Lighting Arnold flames: resonance in doubly forced periodic oscillators

Abstract

We study doubly forced nonlinear planar oscillators:whose forcing frequencies have a fixed rational ratio: ω1 = (m/n)ω2.After some changes of parameter, we arrive at the form we study:We assume has an attracting limit cycle - the unforced planar oscillator - withfrequency ω0, and the twoforcing functions W1 and W2 are period one in their second variables.We consider two parameters as primary: β, an appropriate multipleof the forcing period, and α, the forcing amplitude.The relative forcing amplitude γ∊[1,2] is treated as an auxiliary parameter. The dynamics is studied by considering the stroboscopicmaps induced by sampling the solutions of the differential equationsat time intervals equal to the period of forcing,T = β/ω0. For any fixed γ,these oscillators have a standard form of a periodicallyforced oscillator, and thus exhibit the Arnold resonance tonguesin the primary parameter plane. The special forms at γ = 1and γ = 2 can introduce certain symmetries into the problem. One effect of thesesymmetries is to provide a relatively natural example ofoscillators with multiple attractors. Such oscillators typically haveinteresting bifurcation features within corresponding resonanceregions - features we call `Arnold flames' because of their flame-like appearance in the corresponding bifurcation diagrams. By changing the auxiliary parameter γ, we `melt' onesingly forced oscillator bifurcation diagram into another, and in the process we controlcertain of these `intraresonance region' bifurcation features.