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Pages Tasso J. Kaper, Christopher K.

Multiple-Time-Scale Dynamical Systems | Christopher K.R.T. Jones | Springer

The Phenomenon of Delayed Bifurcation and its Analyses. Synchrony in Networks of Neuronal Oscillators. Metastable dynamics and exponential asymptotics in multi-dimensional domains. Back Matter Pages This strengthens the results of Cartwright and Littlewood and completes research begun by Smale to give a geometric characterization of the dynamics of this system. The chaotic dynamics of the forced van der Pol system described above occur in invariant sets that are not attractors. Indeed, these invariant sets are highly unstable and elusive to approximate with numerical simulations.

The trajectories comprising the invariant sets contain canard segments that lie along unstable sheet of the slow manifold. Numerical solutions which come close to the unstable sheet of the slow manifold fly away from it on the fast time scale of the system. Even round-off errors are amplified fast enough by the instability to prevent trajectories from remaining close to the unstable sheet for a substantial distance. There is a second reason why the chaotic invariant sets in the forced van der Pol system are difficult to display. The invariant sets discovered by Cartwright and Littlewood are not attractors.

This means that trajectories with typical initial conditions do not tend to the invariant set. In the example described by Haiduc, almost all trajectories tend to periodic orbits. Numerical simulations of these examples suggested that there are large sets of parameters for which the maps do have chaotic attractors.


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An extensive body of research has now verified this observation. Qiudong Wang and Lai-Sang Young formulated a list of hypotheses that are sufficient to prove that families of two-dimensional maps satisfying these properties have positive measure sets of parameters for which the maps have chaotic attractors. Thus, there is a positive probability that a map with randomly chosen parameters in one of these families will have a chaotic attractor.

Kathleen Hoffman, Warren Weckesser and I investigated a singularly pertubed system with two slow and two fast variables that is a caricature of a pair of neurons coupled by reciprocal inhibition. We used the computer program AUTO to investigate periodic solutions in this system. The results of this investigation are intriguing and perplexing. With one parameter allowed to vary, AUTO computes a continuous family of periodic orbits.

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