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## “School of Physic”

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Paper   IPM / Physic / 6936
School of Physics
Title:   Hierarchy of One- and Many-Parameter Families of Elliptic Chaotic Maps of cn and sn Types
Author(s):
 1 M.A. Jafarizadeh 2 S. Behnia
Status:   Published
Journal: Phys. Lett. A
No.:  52
Vol.:  310
Year:  2003
Pages:   168-176
Supported by:  IPM
Abstract:
We present a 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 [J. Stat. Phys. 104 (2001); J. Nonlinear Phys. 9 (2002)]. Using this we have been able to obtain the invariant measure of these maps for small k and hence show that these maps have the same KolmogorovSinai entropy or (equivalently) Lyapunov characteristic exponent of the maps [J. Stat. Phys. 104 (2001); J. Nonlinear Phys. 9 (2002)]. Also, 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 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.