An explicit estimate of sojourning time by intermittency with elementary method (Q796159)

The paper studies the 1-parameter ''bifurcation'' family \(F(\mu,x)=\mu +x- x^ 2+\mu xf(\mu,x)+x^ 3g(x),\) where f and g are continuous at \(x=0\), and its random perturbations of the form \((X_{n+1}=F(\mu,X_ n)+| \mu |^{1+\theta}\zeta_ n\) where \(\theta>0\) and \(\zeta_ n\) is a bounded random variable. The author proves that for suitable \(\delta>0\) one gets the following asymptotic behavior of sojourning time in [- \(\delta\),\(\delta]\), \(\lim_{\mu \uparrow 0}\sqrt{-\mu}\min \{n:\quad X_ n<-\delta \}\).