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Bifurcation structure of two coupled periodically driven double-well Duffing oscillators. (English) Zbl 1047.34042

Summary: The bifurcation structure of coupled periodically driven double-well Duffing oscillators is investigated as a function of the strength of the driving force \(f\) and its frequency \(\Omega\). We first examine the stability of the steady-state in linear response, and classify the different types of bifurcation likely to occur in this model. We then explore the complex behavior associated with these bifurcations numerically. Our results show many striking features from the behavior of coupled driven Duffing oscillators with single-well potentials, as characterized by J. Kozowski, U. Parlitz and W. Lauterborn [Phys. Rev. E 51, 1861–1867 (1995)]. In addition to the well-known routes to chaos already encountered in a one-dimensional Duffing oscillator, our model exhibits period-doubling of both types, symmetry-breaking, sudden chaos and a great abundance of Hopf bifurcations, many of which occur more than once for a given driving frequency. We explore the chaotic behavior of our model using two indicators, namely Lyapunov exponents and the power spectrum. Poincaré cross-sections and phase portraits are also plotted to show the manifestation of coexisting periodic and chaotic attractors including the destruction of \(T^2\) tori doubling.

MSC:

34C23 Bifurcation theory for ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
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