Toward Analytical Chaos in Nonlinear Systems

4.2 Nonlinear Vibration Systems

In this section, the analytical solutions of quasi-periodic motion for nonlinear vibration systems are presented, and the stability and bifurcation of approximate solutions of quasi-periodic motions in nonlinear vibration systems will be discussed.

Theorem 4.3

Consider a nonlinear vibration system as

4.68

where is a -continuous nonlinear function vector .

1. If such a dynamical system has a period-m motion with finite norm and period , there is a generalized coordinate transformation with for the periodic motion of Equation (4.68) in a form of
   4.69
2. with ...