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, for the average positive Lyapunov exponent, (lambda(+) - lambda(c)(+)) similar to A - A(c)(beta), where lambda(c)(+) is the value of the positive Lyapunov exponent at crisis, and the average crisis induced mean lifetime tau similar to A - A(c)(-gamma), 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, beta + gamma + 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. (C) 21002 Elsevier Science Ltd. All rights reserved. [References: 19]
展开▼
