MATH332 - Nonlinear Dynamics and Chaos
The remarkable fact that determinism does not guarantee regular or predictable behaviour is having a major impact on many fields of science and engineering as well as mathematics. The discovery of chaos, or of chaotic motions, in simple dynamical systems changes our understanding of the foundations of physics and has many practical applications as well, shedding new light on the workings of lasers, fluids, mechanical structures and chemical reactions.
Dynamical systems involve the study of maps and systems of differential equations. The presence of a nonlinear element usually introduces totally novel phenomena that are absent in linear systems, even at low order. In this unit, the diversity of nonlinear phenomena is explored through the study of second order differential equations, and 1- and 2-dimensional maps. In unforced systems, examples of stable motions include limit cycles and periodic solutions. In harmonically forced systems, we study jump phenomena (where the system jumps discontinuously between two stable states as the frequency is slowly varied), as well as sub- and super-harmonic generation. A parameter-dependent system may radically change behaviour as that parameter is varied, and exhibit bifurcation: manifestations in maps and systems of differential equations will be studied. Chaotic motions will be introduced by a study of the driven pendulum, a second order system that includes nonlinear aspects usually ignored in simpler treatments. An appropriate balance between forcing and damping leads to irregular, but bounded, motions that do not repeat themselves, even approximately - truly chaotic motion in a simple deterministic system.
Prerequisites: MATH235(P); MATH232(P) or MATH236(P).
Corequisites: None.
Not Counted for Credit With: None.