Mixed‐mode oscillations in chemical systems

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)].

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