Discrete and Switching Dynamical Systems

Discrete and Switching Dynamical Systems Downloads

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- Chapter One: Linear Discrete Systems and Stability

pp. 1-62 | DOI: 10.5890/978-1-62155-003-7_1

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In this Chapter, the basic iterative solutions for linear discrete systems will
be presented. The iterative solutions of the linear discrete systems with distinct
and repeated eigenvalues will be discussed. The stability of linear discrete dynamical
systems will be discussed from the oscillatory, monotonic and spiral convergence
and divergence. The invariant, flip and circular critical boundaries of the stability
on the direction of the specific eigenvector will be classified. The lower dimensional
linear discrete systems will be discussed to show stability and stability switching.

- Chapter Two: Stability, Bifurcation and Routes to Chaos

pp. 63-132 | DOI: 10.5890/978-1-62155-003-7_2

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In this Chapter, 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 order singularity and stability for nonlinear discrete systems on the
specific eigenvectors will be presented. A few special cases in the lower dimensional
maps will be presented for a better understanding of generalized theory. The route
to chaos will be discussed briefly, and the intermittency phenomena relative to
specific bifurcations will be presented.

- Chapter Three: Fractality and Complete Dynamics

pp. 133-175 | DOI: 10.5890/978-1-62155-003-7_3

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In this chapter, to understand complexity of chaos in dynamical systems, the multifractality
of chaos generated by period-doubling bifurcation will be discussed via a geometrical
approach and self-similarity. A bouncing ball model will be discussed to show how
to construct discrete maps in practical problems, and the stability and bifurcation
of periodic motions for the bouncing ball will be presented analytically. Positive
and negative dynamics of discrete maps will be discussed, which is a base for the
compete dynamics of discrete dynamical systems. The theory of the complete dynamics
can be developed from the Ying-Yang theory in Luo (2010). The complete dynamics
of a discrete dynamical system with the Henon map will be discussed briefly.

- Chapter Four: Switching Systems and Transports

pp. 177-228 | DOI: 10.5890/978-1-62155-003-7_4

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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 reference points in switching dynamical systems will
be discussed through equi-measuring functions. The G-function of a flow to the equi-measuring
functions in the switching system will be introduced. The local increasing and decreasing
of switching systems to equimeasuring 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
equimeasuring function, the stability of switching systems can be discussed. 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. Stability and bifurcation of periodic
flows in nonlinear switching systems can be discussed in a similar fashion.

- Chapter Five: Mapping Dynamics and Fragmentation

pp. 229-281 | DOI: 10.5890/978-1-62155-003-7_5

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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 systems, and complex motions can
be classified through mapping structure. The nonlinear dynamics of a suspension
system with MR damping will be discussed as an example. With different velocity,
the MR damper will change damping forces to cause the suspension system to be discontinuous.
Using the mapping dynamics, the periodic motions of such suspension systems will
be presented.

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