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Dynamical Systems

Math 60-630-01, Fall 2005

Nonlinear Dynamical Systems

http://www.nd.edu/~malber/Math611.html

MWF 3:00-3:50pm, DBRT 231

Instructor: Mark Alber, 136 Hayes-Healy, malber@nd.edu, 631-8371

 

Theory of nonlinear dynamical systems has applications to a wide variety of fields, from mathematics, 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.

 

Final Grades will be based on a total of 550 points,

distributed as follows:

Exam 1 - 100 points;

Exam 2 - 100 points;

Final - 150 points;

Homework - 100 points;

Projects: 100 points.

Syllabus

Introduction: Review of the linear and nonlinear dynamical systems. Examples: Duffing’s, Van der Pol’s and Lorentz systems. Geometry of the phase space. Variational methods. Symplectic structure. Nonlinear Hamiltonian systems. Integrable systems. Quasiperiodic motion. Averaging method. Discrete dynamical systems. The logistic map.

 

Bifurcation phenomena: Hamiltonian vector fields. Normal forms. Stable and unstable manifolds. Structural stability. Poincare maps. Liapunov exponents. Power spectra. Classification of local and global bifurcations. Strange attractors and basins of attraction. KAM theory.

 

Transition to chaos: Symbolic dynamics. Smale horseshoe map and shift map. Mathematical definition of chaos. Perturbation of homoclinic orbits. Poincare-Melnikov method. Numerical   route to chaos. Stochastic dynamical systems. Chaotic transitions in stochastic dynamical systems. Stochastic resonance.

 Examples from physics, biology and engineering.

References

John Guckenheimer and Philip Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer Verlag; Revised edition (February 20, 1997)

Robert L. Devaney, An Introduction to Chaotic Dynamical Systems, Second edition, Addison-Wesley Publ. Co. (1995)

Steven H. Strogatz, Nonlinear Dynamics and Chaos: With Applications in Physics, Biology, Chemistry, and Engineering (Studies in Nonlinearity), Perseus Publishing (1994)

 

Links of Interest