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Please use this identifier to cite or link to this item: http://hdl.handle.net/1860/1642

Title: Multimodal oscillations in systems with strong contraction
Authors: Medvedev, Georgi
Yoo, Yun
Issue Date: 15-Jan-2007
Publisher: Elsevier Science B.V.
Citation: Physica D: Nonlinear Phenomena, 228(2): pp. 87-106. Retrieved June 21, 2007 from http://www.math.drexel.edu/~medvedev/papers/.
Abstract: One- and two-parameter families of flows in IR3 near an Andronov-Hopf bifurcation (AHB) are investigated in this work. We identify conditions on the global vector field, which yield a rich family of multimodal orbits passing close to a weakly unstable saddle-focus and perform a detailed asymptotic analysis of the trajectories in the vicinity of the saddle-focus. Our analysis covers both cases of sub- and supercritical AHB. For the supercritical case, we find that the periodic orbits born from the AHB are bimodal when viewed in the frame of coordinates generated by the linearization about the bifurcating equilibrium. If the AHB is subcritical, it is accompanied by the appearance of multimodal orbits, which consist of long series of nearly harmonic oscillations separated by large amplitude spikes. We analyze the dependence of the interspike intervals (which can be extremely long) on the control parameters. In particular, we show that the interspike intervals grow logarithmically as the boundary between regions of sub- and supercritical AHB is approached in the parameter space. We also identify a window of complex and possibly chaotic oscillations near the boundary between the regions of suband supercritical AHB and explain the mechanism generating these oscillations. This work is motivated by the numerical results for a finite-dimensional approximation of a free boundary problem modeling solid fuel combustion.
URI: http://dx.doi.org/10.1016/j.physd.2007.02.011
Appears in Collections:Faculty Research and Publications (Mathematics)

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