooth Markov perturbation of a synchronized probabilistic Boolean network is provided to illustrate the intermittency between high-probability synchronization and low-probability desynchronization.Īltman, E., Avrachenkov, K., Queija, RNúñez: Perturbation analysis for denumerable Markov chains with application to queueing models. Ergodicity of the extrinsic noise dynamics is seen to be crucial for the characterization of (de)synchronization sets and their respective relative frequencies. An explicit asymptotic expansion is derived. Existence and uniqueness of invariant distributions are established, as well as their convergence as \(\varepsilon \rightarrow 0\). ![]() If the perturbation is \(C^m\) ( \(m \ge 1\)) in \(\varepsilon \), where \(\varepsilon \) is a perturbation parameter, then the relative frequencies of synchronization with probability \(1-O(\varepsilon ^)\) can both be precisely described for \(1\le \ell \le m\) via an asymptotic expansion of the invariant distribution. Under smooth Markov perturbations, high-probability synchronization and low-probability desynchronization occur intermittently in time. That is, both the probability of synchronization and the proportion of time spent in synchrony are arbitrarily close to one. It is shown that if the deterministic random network is synchronized (resp., uniformly synchronized), then for almost all realizations of its extrinsic noise the stochastic trajectories of the perturbed network synchronize along almost all (resp., along all) time sequences after a certain time, with high probability. On a finite state space and in discrete time, the network allows for unperturbed (or “deterministic”) randomness that represents the extrinsic noise but also for small intrinsic uncertainties modelled by a Markov perturbation. By introducing extrinsic noise as well as intrinsic uncertainty into a network with stochastic events, this paper studies the dynamics of the resulting Markov random network and characterizes a novel phenomenon of intermittent synchronization and desynchronization that is due to an interplay of the two forms of randomness in the system.
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