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Chaos and Control in Singular Biological Economic Systems

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Complexity, Analysis and Control of Singular Biological Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 421))

Preliminaries for Chaotic Behavior and Chaotic Control

Poincaré Surface-of-Section Technique

A traditional approach to gain preliminary insight into the properties of the dynamical system is to carry out a one-dimensional bifurcation analysis. One-dimensional bifurcation diagrams of Poincaré maps present information about the dependence of the dynamics on a certain parameter. The analysis is expected to reveal the type of attractor to which the dynamics will ultimately settle down after passing the initial transient phase and within which the trajectory will then remain forever. On a Poincaré surface of section, the dynamical behavior can be described by a discrete map whose phase-space dimension is less than that of the original continuous flow. Chaotic flows can be understood based on concepts that are convenient for maps such as unstable orbits. The limit sets of the Poincaré map correspond to long-term solutions of the underlying continuous dynamical system in the following way (see references [27, 22]).

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  1. College of Sciences, Northeastern University, WenHua Road, 110819, Shenyang, China, People’s Republic

    Qingling Zhang, Chao Liu & Xue Zhang

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  1. Qingling Zhang
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  2. Chao Liu
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  3. Xue Zhang
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Correspondence to Qingling Zhang .

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Zhang, Q., Liu, C., Zhang, X. (2012). Chaos and Control in Singular Biological Economic Systems. In: Complexity, Analysis and Control of Singular Biological Systems. Lecture Notes in Control and Information Sciences, vol 421. Springer, London. https://doi.org/10.1007/978-1-4471-2303-3_5

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  • DOI: https://doi.org/10.1007/978-1-4471-2303-3_5

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