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2012 | Buch | 1. Auflage

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Regularity and Complexity in Dynamical Systems

verfasst von: Albert C. J. Luo

Verlag: Springer New York

Buchreihe : Nonlinear Systems and Complexity

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Regularity and Complexity in Dynamical Systems describes periodic and chaotic behaviors in dynamical systems, including continuous, discrete, impulsive, discontinuous, and switching systems. In traditional analysis, the periodic and chaotic behaviors in continuous, nonlinear dynamical systems were extensively discussed even if unsolved. In recent years, there has been an increasing amount of interest in periodic and chaotic behaviors in discontinuous dynamical systems because such dynamical systems are prevalent in engineering. Usually, the smoothening of discontinuous dynamical system is adopted in order to use the theory of continuous dynamical systems. However, such technique cannot provide suitable results in such discontinuous systems. In this book, an alternative way is presented to discuss the periodic and chaotic behaviors in discontinuous dynamical systems.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Nonlinear Continuous Dynamical Systems

Abstract

In this Chapter, basic concepts of nonlinear dynamical systems will be presented as a review material. Local theory, global theory and bifurcation theory of nonlinear dynamical systems will be discussed. The stability switching and bifurcation on specific eigenvectors of the linearized system at equilibrium will be presented. The higher singularity and stability for nonlinear systems on the specific eigenvectors will be developed. In addition, a periodically excited Duffing oscillator with cubic damping and constant force will be discussed as an application. The stability of approximate periodic solutions of such a Duffing oscillator will be discussed.

Albert C. J. Luo

Chapter 2. Nonlinear Discrete Dynamical Systems

Abstract

In this chapter, the basic concepts of nonlinear discrete systems will be presented. The local and global theory of stability and bifurcation for nonlinear discrete systems will be discussed. The stability switching and bifurcation on specific eigenvectors of the linearized system at fixed points under specific period will be presented. The higher singularity and stability for nonlinear discrete systems on the specific eigenvectors will be developed. A few special cases in the lower dimensional maps will be presented for a better understanding of the generalized theory. The route to chaos will be discussed briefly, and the intermittency phenomena relative to specific bifurcations will be presented. The normalization group theory for 2-D discrete systems will be presented via Duffing discrete systems.

Albert C. J. Luo

Chapter 3. Chaos and Multifractality

Abstract

In this Chapter, basic concepts of fractal in nonlinear dynamical systems will be presented as an introduction. The fractal generation rules will be presented for nonrandom and random fractals. The multifractals based on the single- and joint-multifractal measures will be presented. Multifractality of chaos generated by period-doubling bifurcation will be presented via a geometrical approach and self-similarity. Fractality of hyperbolic chaos will be discussed.

Albert C. J. Luo

Chapter 4. Complete Dynamics and Synchronization

Abstract

This chapter presents a Ying–Yang theory for nonlinear discrete dynamical systems with consideration of positive and negative iterations of discrete iterative maps. In existing analysis, the solutions relative to “Yang” in nonlinear dynamical systems are extensively investigated. However, the solutions pertaining to “Ying” in nonlinear dynamical systems are presented. A set of concepts on “Ying” and “Yang” in discrete dynamical systems are introduced. Based on the Ying–Yang theory, the complete dynamics of discrete dynamical systems can be discussed. A discrete dynamical system with the Henon map is investigated as an example. The companion and synchronization of discrete dynamical systems will be introduced, and the corresponding conditions are developed. The synchronization dynamics of Duffing and Henon maps will be discussed.

Albert C. J. Luo

Chapter 5. Switching Dynamical Systems

Abstract

In this chapter, dynamics of switching dynamical systems will be presented. A switching system of multiple subsystems with transport laws at switching points will be discussed. The existence and stability of switching dynamical systems will be discussed through equi-measuring functions. The G-function of the equi-measuring functions will be introduced. The local increasing and decreasing of switching systems to equi-measuring functions will be presented. The global increasing and decreasing of the switching systems to equi-measuring functions will be discussed. Based on the global and local properties of the switching dynamical systems to the equi-measuring function, the stability of switching systems can be discussed. To demonstrate flow regularity and complexity of switching systems, the impulsive system is as a special switching system to present, and the quasi-periodic flows and chaotic diffusion of impulsive systems will be presented. A frame work for periodic flows in switching systems will be presented. The periodic flows and stability for linear switching systems will be discussed. This framework can be applied to nonlinear switching systems. The further results on stability and bifurcation of periodic flows in nonlinear switching systems can be discussed.

Albert C. J. Luo

Chapter 6. Mapping Dynamics and Symmetry

Abstract

In the previous chapter, dynamics of switching systems were discussed. In this chapter, mapping dynamics and symmetry in discontinuous dynamical systems will be discussed. The G-function of the discontinuous boundary will be presented first. To understand of nonlinear dynamics of a flow from one domain to another domain, mapping dynamics of discontinuous dynamics systems will be presented, which is a generalized symbolic dynamics. Using the mapping dynamics, one can determine periodic and chaotic dynamics of discontinuous dynamical systems, and complex motions can be classified through mapping structure. The nonlinear dynamics of the Chua’s circuit system will be presented as an example. The flow grazing property of a flow will be discussed, which is a source to generate the complex motions in discontinuous dynamical systems. The flow symmetry in discontinuous dynamical systems will be discussed through the mapping dynamics and grazing. The strange attractor fragmentation generated by the grazing of flows to the boundary will be presented.

Albert C. J. Luo

Backmatter

Titel: Regularity and Complexity in Dynamical Systems
verfasst von: Albert C. J. Luo
Verlag: Springer New York
Electronic ISBN: 978-1-4614-1524-4
Print ISBN: 978-1-4614-1523-7
DOI: https://doi.org/10.1007/978-1-4614-1524-4

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