Multimodal oscillations in systems with strong contraction
The authors consider a 3-dimension system of ordinary differential equations, which is obtained by a finite dimensional approximation for the interface dynamics in a free boundary problem modeling solid fuel combustion. By varying two system parameters, Hopf bifurcation boundaries with both super- and subcritical postbifurcation behaviour are found. In the supercritical regime a series of periodic doubling bifurcations is observed, where the double frequency is primarly generated by the center manifold reduction for the stable mode. In the subcritical case a Shilnikov-like scenario is found with multimodal solutions and arbitrarily long intervals between successive large amplitude oscillations. Also a chaotic attractor followed by a reverse period-doubling cascade is numerically observed in this parameter region. The existence of these solutions is analytically established by constructing an appropriate return map close to the unstable stationary point.
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