Toward Analytical Chaos in Nonlinear Systems

Chapter 4Analytical Periodic to Quasi-Periodic Flows

In this chapter, from the idea of Luo (2012, 2013), period-m flows to quasi-periodic flows in nonlinear dynamical systems will be presented. The analytical solutions of quasi-periodic flows in autonomous systems will be discussed, and the analytical solutions of quasi-periodic flows in periodically forced nonlinear dynamical systems will be presented. The analytical solutions of quasi-periodic motions in free and periodically forced vibration systems will be presented. The analytical solutions of quasi-periodic flows for time-delayed nonlinear systems will be presented with/without periodic excitations, and time-delayed nonlinear vibration systems will be discussed as well.

4.1 Nonlinear Dynamical Systems

In this section, analytical quasi-periodic flows in autonomous and periodically forced, nonlinear dynamical systems will be discussed. Consider analytical quasi-periodic flows in autonomous nonlinear systems first.

Theorem 4.1

Consider a nonlinear dynamical system as

4.1

where is a -continuous nonlinear function vector .