In this paper, we discuss an efficient iterative method for the estimation of the chief dynamical invariants of chaotic systems based on stochastically stable piecewise affine maps (e.g. the invariant measure, the Lyapunov exponent as well as the Kolmogorov–Sinai entropy). The proposed method represents an alternative to the Monte-Carlo methods and to other methods based on the discretization of the Frobenius–Perron operator, such as the well known Ulam's method. The proposed estimation method converges not slower than exponentially and it requires a computation complexity that grows linearly with the iterations. Referring to the theory developed by C. Liverani, we discuss a theoretical tool for calculating a conservative estimation of the convergence rate of the proposed method. The proposed approach can be used to efficiently estimate any order statistics of a symbolic source based on a piecewise affine mixing map.

Addabbo, T., Fort, A., Papini, D., Rocchi, S., Vignoli, V. (2009). An efficient and accurate method for the estimation of entropy and other dynamical invariants for piecewise affine chaotic maps. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS IN APPLIED SCIENCES AND ENGINEERING, 19(12), 4175-4195 [10.1142/S0218127409025286].

An efficient and accurate method for the estimation of entropy and other dynamical invariants for piecewise affine chaotic maps

ADDABBO, TOMMASO;FORT, ADA;PAPINI, DUCCIO;ROCCHI, SANTINA;VIGNOLI, VALERIO
2009-01-01

Abstract

In this paper, we discuss an efficient iterative method for the estimation of the chief dynamical invariants of chaotic systems based on stochastically stable piecewise affine maps (e.g. the invariant measure, the Lyapunov exponent as well as the Kolmogorov–Sinai entropy). The proposed method represents an alternative to the Monte-Carlo methods and to other methods based on the discretization of the Frobenius–Perron operator, such as the well known Ulam's method. The proposed estimation method converges not slower than exponentially and it requires a computation complexity that grows linearly with the iterations. Referring to the theory developed by C. Liverani, we discuss a theoretical tool for calculating a conservative estimation of the convergence rate of the proposed method. The proposed approach can be used to efficiently estimate any order statistics of a symbolic source based on a piecewise affine mixing map.
Addabbo, T., Fort, A., Papini, D., Rocchi, S., Vignoli, V. (2009). An efficient and accurate method for the estimation of entropy and other dynamical invariants for piecewise affine chaotic maps. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS IN APPLIED SCIENCES AND ENGINEERING, 19(12), 4175-4195 [10.1142/S0218127409025286].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11365/21649
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