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

Abstract

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 A c, for the average positive Lyapunov exponent, ( λ +− λ c +)∼| A− A c| β , where λ c + is the value of the positive Lyapunov exponent at crisis, and the average crisis induced mean lifetime τ∼| A− A c| − 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 D c at the critical point A c, namely, β+ γ+ D c=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.