Abstract
We give hierarchy of one-parameter family Φ(α, x) of maps at the interval [0, 1] with an invariant measure. Using the measure, we calculate Kolmogorov-Sinai entropy, or equivalently Lyapunov characteristic exponent of these maps analytically, where the results thus obtained have been approved with the numerical simulation. In contrary to the usual one-parameter family of maps such as logistic and tent maps, these maps do not possess period doubling or period-n-tupling cascade bifurcation to chaos, but they have single fixed point attractor for certain values of the parameter, where they bifurcate directly to chaos without having period-n-tupling scenario exactly at those values of the parameter whose Lyapunov characteristic exponent begins to be positive.
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Jafarizadeh, M.A., Behnia, S., Khorram, S. et al. Hierarchy of Chaotic Maps with an Invariant Measure. Journal of Statistical Physics 104, 1013–1028 (2001). https://doi.org/10.1023/A:1010449627146
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DOI: https://doi.org/10.1023/A:1010449627146