ABSTRACT

This chapter focuses on nonlinear oscillations and variety of systems, with emphases on behaviors in phase-space plots and bifurcation diagrams. The Duffing oscillator is an example of a damped, driven nonlinear oscillator. The chapter analyzes the output from the simulations using the discrete Fourier transform. Fourier analysis and phase-space plots indicate that a chaotic system contains a number of dominant frequencies, and that the system tends to "jump" from one frequency to another. The chapter examines projectile motion, bound states of three-body systems, and Coulomb and chaotic scattering. It also focuses on some of the unusual behavior of billiards, which are a mix of scattering and bound states. Problems related to Lagrangian and Hamiltonian dynamics follow, with the actual computation of Hamilton's principle. The chapter concludes with the problem of several weights connected by strings: a simple problem that requires a complex solution involving both a derivative algorithm and a search algorithm.