Bifurcation scenarios of the noisy duffing-van der pol oscillator

This paper presents a numerical study of the bifurcation behavior of the noisy Duffing-van der Pol oscillator% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4baFfea0dXde9vqpa0lb9% cq0dXdb9IqFHe9FjuP0-iq0dXdbba9pe0lb9hs0dXda91qaq-xfr-x% fj-hmeGabaqaciGacaGaaeqabaWaaeaaeaaakeaatCvAUfKttLeary% qr1ngBPrgaiuGacuWF4baEgaWaaiaaiccacqWF9aqpcaaIGaGaaiik% aerbtLhBMfwzUbacgiGaa4xSdiaaiccacqGHRaWkcaaIGaGaeq4Wdm% 3ccaaIXaGcceqGxbGbaiaaliaaigdakiGacMcacqWF4baEcaaIGaGa% ci4kaiaaiccacqaHYoGycuWF4baEgaGaaiaaiccacqGHsislcaaIGa% Gae8hEaG3aaWbaaSqabeaacaaIZaaaaOGaaGiiaiabgkHiTiaaicca% cqWF4baEdaahaaWcbeqaaiaaikdaaaGccuWF4baEgaGaaiaaiccaci% GGRaGaaGiiaiabeo8aZTGaaGOmaOGabe4vayaacaGaaeOmaiaabYca% aaa!5F62!\\ddot x = (\alpha + \sigma 1{\rm{\dot W}}1)x + \beta \dot x - x^3 - x^2 \dot x + \sigma 2{\rm{\dot W2,}}\]where , are bifurcation parameters, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4baFfea0dXde9vqpa0lb9% cq0dXdb9IqFHe9FjuP0-iq0dXdbba9pe0lb9hs0dXda91qaq-xfr-x% fj-hmeGabaqaciGacaGaaeqabaWaaeaaeaaakeaaceqGxbGbaiaali% aaigdakiqabEfagaGaaSGaciOmaaaa!35B4!\{\rm{\dot W}}1{\rm{\dot W}}2\] are independent white noise processes, and ₁, ₂ are intensity parameters. A stochastic bifurcation here means (a) the qualitative change of stationary measures or (b) the change of stability of invariant measures and the occurrence of new invariant measures for the random dynamical system generated by (1). The first type of bifurcation can be observed when studying the solution of the Fokker-Planck equation, this stationary measure is a quantity corresponding to the one-point motion. More generally, if one is interested in the simultaneous motion of n points (n1) forward and backward in time, then the second type of bifurcation arises naturally, capturing all the stochastic dynamics of (1). Based on the numerical results, we propose definitions of the stochastic pitchfork and Hopf bifurcations.