16 research outputs found
From random Poincar\'e maps to stochastic mixed-mode-oscillation patterns
We quantify the effect of Gaussian white noise on fast--slow dynamical
systems with one fast and two slow variables, which display mixed-mode
oscillations owing to the presence of a folded-node singularity. The stochastic
system can be described by a continuous-space, discrete-time Markov chain,
recording the returns of sample paths to a Poincar\'e section. We provide
estimates on the kernel of this Markov chain, depending on the system
parameters and the noise intensity. These results yield predictions on the
observed random mixed-mode oscillation patterns. Our analysis shows that there
is an intricate interplay between the number of small-amplitude oscillations
and the global return mechanism. In combination with a local saturation
phenomenon near the folded node, this interplay can modify the number of
small-amplitude oscillations after a large-amplitude oscillation. Finally,
sufficient conditions are derived which determine when the noise increases the
number of small-amplitude oscillations and when it decreases this number.Comment: 56 pages, 14 figures; revised versio
Dynamical Systems; Proceedings of an IIASA Workshop, Sopron, Hungary, September 9-13, 1985
The investigation of special topics in systems dynamics -- uncertain dynamic processes, viability theory, nonlinear dynamics in models for biomathematics, inverse problems in control systems theory -- has become a major issue at the System and Decision Sciences Research Program of IIASA.
The above topics actually reflect two different perspectives in the investigation of dynamic processes. The first, motivated by control theory, is concerned with the properties of dynamic systems that are stable under variations in the systems' parameters. This allows us to specify classes of dynamic systems for which it is possible to construct and control a whole "tube" of trajectories assigned to a system with uncertain parameters and to resolve some inverse problems of control theory within numerically stable solution schemes.
The second perspective is to investigate generic properties of dynamic systems that are due to nonlinearity (as bifurcations theory, chaotic behavior, stability properties, and related problems in the qualitative theory of differential systems). Special stress is given to the applications of nonlinear dynamic systems theory to biomathematics and ecology.
The proceedings of a workshop on the "Mathematics of Dynamic Processes", dealing with these topics is presented in this volume