Detection of chaos in some local regions of phase portraits using Shannon entropies

Ling, Wing-Kuen and Ho, Charlotte Yuk-Fan and Tam, Peter Kwong-Shun (2004) Detection of chaos in some local regions of phase portraits using Shannon entropies. International Journal of Bifurcation and Chaos, 14 (4). pp. 1493-1499. ISSN 0218-1274

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Full text URL: http://dx.doi.org/10.1142/S0218127404010023

Abstract

This letter demonstrates the use of Shannon entropies to detect chaos exhibited in some local regions on the phase portraits. When both the eigenvalues of the second-order digital filters with two’s complement arithmetic are outside the unit circle, the Shannon entropies of the state variables are independent of the initial conditions and the filter parameters, except for some special values of the filter parameters. At these special values, the Shannon entropies of the state variables are relatively small. The state trajectories corresponding to these filter parameters either exhibit random-like chaotic behaviors in some local regions or converge to some fixed points on the phase portraits. Hence, by measuring the Shannon entropies of the state variables, these special state trajectory patterns can be detected. For completeness, we extend the investigation to the case when the eigenvalues of the second-order digital filters with two’s complement arithmetic are complex and inside or on the unit circle. It is found that the Shannon entropies of the symbolic sequences for the type II trajectories may be higher than that for the type III trajectories, even though the symbolic sequences of the type II trajectories are periodic and have limit cycle behaviors, while that of the type III trajectories are aperiodic and have chaotic behaviors.

Item Type:Article
Additional Information:This letter demonstrates the use of Shannon entropies to detect chaos exhibited in some local regions on the phase portraits. When both the eigenvalues of the second-order digital filters with two’s complement arithmetic are outside the unit circle, the Shannon entropies of the state variables are independent of the initial conditions and the filter parameters, except for some special values of the filter parameters. At these special values, the Shannon entropies of the state variables are relatively small. The state trajectories corresponding to these filter parameters either exhibit random-like chaotic behaviors in some local regions or converge to some fixed points on the phase portraits. Hence, by measuring the Shannon entropies of the state variables, these special state trajectory patterns can be detected. For completeness, we extend the investigation to the case when the eigenvalues of the second-order digital filters with two’s complement arithmetic are complex and inside or on the unit circle. It is found that the Shannon entropies of the symbolic sequences for the type II trajectories may be higher than that for the type III trajectories, even though the symbolic sequences of the type II trajectories are periodic and have limit cycle behaviors, while that of the type III trajectories are aperiodic and have chaotic behaviors.
Keywords:second-order digital filter, two’s complement arithmetic, Shannon entropy
Subjects:H Engineering > H310 Dynamics
Divisions:College of Science > School of Engineering
ID Code:3059
Deposited By:INVALID USER
Deposited On:06 Aug 2010 13:26
Last Modified:13 Mar 2013 08:43

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