Chapter 10 - Quasiperiodicity, resonance overlap and chaos

Abstract

This chapter discusses the concepts of quasiperiodicity, resonance, and overlap. It begins by explaining periodic and quasiperiodic motions using equations. For a periodic function f with period T, the frequency function, Dirac function, quasiperiodic function, and Fourier transformed function are derived. The dynamics of a system of two frequencies with two corresponding phase angles are demonstrated using a torus. Sine circle map is then described. The formation of an Arnold tongue under a sine circle map is used to demonstrate quasiperiodic motion, periodic motion, the overlap of resonances, and route to chaos. The phase-locking effect is also discussed. The chapter then provides information on resonance overlap and birth of chaos. Chirikov's conjecture that overlap of resonances will lead to chaos and Oxtoby and Rice's assumption that the chaotic motion is related to the overlapping of resonances are tested by employing the vibration of H2O. The role of Arnold diffusion is also discussed. The coincidence of chaotic and barrier regions is the next section. The section demonstrates that the barrier region in the chaotic region is composed of periodic and quasiperiodic motions. The periodic and quasiperiodic motions cannot deliver energy out as the chaotic motion. Hence, they stop the energy transfer.