Lighting Arnold Flames: Resonance in Doubly Forced Periodic Oscillators

We study doubly forced nonlinear planar oscillators: ẋ = V(x) + α1W1(x, ω1t) + α2W2(x, ω2t), whose forcing frequencies have a fixed rational ratio: ω1 = m n ω2. After some changes of parameter, we arrive at the form we study: ẋ = V(x) + α{(2 − γ)W1(x, m ω0 β t) + (γ − 1)W2(x, n ω0 β t)}. We assume ẋ = V(x) has an attracting limit cycle — the unforced planar oscillator — with frequency ω0, and the two forcing functions W1 and W2 are period one in their second variables. We consider two parameters as primary: β, an appropriate multiple of the forcing period, and α, the forcing amplitude. The relative forcing amplitude γ ∈ [1, 2] is treated as an auxiliary parameter. The dynamics are studied by considering the stroboscopic maps induced by sampling the solutions of the differential equations at time intervals equal to the period of forcing, T = β ω0 . For any fixed γ, these oscillators have a standard form of a periodically forced oscillator, and thus exhibit the Arnold resonance tongues in the primary parameter plane. The special forms at γ = 1 and γ = 2 can introduce certain symmetries into the problem. One effect of these symmetries is to provide a relatively natural example of oscillators with multiple attractors. Such oscillators typically have interesting bifurcation features within corresponding resonance regions — features we call “Arnold flames” because of their flamelike appearance in the corresponding bifurcation diagrams. By changing the auxiliary parameter γ we “melt” one singly forced oscillator bifurcation diagram into another, and in the process we control certain of these “intraresonance region” bifurcation features.