Arnold Flames and Resonance Surface Folds

Abstract

Periodically forced planar oscillators are often studied by varying the two parameters of forcing amplitude and forcing frequency. For low forcing amplitudes, the study of the essential oscillator dynamics can be reduced to the study of families of circle maps. The primary features of the resulting parameter plane bifurcation diagrams are "(Arnold) resonance horns" emanating from zero forcing amplitude. Each horn is characterized by the existence of a periodic orbit with a certain period and rotation number. In this paper we investigate divisions of these horns into subregions -different subregions corresponding to maps having different numbers of periodic orbits. The existence of subregions having more than the "usual" one pair of attracting and repelling periodic orbits implies the existence of "extra folds" in the corresponding surface of periodic points in the cartesian product of the phase and parameter spaces. The existence of more than one attracting and one repelling periodic orbit is shown to be generic. For some of the families we create, the resulting parameter plane bifurcation pictures appear in shapes we call "Arnold flames." Results apply both to circle maps and forced oscillator maps.

Computing Arnol′d tongue scenarios

“…This would allow us to visualise the full geometry of these resonance surfaces in a three-dimensional space. See a related study in [29].…”

“…In contrast, quasi-periodicity is a codimension-one phenomenon which is thus generically destroyed by perturbation. The result is a well-known bifurcation diagram in the two-parameter plane called the "Arnol d tongue" scenario [1,2,3,4,5,6,14,17,18,21,26,28,29,37]. It has a countable collection of Arnol d tongues, emanating from "rational" points on the zero forcing/coupling axis, and opening up into regions where the coupling strength is turned on.…”

Geometry and dynamics of mildly degenerate Hopf–Neĭmarck–Sacker families near resonance

“…The research programme of Peckham et al reflected in [24,25,[30][31][32] views resonance sets as projections on a 'traditional' parameter plane of saddle-node bifurcation sets in the product of parameter and phase space. This approach has the same spirit as ours and many interesting geometric properties of resonance sets are discovered and explained in this way.…”

“…Moreover, the condition sign(σ 2 ) = −sign(τ 2 ) implies that the stratum of type A 2 is attached to a stratum of type ∂A 2 given in (25).…”

“…This parabola gets narrower as τ 2 tends to 0, so comparing with the Whitney umbrella we conclude that for (24)). These points only occur for τ 1 < 0, because u > 0, so the self-intersection points (A 2 2 ) are given by (25).…”