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Hierarchy of chaotic maps with an invariant measure. (English) Zbl 1143.82308

Summary: We give hierarchy of one-parameter family \((\alpha, x)\) 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.

MSC:

82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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