Stochastic Systems
Two basis approaches describe the behavior of dynamical systems under perturbations: The deterministic and the stochastic approach. The deterministic model is used to account for all uncertainties within a given range (robustness, see Control of Dynamical Systems), while the stochastic model assumes that certain statistical characteristics of the perturbation are known. The problem is then to study the random response behavior of the system under these perturbations. our group is interested in stochastic systems that can be modeled as Markov diffusion processes or as stochastic flows. In both cases, so-called support theorems provide a powerful connection to the study of control systems, which allow us to obtain results on the global behavior of stochastic systems. Numerical algorithms are developed for the simulation and application of the theory. The figures above indicate the coexistence region for a population of two interacting species and the probability of extinction of the prey. (Supported by NSF, ONR, DFG, Fundacion Audes, DGICT)
- Collaborators
- Dr. Fritz Colonius, University of Augsburg, Germany
- Dr. Gerhard Häckl, Germany
- Dr. Helmut Pradlwarter, University of Innsbruck, Austria
- Dr. Javier de la Rubia, UNED, Madrid, Spain
- Students
- Dr. Patrick Homblé
- Dr. Kaisheng Fau
- Dr. Brian O'Donnel
- Dr. Ruey-Gang Lai
- Recent Publications
- Books
- 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.
- 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.
- Kliemann, W., G. Koch and F. Marchetti, On the unnormalized solution of the filtering problem
with counting process observations, IEEE Trans. IT 36 (1990), 1415-1425.
- Colonius, F. and W. Kliemann, Remarks on Ergodic Theory of Stochastic Flows and Control
Flows, in: Diffusion Processes and Related Problems in Analysis, Vol. II, (M. Pinsky and V.
Wihstutz, eds.), Birkhäuser (1992), 203-240.
- Colonius, F. and W. Kliemann, Controlling the Dynamics of a Random System, in: Nonlinear
Stochastic Mechanics (N. Bellomo, F. Casciati, eds.), Springer (1992), 333-346.
- Colonius, F. and W. Kliemann, Random perturbations of bifurcation diagrams, Nonlinear
Dynamics 5 (1994), 353-373.
- Colonius, F. and W. Kliemann, Reliability assessment of dynamical systems with random
excitation, Proceedings of 32nd IEEE Conference on Decision and Control 1993, San Antonio
(1993), 3879-3884.
- Colonius, F. and W. Kliemann, Local robust stabilization of nonlinear oscillators under
parametric excitation, in: Stochastic Dynamics and Reliability of Nonlinear Ocean Systems (R.
A. Ibrahim, Y. K. Lin, eds.), ASME DE-Vol. 77, (1994), 1-5.
- Colonius, F., F. J. de la Rubia, and W. Kliemann, Stochastic models with multistability and
extinction levels, SIAM J. Applied Mathematics 56(1996), 919-945.
- Colonius, F., G. Häckl, and W. Kliemann, Dynamic reliability of nonlinear systems under
random excitation, in: Vibrations of Nonlinear, Random, and Time-Varying Systems, ASME
DE-Vol. 84-1, (1995), 1007-1024.
- Kliemann, W., Nonlinear time series - bifurcation, chaos, and stationarity, Modelling and
Prediction Honoring Seymour Geisser, J. C. Lee, W. O. Johnson, and A. Zellner (eds),
Springer (1996), 389-401.
- Papers, not refereed
- Colonius, F. and W. Kliemann, Lyapunov exponents in nonlinear stochastic dynamics,
Proceedings of 32nd Annual Technical Meeting of the Society of Engineering Science, New
Orleans, Oct. 1995, 273-275.
- Pradlwarter, H. J., and W. Kliemann, First exit times in nonlinear dynamical systems by
advanced Monte Carlo simulation, Proceedings of the 11th Conference on Engineering
Mechanics, ASCE, Fort Lauderdale, May 1996, Vol. I, 523-526.