Devil's terraces: determining the organization of resonance tongues in a periodically forced dynamical system

In periodically forced dynamical systems, resonance tongues are open regions of a parameter plane in which the dynamics on an invariant torus locks to a stable periodic orbit. While individual resonance tongues are well understood, the principles governing their global arrangement remain largely unexplored. We develop a topological framework, grounded in applied topology and Morse theory, whose central object is the two-dimensional resonance surface, defined as the graph of the rotation number $\rho$ over a parameter plane. Within this framework, resonance tongues appear as terraces of the resonance surface at rational values of $\rho$, and their global arrangement is determined by the singularities of this surface. Resolving the resonance surface requires the accurate computation of $\rho$, and we present an algorithm that does so efficiently and at high resolution. As a specific example, we examine a periodically forced model of vertical mixing in the North Atlantic, a process relevant to the Atlantic Meridional Overturning Circulation, and study how its resonance surface changes under variation of a third parameter. We identify six distinct resonance-tongue arrangements and show that the resonance transitions between them are due to changes in the number and type of singularities on the boundary of the resonance surface.