Heteroclinic chaotic behavior driven by a Brownian motion

Limiting dynamical behavior of random fractional FitzHugh-Nagumo systems driven by a Wong-Zakai approximation process

“…It is worth mentioning that there are numerous works in the literature on the existence of attractors for standard stochastic equations driven by a white noise, see, [1,2,3,4,5,6,7,8,9,11,13,20,22,23,24,33,34,35,36,38,41]. The dynamical behavior of standard stochastic equations driven by colored noise or Wong-Zakai approximation process have been examined in [15,16,29,30,31,32,40]. Recently, the random dynamics is investigated for fractional nonclassical diffusion equations driven by colored noise [39].…”

Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations

“…In the pioneer work about approximations, Wong and Zakai [40,41] studied the approximations for one-dimensional Brownian motion. Their work was later extended to stochastic differential equations of higher dimension ( [18,19,22,27,35,33,34,36]). The Wong-Zakai approximations have also been generalized to stochastic differential equations driven by martingales and semi-martingales ( [24,25,29,30]).…”

Wong-Zakai approximations and asymptotic behavior of stochastic Ginzburg-Landau equations

“…Using deterministic differential equations to approximate stochastic differential equations was introduced by Wong and Zakai in their pioneer work [60,59] in which they discussed both piecewise linear approximations and piecewise smooth approximations for one-dimensional Brownian motions. Their work was later generalized to stochastic differential equations of higher dimensions, for example, by McShane [37], Stroock and Varadhan [45], Sussmann [46,47], Ikeda et al [22], Ikeda and Watanabe [23], and recently by Kelly and Melbourne [24], and Shen and Lu [43] in which the same approximations as this paper were studied. The results of the Wong-Zakai approximations have also been extended to stochastic differential equations driven by martingales and semimartingales, see for example, Nakao and Yamato [39], Konecny [25], Protter [41], Nakao [38], and Kurtz and Protter [26,27].…”

“…(3) generates a random dynamical system, which allows us to investigate the pathwise dynamics such as random attractors. Such approximation was also used in Lu and Wang [36,35], Wang et al [58], and Shen et al [43] where they studied the chaotic behavior of random differential equations driven by a multiplicative noise of G δ (θ t ω) and long term behavior of stochastic reaction-diffsion equations driven by multiplicative noise .…”