Bifurcation in Dynamical Systems with Multiple Time Scales

Cornell University, Ithaca NY

Investigators

Abstract

Guckenheimer 0101208 The investigator studies slow-fast decompositions and bifurcations of trajectories in dynamical systems with multiple time scales. This extends the theory of bifurcation in generic families of dynamical systems to those with two time scales. Emphasis is placed upon relaxation oscillations, periodic orbits that have both slow and fast segments. The initial stages of the work seek a classification of degenerate decompositions appearing in periodic orbits of one parameter families with relaxation oscillations. Geometric methods are used to determine bifurcations associated with each degenerate decomposition. Numerical investigation of examples is used to motivate the work and to ensure that the results are directly applicable to biological models. Together with collaborators Kathleen Hoffman and Warren Weckesser, the investigator is reexamining a classical example, the forced van der Pol system that gave birth to the discovery of chaos for dissipative dynamical systems, and expects to give a full description of the bifurcations that occur within this system. He also develops algorithms for the computation of structures that are difficult to compute with existing methods. Rhythmic phenomena are ubiquitous in biological systems. Examples include the heartbeat, the cell cycle, circadian rhythms, legged locomotion, and electrical signals in the nervous system. Most of these involve multiple time scales. The investigator pursues new mathematical theory and computational methods that apply to dynamical systems with multiple time scales. Emphasis is given to models of neural systems, an area in which the presence and importance of complex dynamics are manifest. The investigations draw upon decades of research in characterizing generic phenomena observed in dynamical systems with a single time scale. The resulting body of mathematics, sometimes called chaos theory, needs extension and modification to fully explain the behavior of systems with multiple time scales. Those extensions are the goal of this project. On a longer time frame, the project lays foundations for coming generations of biological models for cellular processes such as gene expression and signal transduction. New biotechnology leads to ever more complicated reaction networks that are simulated as dynamical systems. This project produces results that aid the implementation and interpretation of such simulations of complex, multiple time scale dynamical systems.