Hierarchy of one and many-parameter families of elliptic chaotic maps of cn and sn types

Abstract

We present hierarchy of one and many-parameter families of elliptic chaotic maps of cn and sn types at the interval [0,1]. It is proved that for small values of k the parameter of the elliptic function, these maps are topologically conjugate to the maps of references [1,2], where using this we have been able to obtain the invariant measure of these maps for small k and thereof it is shown that these maps have the same Kolmogorov-Sinai entropy or equivalently Lyapunov characteristic exponent of the maps of references [1,2]}. As this parameter vanishes, the maps are reduced to the maps presented in above-mentioned reference. Also in contrary to the usual family of one-parameter maps, such as the logistic and tent maps, these maps do not display period doubling or period-n-tupling cascade transition to chaos, but they have single fixed point attractor at certain parameter values where they bifurcate directly to chaos without having period-n-tupling scenario exactly at these values of parameters whose Lyapunov characteristic exponent begin to be positive.

14 pages, Latex

References in corpus (5)

• Composition of Chaotic Maps with an Invariant Measure
• Superposition of chaotic process with convergence to Levy's stable law
• Hierarchy of Chaotic Maps with an Invariant Measure
• Hierarchy of chaotic maps with an invariant measure and their compositions
• Exactly solvable chaos and addition theorems of elliptic functions

Cited by in corpus (3)

• Hierarchy of piecewise non-linear maps with non-ergodicity behavior
• Hierarchy of random deterministic chaotic maps with an invariant measure
• Chaos suppression via adaptive feedback control of intermittency: From exactly solvable ergodic maps to interacting microbubble clusters