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.

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Mixed-Mode Oscillations with Multiple Time Scales

Chemical reactions with nonlinear kinetic behavior can give rise to a remarkable set of spatiotemporal phenomena. These include periodic and chaotic changes in concentration, traveling waves of chemical reactivity, and stationary spatial (Turing) patterns. Although chemists were initially skeptical of the existence and the relevance of these phenomena, much progress has been made in the past two decades in characterizing, designing, modeling, and understanding them. Several nonlinear dynamical phenomena in chemical systems provide simpler analogues of behaviors found in biological systems.

http://hopf.chem.brandeis.edu/pubs/pub236%20rep.pdf

Nonlinear Chemical Dynamics: Oscillations, Patterns, and Chaos

Electrochemical Reaction Dynamics - A Review

Oscillation quenching mechanisms: Amplitude vs. oscillation death

Topological and phenomenological classification of bursting oscillations

Experiments have been carried out on the Belousov–Zhabotinskii reaction in an isothermal continuous stirred tank reactor. The flow rate was varied from low values where simple single peak oscillations occur to large values where the reactor is in steady state. Several types of periodic multipeak oscillations were observed. Three regions of nonperiodic behavior were found, each of which occurred in a well defined range of flow rates bracketed by periodic oscillations.

D. Marinko

Papers

An experimental study of multiple peak periodic and nonperiodic oscillations in the Belousov–Zhabotinskii reaction