A bifurcation theory for a class of discrete time Markovian stochastic systems
type: Item
instance of: scholarly article
title: A bifurcation theory for a class of discrete time Markovian stochastic systems
zbMATH Open document ID: 1167.37028
author: Cees Diks
author: Florian Wagener
published in: Physica D
publication date: 2009-02-10T00:00:00Z
review text: A bifurcation theory of smooth stochastic dynamical systems \[ X_{t+1} = g(X_t,\varepsilon_t),\quad t \in \mathbb{N} \] with everywhere positive transition densities is presented. The ``dependence ratio'' of joint and marginal stationary probability densities is used to classify its behavior with respect to possible bifurcations. This ``dependence ratio'' is a geometric invariant of the system. The authors introduce an equivalence relation defined on those ratios in order to obtain a bifurcation theory for which, in the compact case, the set of stable (non-bifurcating) systems is open and dense. Two simple, but illustrative examples support their theory. An appendix states the mathematical proof-steps of their major propositions.
reviewed by: Henri Schurz
Mathematics Subject Classification ID: 37H20
Mathematics Subject Classification ID: 37H10
Mathematics Subject Classification ID: 39A28
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 60J05
zbMATH DE Number: 5506328
zbMATH Keywords: bifurcation theory
zbMATH Keywords: discrete time stochastic dynamical systems
zbMATH Keywords: geometric invariants
zbMATH Keywords: dependence ratio
zbMATH Keywords: stationary probability densities
zbMATH Keywords: Markovian systems
MaRDI profile type: Publication
full work available at URL: j.physd.2008.07.021
OpenAlex ID: W2080647171
cites work: Attractors and time averages for random maps
cites work: Q4210478
cites work: A stochastic Hopf bifurcation
cites work: Q4263367
cites work: Exponential growth of fixed-mix strategies in stationary asset markets
cites work: Phenomenological and ratio bifurcations of a class of discrete time stochastic processes
cites work: Q3679908
cites work: Differential Topology
cites work: Dependence function for continuous bivariate densities
cites work: Hard bifurcations in dynamical systems with bounded random perturbations
cites work: Q4263374
cites work: Miscellanea. The local dependence function
cites work: Q4263372
cites work: Local dependence functions for extreme value distributions
cites work: An introduction to copulas. Properties and applications
cites work: Q3874454
cites work: Q3034619
cites work: Q3281461
cites work: Q4058507
cites work: Transformation invariant stochastic catastrophe theory
cites work: Q3771283
cites work: What are SRB measures, and which dynamical systems have them?
cites work: Stability of dynamical systems
cites work: Bifurcations of stationary measures of random diffeomorphisms
cites work: Dynamics and bifurcations of random circle diffeomorphism
DOI: 10.1016/J.PHYSD.2008.07.021
description: scientific article; zbMATH DE number 5506328