Blowout bifurcation of chaotic saddles (Q1585745)

The authors consider the dynamical system generated by two coupled logistic maps, \[ (x_{n+1},y_{n+1})=F(x_n,y_n), \] where \(F(x,y)=\left(\rho x(1-x)+d(y-x), \rho y(1-y)+d(x-y)\right)\), and the parameters are real and near \(\rho=3.64\) and \(d=-1.1\). For these values it has been noticed in previous papers that on the invariant manifold \(S=\{(x,y): y=x\}\), \(F\) exhibits chaotic behavior. On the other hand if \(d\) increases until \(d_c\simeq-0.92\) the sign of the Lyapunov exponents tranverse to \(S\) changes from negative to positive (this phenomenon is known as blowout bifurcation). The aim of the paper is to study a similar bifurcation, but when in the invariant manifold \(S\) there is a different behavior. In particular the authors fix \(\rho=3.63\) (for this value the dynamical system has a chaotic saddle and a period-6 attractor in \(S\)) and study the evolution of the attractor when \(d\) varies. From the text: ``\dots{} the model studied was used only for the purpose of illustrating the blowout bifurcation of a chaotic saddle. The logistic map used to describe the dynamics in the invariant space is considered as a paradigm in the study of nonlinear systems. We believe that our results are typical for a class of two coupled chaotic systems''.