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Toward Analytical Chaos in Nonlinear Systems by Albert C. J. Luo

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Chapter 3An Analytical Method for Periodic Flows

In this chapter, from Luo (2012a), the analytical dynamics of periodic flows and chaos in nonlinear dynamical systems will be presented. The analytical solutions of periodic flows and chaos in autonomous systems will be discussed first, and the analytical dynamics of non-autonomous nonlinear dynamical systems will be presented. The analytical solutions of periodic motions in free and periodically excited vibration systems will be presented. In a similar fashion, the analytical solutions of periodic flows for time-delayed nonlinear systems will be presented with/without periodic excitations, and time-delayed nonlinear vibration systems will also be discussed for engineering applications. The analytical solutions of periodic flows and chaos are independent of the small parameters, which are different from the traditional perturbation methods. The methodology presented herein will end the history of chaos being numerically simulated only.

3.1 Nonlinear Dynamical Systems

In this section, analytical periodic flows in autonomous and periodically forced, nonlinear dynamical systems will be discussed. A generalized harmonic balance method with the Fourier series expressions will be presented for such analytical, periodic flows, and chaos in nonlinear dynamical systems. The local stability and bifurcation theory of equilibriums in nonlinear autonomous systems of coefficients in the Fourier series solutions will be employed to classify ...

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