Toward Analytical Chaos in Nonlinear Systems

Chapter 5Quadratic Nonlinear Oscillators

In this chapter, analytical solutions for period-m motions in a periodically forced, quadratic nonlinear oscillator will be presented through the Fourier series solutions with finite harmonic terms, and the stability and bifurcation analyses of the corresponding period-1 motions will be carried out. There are many period-1 motions in such a nonlinear oscillator, and the parameter map for excitation amplitude and frequency will be developed for different period-1 motions. For each period-1 motion branch, analytical bifurcation trees of period-1 motions to chaos will be presented. For a better understanding of complex period-m motions in such a quadratic nonlinear oscillator, trajectories, and amplitude spectrums will be illustrated numerically.

5.1 Period-1 Motions

In this section, period-1 motions in a periodically forced, quadratic nonlinear oscillator will be discussed. The analytical solutions with only two harmonic terms in the Fourier series expressions will be discussed first as an introduction. The appropriate analytical solutions will be presented with finite harmonic terms based on the prescribed accuracy of harmonic amplitudes. From appropriate solutions, the analytical bifurcation trees for period-1 motions to chaos can be found. Infinite, countable period-1 motions that exist in such an oscillator will be presented, and the corresponding parameter maps will be presented, and complex period-1 motions will be illustrated.