Universality and scaling in chaotic attractor-to-chaotic attractor transitions

In this paper we discuss chaotic attractor-to-chaotic attractor transitions in two-dimensional multiparameter maps as an external parameter is varied. We show that the transitions are sharply defined and may be classed as second-order phase transitions. We obtain scaling laws, about the critical point Ac, for the average positive Lyapunov exponent, (λ+−λc+) A−Acβ, where λc+ is the value of the positive Lyapunov exponent at crisis, and the average crisis induced mean lifetime τ A−Ac−γ, where A is the parameter that is varied. Here average means averaged over many initial conditions. Furthermore we find that there is an algebraic relationship between the critical exponents and the correlation dimension Dc at the critical point Ac, namely, β+γ+Dc=constant. We find this constant to be approximately 2.31. We postulate that this is a universal relationship for second-order phase transitions in two-dimensional multiparameter non-hyperbolic maps.