Nonlinear Dynamical Systems

(3 Credit Hours) · Undergraduate

Theory of nonlinear dynamical systems has applications to a wide variety of fields, from physics, biology, and chemistry, to engineering, economics, and medicine. This is one of its most exciting aspects - that it brings researchers from many disciplines together with a common language. A dynamical system consists of an abstract phase space or state space, whose coordinates describe the dynamical state at any instant; and a dynamical rule which specifies the immediate future trend of all state variables, given only the present values of those same state variables. Dynamical systems are "deterministic" if there is a unique consequent to every state, and "stochastic" or "random" if there is more than one consequent chosen from some probability distribution. A dynamical system can have discrete or continuous time. The discrete case is defined by a map and the continuous case is defined by a "flow. Nonlinear dynamical systems have been shown to exhibit surprising and complex effects that would never be anticipated by a scientist trained only in linear techniques. Prominent examples of these include bifurcation, chaos, and solitons. This course will be self-contained.