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

A prototype model is exploited to reveal the origin of mixed‐mode oscillations The initial oscillatory solution is born at a supercritical Hopf bifurcation and exhibits subsequent period doubling as some parameter is varied This period‐2 solution subsequently loses stability, but continues to exist−regaining stability to form the 11 mixed‐mode state (one large plus one small excursion) Other mixed‐mode states lie on isolated branches or ‘‘isolas’’ of limit cycles in the one‐parameter bifurcation diagram and are separated by regions of chaos As a second parameter is varied, the number of isola solutions increases and the ‘‘gaps’’ between them become narrower, leading to correspondingly more complete Devil’s staircases An exactly comparable scenario is shown to arise in the three variable model of the Belousov–Zhabotinsky reaction proposed recently by Gyorgyi and Field [Nature 335, 808 (1992)]

http://heracles.chem.wvu.edu/Garbage/current_papers/MixedMode.JCP.pdf

Mixed‐mode oscillations in chemical systems

The interaction of chemical waves propagating through capillary tubes is studied experimentally and numerically. Certain combinations of two or more tubes give rise to logic gates based on input and output signals in the form of chemical waves and wave initiations. The geometrical configuration, the temporal synchronization of the waves, and the ratio of the tube radius to the critical radius of the excitable medium determine the features of the logic gates.

http://heracles.chem.wvu.edu/papers/logicgates.jcp.pdf

Logic gates in excitable media

Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram

Our experiments on the manganese‐catalyzed Belousov–Zhabotinskii reaction in a stirred flow reactor reveal many sequences of distinct multipeaked periodic states. In the parameter ranges studied the waveform for each periodic state consists of an admixture of small and large amplitude oscillations. No chaos is discernible, and in many cases the transitions from one periodic state to another occur without any observable hysteresis. Two types of sequences were studied in detail, one with waveforms consisting of concatenations of two basic patterns and another with waveforms consisting of concatenations of three basic patterns. The sequences of states with two patterns are described well by Farey arithmetic, which provides rational approximations of irrational numbers. These states can be characterized by a firing number, the ratio of the number of small amplitude oscillations to the total number of oscillations per period. For our data this ratio is a monotone stepwise‐increasing function of flow rate, and the steps have a fractal dimension. The relationship between the observed sequence and the Farey arithmetic and the observation of a fractal dimension for the steps in the firing number suggest that the states formed by concatenating two patterns can be interpreted in terms of frequency locking on a 2 torus in phase space. The sequences of states with three basic patterns are described by a generalized Farey arithmetic that provides rational approximations for pairs of irrational numbers that are mutually irrational; this suggests that these states can be interpreted in terms of frequency locking on a 3 torus. A piecewise‐linear two‐dimensional map is shown to yield a phase diagram in qualitative accord with the measured phase diagram for these sequences.

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Papers

Complex periodic oscillations and Farey arithmetic in the Belousov–Zhabotinskii reaction