Constructing an opposite map to a specified chaotic map. (English) Zbl 1125.37313

Huang, Weihong

Nonlinearity 18, No. 3, 1375-1391 (2005).

Summary: For a two-segmental complete chaotic map \(F: [0, 1]\to [0, 1]\) that preserves an invariant density \(\varphi\) and has a partitioning point at \(x_c\), its opposite map is defined to possess the following four characteristics: (i) \(\widetilde {F}\) has the same metric structure; (ii) \(\widetilde {F}\) preserves an invariant density \(\widetilde {\varphi}(x)= \varphi(1-x)\); (iii) both \(F\) and \(\widetilde {F}\) have the same degree of chaoticity in the sense of identical Lyapunov exponent and (iv) the partitioning point of \(\widetilde {F}\) is at \(\widetilde {x}_c= 1-x_c\). An approach for constructing opposite maps analytically for all four types of two-segmental complete chaotic maps is provided. Meanwhile, a mutual implication relationship that is invariant with respect to conjugation (metric equivalence) is defined for all two-segmental complete chaotic maps that share an identical invariant measure, an identical Lyapunov exponent and an identical partitioning point. Through this relationship, a unique implied family of chaotic maps is formed so that as long as any member of this family is identified, the rest can be constructed analytically, which makes it possible for all known statistical properties originally established for a particular class of chaotic maps to be generalized to all two-segmental chaotic maps. Numerical simulations conducted are in good agreement with theoretical results.

MSC:

37E05	Dynamical systems involving maps of the interval
37C30	Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior