DYNAMICAL SYSTEMS, NONLINEAR PHENOMENA and CHAOS

MAGN 421/MAGN 510

Fall 1995, 9:30-10:45 Tuesday and Thursday, Alderson Hall 162

Instructor: Erik S. Van Vleck, Stratton 216, x3553

Topics

Existence, uniqueness and continuous dependence of solutions.
Differential inequalities and the implicit function theorem.
Linear systems and stability from linearization.
Mappings on the interval and the plane.
Period doubling bifurcations, the Feigenbaum cascade and chaos.
Saddle-node, pitchfork and Hopf bifurcations.
Lyapunov functions and the invariance principle.
Imaginary eigenvalues and center manifolds.
Periodic orbits and the Poincare-Bendixson theorem.
Conservative, gradient and dissipative systems.
Lyapunov exponents, entropy and dimension.

Text

J. K. Hale and H. Kocak, ``Dynamics and Bifurcations,'' Springer-Verlag (1992).

References

1. E. A. Coddington and N. Levinson, ``Theory of Ordinary Differential
Equations,'' McGraw-Hill (1955).
2. P. Hartman, ``Ordinary Differential Equations,'' Wiley (1964).
3. J. K. Hale, ``Ordinary Differential Equations,'' Krieger (1980).
4. J. Guckenheimer and P. Holmes, ``Nonlinear Oscillations,
Dynamical Systems, and Bifurcations of Vector Fields,'' Springer-Verlag
(1983).
5. R. L. Devaney, ``An Introduction to Chaotic Dynamical Systems,''
Benjamin/Cummings (1986).
6. C. Robinson, ``Dynamical Systems: Stability, Symbolic Dynamics,
and Chaos,'' CRC Press (1995).
7. D. K. Arrowsmith and C. M. Place, ``An Introduction to
Dynamical Systems,'' Cambridge (1990).
8. E. Ott, ``Chaos in Dynamical Systems,'' Cambridge (1993).
9. P. G. Drazin, ``Nonlinear Systems,'' Cambridge (1992).
10. D. Ruelle, ``Chaotic Evolution and Strange Attractors,'' Cambridge (1989).
11. P. Collet and J.-P. Eckmann, ``Iterated Maps on the Interval as Dynamical
Systems,'' Birkhauser (1980).
12. M. Barnsley, ``Fractals Everywhere,'' Academic Press (1988).
13. H.-O. Peitgen and P. H. Richter, ``The Beauty of Fractals,'' Springer (1986).

Computer Packages: (PHASER, DYNAMICS, KAOS, etc.)