Amplitude-modulated spiking as a novel route to bursting: Coupling-induced mixed-mode oscillations by symmetry breaking.

Mixed-mode oscillations consisting of alternating small- and large-amplitude oscillations are increasingly well understood and are often caused by folded singularities, canard orbits, or singular Hopf bifurcations. We show that coupling between identical nonlinear oscillators can cause mixed-mode oscillations because of symmetry breaking. This behavior is illustrated for diffusively coupled FitzHugh-Nagumo oscillators with negative coupling constant, and we show that it is caused by a singular Hopf bifurcation related to a folded saddle-node (FSN) singularity. Inspired by earlier work on models of pancreatic beta-cells [Sherman, Bull. Math. Biol. 56, 811 (1994)], we then identify a new type of bursting dynamics due to diffusive coupling of cells firing action potentials when isolated. In the presence of coupling, small-amplitude oscillations in the action potential height precede transitions to square-wave bursting. Confirming the hypothesis from the earlier work that this behavior is related to a pitchfork-of-limit-cycles bifurcation in the fast subsystem, we find that it is caused by symmetry breaking. Moreover, we show that it is organized by a FSN in the averaged system, which causes a singular Hopf bifurcation. Such behavior is related to the recently studied dynamics caused by the so-called torus canards.

Stabilization of vortices in the wake of a circular cylinder using harmonic forcing.

We explore whether vortex flows in the wake of a fixed circular cylinder can be stabilized using harmonic forcing. We use Föppl's point vortex model augmented with a harmonic point source-sink mechanism which preserves conservation of mass and leaves the system Hamiltonian. We discover a region of Lyapunov-stable vortex motion for an appropriate selection of parameters. We identify four unique parameters that affect the stability of the vortices: the uniform flow velocity, vortex equilibrium positions, forcing amplitude, and forcing frequency. We assess the robustness of the controller using a Poincaré section.

Canard explosion of limit cycles in templator models of self-replication mechanisms.

Templators are differential equation models for self-replicating chemical systems. Beutel and Peacock-López [J. Chem. Phys. 126, 125104 (2007)] have numerically analyzed a model for a cross-catalytic self-replicating system and found two cases of canard explosion, that is, a substantial change of amplitude of a limit cycle over a very short parameter interval. We show how the model can be reduced to a two-dimensional system and how canard theory for slow-fast equations can be applied to yield analytic information about the canard explosion. In particular, simple expressions for the parameter value where the canard explosion occurs are obtained. The connection to mixed-mode oscillations also observed in the model is briefly discussed.

Bifurcation analysis in a vortex flow generated by an oscillatory magnetic obstacle.

A numerical simulation and a theoretical model of the two-dimensional flow produced by the harmonic oscillation of a localized magnetic field (magnetic obstacle) in a quiescent viscous, electrically conducting fluid are presented. Nonuniform Lorentz forces produced by induced currents interacting with the oscillating magnetic field create periodic laminar flow patterns that can be characterized by three parameters: the oscillation Reynolds number, Reomega, the Hartmann number, Ha, and the dimensionless amplitude of the magnetic obstacle oscillation, D. The analysis is restricted to oscillations of small amplitude and Ha=100. The resulting flow patterns are described and interpreted in terms of position and evolution of the critical points of the instantaneous streamlines. It is found that in most of the cycle, the flow is dominated by a pair of counter rotating vortices that switch their direction of rotation twice per cycle. The transformation of the flow field present in the first part of the cycle into the pattern displayed in the second half occurs via the generation of hyperbolic and elliptic critical points. The numerical solution of the flow indicates that for low frequencies (v.e. Reomega=1), two elliptic and two hyperbolic points are generated, while for high frequencies (v.e. Reomega=100), a more complex topology involving four elliptic and two hyperbolic points appear. The bifurcation map for critical points of the instantaneous streamline is obtained numerically and a theoretical model based on a local analysis that predicts most of the qualitative properties calculated numerically is proposed.

Canards and mixed-mode oscillations in a forest pest model.

We consider a three-variable forest pest model, proposed by Rinaldi & Muratori (1992) [Rinaldi, S., Muratori, S., 1992. Limit cycles in slow-fast forest-pest models. Theor. Popul. Biol. 41, 26-43]. The model allows relaxation oscillations where long pest-free periods are interspersed with outbreaks of high pest concentration. For small values of the timescale of the young trees, the model can be reduced to a two-dimensional model. By a geometrical analysis we identify a canard explosion in the reduced model, that is, a change over a narrow parameter interval from outbreak dynamics to small oscillations around an endemic state. For larger values of the timescale of the young trees the two-dimensional approximation breaks down, and a broader parameter interval with mixed-mode oscillations appear, replacing the simple canard explosion. The analysis only relies on simple and generic properties of the model, and is expected to be applicable in a larger class of multiple timescale dynamical models.

Introduction to focus issue: mixed mode oscillations: experiment, computation, and analysis.

Mixed mode oscillations (MMOs) occur when a dynamical system switches between fast and slow motion and small and large amplitude. MMOs appear in a variety of systems in nature, and may be simple or complex. This focus issue presents a series of articles on theoretical, numerical, and experimental aspects of MMOs. The applications cover physical, chemical, and biological systems.