Greene’s residue criterion for the breakup of invariant tori of volume-preserving maps

Invariant tori play a fundamental role in the dynamics of symplectic and volume-preserving maps. Codimension-one tori are particularly important as they form barriers to transport. Such tori foliate the phase space of integrable, volume-preserving maps with one action and d angles. For the area-preserving case, Greene’s residue criterion is often used to predict the destruction of tori from the properties of nearby periodic orbits. Even though KAM theory applies to the three-dimensional case, the robustness of tori in such systems is still poorly understood. We study a three-dimensional, reversible, volume-preserving analogue of Chirikov’s standard map with one action and two angles. We investigate the preservation and destruction of tori under perturbation by computing the “residue ” of nearby periodic orbits. We find tori with Diophantine rotation vectors in the “spiral mean ” cubic algebraic field. The residue is used to generate the critical function of the map and find a candidate for the most robust torus.