Continuous Dynamical Systems

Continuous Dynamical Systems Downloads

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- Chapter One: Linear Systems and Stability

pp. 1-54 | DOI: 10.5890/978-1-62155-001-3_1

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In this Chapter, the theory of linear systems will be presented. Separated linear
systems and diagonalization of square matrix will be discussed first. The linear
operator exponentials will be presented. The fundamental solutions of autonomous
linear systems will be given with the matrix possessing real eigenvalues, complex
eigenvalues and repeated eigenvalues. The stability theory for autonomous linear
systems will be discussed. The solutions of non-autonomous linear systems will be
discussed and steady state solutions will be presented. A generalized “resonance”
concept will be introduced, and the resonant solutions will be presented. Solutions
and stability for lower dimensional linear systems will be discussed in details.

- Chapter Two: Stability Switching and Bifurcation

pp. 55-108 | DOI: 10.5890/978-1-62155-001-3_2

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In this Chapter, basic concepts of nonlinear dynamical systems will be introduced.
Local theory, global theory and bifurcation theory of nonlinear dynamical systems
will be briefly discussed. The stability switching and bifurcation on specific eigenvectors
of the linearized system at equilibrium will be discussed. The higherorder singularity
and stability for nonlinear systems on the specific eigenvectors will be developed.
The lower-dimensional dynamical systems will be discussed to help one understand
the stability and bifurcation theory, and the Lyapunov function stability will be
briefly discussed.

- Chapter Three: Analytical Periodic Flows and Chaos

pp. 109-166 | DOI: 10.5890/978-1-62155-001-3_3

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In this Chapter, the analytical dynamics of periodic flows and chaos in nonlinear
dynamical systems will be presented. The analytical solutions of periodic flows
and chaos for autonomous systems will be discussed first, and the analytical dynamics
of periodically forced nonlinear dynamical systems will be presented. The analytical
solutions for free and forced vibration systems will be discussed. A periodically
forced Duffing oscillator will be discussed as an application to demonstrate the
period-m motions in such a nonlinear system. The analytical solutions of periodic
flows and chaos are independent on small parameters. The method presented herein
will end the history of chaos being numerically simulated only.

- Chapter Four: Global Transversality and Chaos

pp. 167-236 | DOI: 10.5890/978-1-62155-001-3_4

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In this Chapter, the global tangency and transversality of a perturbed flow in dynamical
systems with at least one separatrix will be discussed. The local and global flows
in perturbed dynamical systems will be introduced through the geometrical intuition.
The G(k ) -function ( k = 0,1,2,...) for a perturbed flow to a specified, first
integral surface will be introduced. The definitions of the global and tangential
flows will be presented, and the sufficient and necessary conditions of such global
and tangential flows to the separatrix will be developed via the Gfunctions. Such
conditions give the global tangency and transversality of flows to the separatrix
surface in nonlinear dynamical systems. The perturbed Hamiltonian system will be
discussed. A periodically forced, damped Duffing oscillator with a separatrix will
be presented as an example.

- Chapter Five: Resonance and Hamiltonian Chaos

pp. 237-284 | DOI: 10.5890/978-1-62155-001-3_5

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In this Chapter, nonlinear Hamiltonian chaos including stochastic and resonant layers
in two-dimensional nonlinear Hamiltonian systems will be presented. The chaos and
resonance mechanism in the stochastic layer of generic separatrix will be discussed
that is formed by the primary resonance interaction in nonlinear Hamiltonians systems.
However, the chaos in the resonant layer of the resonant separatrix will be presented
that is formed by the sub-resonance interaction.

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