Periodicity and chaos in a modulated logistic map

We study the onset of chaos in a logistic map whose parameter is modulated nonlinearly. The bifurcation pattern with respect to a parameter is obtained and the critical value of is seen to be 0.89, where periodicity just ends. Further evidence for this regime is obtained from the analysis of the intermittency pattern. The stability in the different ranges of under repeated iteration is exhibited by the values of Lyapunov exponents. Beyond=0.89, the largest Lyapunov exponent becomes positive and the situation turns out to be unstable. Confirmation comes from a functional analysis of the stable and unstable manifolds which touch at=0.89.