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Paper   IPM / P / 6936
School of Physics
  Title:   Hierarchy of One- and Many-Parameter Families of Elliptic Chaotic Maps of cn and sn Types
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
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 KolmogorovSinai 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.

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