Mathematical Sciences: Applied Dynamics of Near Integrable Systems

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9403750 Kovacic This proposal describes intended research concerning irregular dynamical behavior in two classes of physical systems. The first class will be mechanical systems described by ordinary differential equations. The proposed work will show the existence of special trajectories, called homoclinic orbits, that connect equilibrium positions to each other in an infinite amount of time. These trajectories will consist of a sequence of slow and fast stretches following one another, and may be responsible for irregular behavior of nearby trajectories, and thus of the mechanical systems under investigation. One such system describes irregular beats in the amplitudes of two coupled mathematical pendula. The second class of physical systems under investigation is laser optics. The proposed research will investigate the occurrence of irregular dynamics of soliton-like optical pulses in the Maxwell-Bloch partial differential equations describing ring lasers. These dynamics will result in intermittent blinking, on a very short time scale, of the laser output. Both analytical and computational methods will be used in the course of this research. This proposal describes intended research in two problem areas concerning near-integrable dynamical systems. The first proposed area concerns multi-bump orbits homoclinic to resonance bands. The work proposed here will utilize the Melnikov method, geometric singular perturbation theory, and the exchange lemma to describe new classes of such orbits as well as new techniques that can be used to find these orbits in concrete physical problems. Multi-bump orbits homoclinic to resonance bands consist of a succession of fast bumps interspersed with segments on which trajectories move very slowly in time. Besides these theoretical studies, a computational verification and investigations of physically interesting examples that may exhibit the phenomenon of orbits homoclinic to resonance bands are also proposed. The second propos ed are concerns chaotic dynamics in the near-integrable regime of the Maxwell-Bloch partial differential equations describing samples of lasing material in a ring cavity. These dynamics may be present due to the existence of space-dependent homoclinic orbits in the corresponding infinite-dimensional phase space. The existence of these homoclinic orbits and chaotic dynamics will be ascertained with the use of a combination of numerical and analytical techniques. The numerical techniques will include simulations using split-step and pseudo-spectral methods, as well as calculations of various diagnostics such as power spectra, Lyapunov exponents, and Lyapunov attractor dimension. The analytical techniques will include the subharmonic Melnikov method, perturbation theory, Floquet theory, invariant manifold theory, inverse spectral theory, and the homoclinic Melnikov method for near-integrable partial differential equations.