Complex Dynamical Systems
The basic theme of dynamical systems theory is the study of a 'space of dynamical systems,' here understood as the space of vector fields over a manifold. What one would like to have is a classification of the systems in this space according to their qualitative behavior. Fortunately, this is a very difficult problem, so one starts with the less ambitious concepts of limit sets (or chain limit sets), behavior on limit sets (ergodic theory, chain recurrence), behavior around limit sets (attractors, repellers, linearlization, Lyapunov exponents), and global behavior (Morse decompositions, Morse index). Having analyzed a vector field in this way, one would like to know whether 'nearby' systems behave 'in a similar way,' i.e. one studies persistence and convergence properties (bifurcation theory, structural stability, perturbations).
Our work emphasizes dynamical systems over function spaces, as they occur in uncertain systems theory, control theory, or random systems. Several techniques from these area allow us to study the complex behavior of these systems (chaos, Lyapunov spectra, etc.) in a rather straightforward and unified manner, yielding new results on local and global behavior. As an example we show 'outer approximations' of the strange attractor in the Lorenz system above. (Supported by NSF, ONR, DFG, Fundacion Audes.)
- Collaborators
- Dr. Fritz Colonius, University of Augsburg, Germany
- Dr. Lars Grüne, University of Augsburg, Germany
- Dr. Gerhard Häckl, Germany
- Students
- Dr. Lisa Joseph
- Dr. Chung-Ming Ou
- Recent Publications
- Books
- Fink, A.M., R. K. Miller and W. Kliemann (eds.), Delay and Differential Equations, World
Scientific, Singapore, 1992, 166 pp.
- Kliemann, W. and N. S. Namachchivaya (eds.), Nonlinear Dynamics and Stochastic Mechanics,
CRC Press, Boca Raton, 1995, 530 pp.
- Kliemann, W., W. F. Lanford and N. S. Namachchivaya (eds.) Nonlinear Dynamics and Stochastic Mechanics, AMS Press, Fields Institute Communication Vol. 9, 1996, 238 pp.
- Chapters
- Kliemann, W., Analysis of Nonlinear Stochastic Systems, in Analysis and Estimation of
Stochastic Mechanical Systems, (Schiehlen, W. and W. Wedig, eds.), CISM Courses and
Lecture, No. 303, Springer, New York (1988), 43-102.
- Papers, refereed
- Colonius, F. and W. Kliemann, Stability radii and Lyapunov exponents, in: Control of
Uncertain Systems, D. Hinrichsen and B. Martensson (eds.), Birkhäuser (1990), 19-56.
- Colonius, F. and W. Kliemann, Minimal and maximal Lyapunov exponents of bilinear control
systems, J. Diff. Equations 101 (1993), 232-275.
- Colonius, F. and W. Kliemann, Limit Behavior and Genericity for Nonlinear Control Systems,
J. Diff. Equations 109 (1994), 8-41.
- Colonius, F. and W. Kliemann, A Dynamical Systems Approach to Control, in Proceedings of
IFAC NOLCOS '92, Bordeaux, France (1992), 361-367.
- Colonius, F., G. Häckl and W. Kliemann, Controllability near a Hopf bifurcation, Proceedings
of IEEE Conference and Decision and Control 1992, Tucson (1992), 2113-2118.
- Colonius, F. and W. Kliemann, Control properties of linear semigroups on projective spaces, J.
Dynamics and Differential Equations 5 (1993), 495-528.
- Colonius, F. and W. Kliemann, Some aspects of control systems as dynamical systems, J.
Dynamics and Differential Equations 5 (1993), 469-494.
- Colonius, F. and W. Kliemann, The Morse spectrum of linear flows on vector bundles, Transa.
AMS. 348(1996), 4355-4388.
- Colonius, F. and W. Kliemann, The Lyapunov spectrum of families of time-varying matrices,
Transa. AMS. 348(1996), 4389-4408.
- Colonius, F. and W. Kliemann, A stability radius for nonlinear differential equations subject to
time varying perturbations, in: Proceedings of IFAC NOLCOS '95, (1995) 44-46.
- Treinen, R., V. Vittal, and W. Kliemann, An improved technique to determine the controlling
unstable equilibrium point in a power system, IEEE Transactions on CS 43(1996), 313-323.
- Colonius, F. and W. Kliemann, Stability of time varying systems, in: Vibrations of Nonlinear,
Random, and Time-Varying Systems, ASME DE-Vol. 84-1, (1995), 365-373.
- Papers, not refereed
- Colonius, F., W. Kliemann, and S. Krull, Stability and stabilization of linear, uncertain systems
-A Lyapunov exponents approach, Report No. 372 of the Schwerpunktprogramm der Deutschen
Forschungsgemeinschaft 'Anwendungsbezogene Optimierung und Steuerung' (1992), 38 pp.
- Lin, S., V. Ajjarapu, B. Lee, V. Vittal, and W. Kliemann, Control of voltage collapse in an
electrical power system using center manifold theory, in: Proceedings of the Midwest Power
Symposium 95, 4 pp.