A detailed study of a forced chemical oscillator: Arnol’d tongues and bifurcation sets

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Abstract
We investigate in detail the dynamics of a time-periodically forced chemical oscillator in the parameter plane of forcing amplitude and forcing period. In particular, we present computed bifurcation sets for two typical cases of a forced, autonomously oscillating continuous stirred tank reactor system. The total mass flow rate j is used as the forcing variable by varying it sinusoidally in time about the autonomous system’s value. We find a wide variety of new nonlinear phenomena, including a global bifurcation structure—the skeletal bifurcation structure—that is common to the two cases presented and to other forced oscillator systems. The skeletal bifurcation structure is periodic along the forcing period axis and is mainly composed of the boundaries of Arnol’d tongues, which terminate at finite forcing amplitudes. In one of the cases studied, the invariant torus is destroyed between two critical curves and cascades of period doubling occur within the Arnol’d tongues; we relate this destruction of the torus to the discontinuous bifurcation that destroys the limit cycle in the autonomous system as j is varied.

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