Abstract. Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale "mechanism." Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system.

Many systems in science and engineering can be modelled as coupled or forced nonlinear oscillators, which may possess quasi-periodic or phase-locked invariant tori. Since there exist routes to chaos involving the breakdown of invariant tori, these phenomena attract considerable attention. This paper presents a new algorithm for the computation and continuation of quasiperiodic invariant tori of ordinary differential equations that is based on a natural parametrisation of such tori. Since this parametrisation is uniquely defined, the proposed method requires neither the computation of a base of a transversal bundle, nor re-meshing during continuation. It is independent of the stability type of the torus and examples of attracting and saddle-type tori are given. The algorithm is robust in the sense that it can compute approximations to weakly resonant tori. The performance of the method is demonstrated with examples.

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Abstract. Many applications give rise to dynamical systems in the form of a vector field with a phase space of moderate dimension. Examples are the Lorenz equations, mechanical and other oscillators, and models of spiking neurons. The global dynamics of such a system is organized by the stable and unstable manifolds of the saddle points, of the saddle periodic orbits, and, more generally, of all compact invariant manifolds of saddle type. Except in very special circumstances the (un)stable manifolds are global objects that cannot be found analytically but need to be computed numerically. This is a nontrivial task when the dimension of the manifold is larger than one. In this paper we present an algorithm to compute the k-dimensional unstable manifold of an equilibrium or periodic orbit (or a more general normally hyperbolic invariant manifold) of a vector field with an n-dimensional phase space, where 1 < k < n. Stable manifolds are computed by considering the flow for negative time. The key idea is to view the unstable manifold as a purely geometric object, hence disregarding the dynamics on the manifold, and compute it as a list of approximate geodesic level sets, which are (topological) (k − 1)-spheres. Starting from a (k − 1)-sphere in the linear eigenspace of the equilibrium or periodic orbit, the next geodesic level set is found in a local (and changing) coordinate system given by hyperplanes perpendicular to the last geodesic level set. In this setup the mesh points defining the approximation of the next geodesic level set can be found by solving boundary value problems. By appropriately adding or removing mesh points it is ensured that the mesh that represents the computed manifold is of a prescribed quality.The algorithm is presently implemented to compute two-dimensional manifolds in a phase space of arbitrary dimension. In this case the geodesic level sets are topological circles and the manifold is represented as a list of bands between consecutive level sets. We use color to distinguish between consecutive bands or to indicate geodesic distance from the equilibrium or periodic orbit, and we also show how geodesic level sets change with increasing geodesic distance. This is very helpful when one wants to understand the often very complicated embeddings of two-dimensional (un)stable manifolds in phase space.The properties and performance of our method are illustrated with several examples, including the stable manifold of the origin of the Lorenz system, a two-dimensional stable manifold in a fourdimensional phase space arising in a problem in optimal control, and a stable manifold of a periodic orbit that is a Möbius strip. Each illustration is accompanied by an animation (supplied with this paper).

We describe an efficient algorithm for computing two-dimensional stable and unstable manifolds of three-dimensional vector fields. Larger and larger pieces of a manifold are grown until a sufficiently long piece is obtained. This allows one to study manifolds geometrically and obtain important features of dynamical behavior. For illustration, we compute the stable manifold of the origin spiralling into the Lorenz attractor, and an unstable manifold in 3 -model converging to an attracting limit cycle. © 1999 American Institute of Physics. ͓S1054-1500͑99͒02403-9͔The computation of stable and unstable manifolds for vector fields is of direct practical use in applications, because these manifolds form boundaries of basins of attraction and their intersection gives rise to chaotic dynamics. So far only one-dimensional manifolds of vector fields are readily computable, which limits the study to two-dimensional vector fields. However, many interesting systems, for example the well-known Lorenz and Chua systems, are three-dimensional vector fields. In this paper we are concerned with the technically challenging case of computing two-dimensional stable and unstable manifolds of three-dimensional vector fields. Combined with visualization techniques, this constitutes a new powerful tool for a complete exploration of the dynamical behavior of three-dimensional physical models. Our method starts near a saddle point or periodic orbit and grows the manifold as a sequence of closed curves until a prespecified arclength is reached. We take special care of maintaining a prescribed mesh quality, which is achieved by solving suitable boundary value problems.

Plateau bursting is typical of many electrically excitable cells, such as endocrine cells that secrete hormones and some types of neurons that secrete neurotransmitters. Although in many of these cell types the bursting patterns are regulated by the interplay between voltage-gated calcium channels and calcium-sensitive potassium channels, they can be very different. We investigate so-called square-wave and pseudo-plateau bursting patterns found in endocrine cell models that are characterized by a super- or subcritical Hopf bifurcation in the fast subsystem, respectively. By using the polynomial model of Hindmarsh and Rose (Proceedings of the Royal Society of London B 221(1222), 87–102), which preserves the main properties of the biophysical class of models that we consider, we perform a detailed bifurcation analysis of the full fast-slow system for both bursting patterns. We find that both cases lead to the same possibility of two routes to bursting, that is, the criticality of the Hopf bifurcation is not relevant for characterizing the route to bursting. The actual route depends on the relative location of the full-system’s fixed point with respect to a homoclinic bifurcation of the fast subsystem. Our full-system bifurcation analysis reveals properties of endocrine bursting that are not captured by the standard fast-slow analysis.

This paper is concerned with the geometry of slow manifolds of a dynamical system with two slow and one fast variable. Specifically, we study the dynamics near a folded node singularity, which is known to give rise to so-called canard solutions. Geometrically, canards are intersection curves of two-dimensional attracting and repelling slow manifolds, and they are a key element of slow-fast dynamics. For example, canard solutions are associated with mixed-mode oscillations, where they organize regions with different numbers of small oscillations. We perform a numerical study of the geometry of two-dimensional slow manifolds in the normal form of a folded node in R 3. Namely, we view the part of a slow manifold that is of interest as a one-parameter family of orbit segments up to a suitable cross-section. Hence, it is the solution of a two-point boundary value problem, which we solve by numerical continuation with the package AUTO. The computed family of orbit segments is used to obtain a mesh representation of the manifold as a surface. With this approach we show how the attracting and repelling slow manifolds change in dependence on the eigenvalue ratio µ of the reduced flow. At µ = 1 two primary canards bifurcate and secondary canards are created at odd integer values of µ. We compute 24 secondary canards to investigate how they spiral more and more around one of the primary canards. The first twelve secondary canards are continued in µ to obtain a numerical bifurcation diagram.