Math 6063001, Fall 2005
Nonlinear Dynamical Systems
http://www.nd.edu/~malber/Math611.html
MWF 3:003:50pm, DBRT 231
Instructor: Mark Alber, 136 HayesHealy,
malber@nd.edu, 6318371
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 aspectsthat 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
selfcontained.
Final
Grades will be based on a total of 550 points,
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.
PoincareMelnikov 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, AddisonWesley 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
