Computing the Conjugacy of Invariant Tori for Volume-Preserving Maps

In this paper we implement a numerical algorithm to compute codimension-one tori in three-dimensional, volume-preserving maps. A torus is defined by its conjugacy to rigid rotation, which is in turn given by its Fourier series. The algorithm employs a quasi-Newton scheme to find the Fourier coefficients of a truncation of the series. We show that this method converges for tori of two example maps by continuation from an integrable case, and discuss the scaling of computational resources required for accurate computations. We demonstrate that the growth of the largest singular value of the derivative of the conjugacy predicts the threshold for the destruction of the torus. We use these singular values to examine the mechanics of the breakup of tori, identifying its onset with the formation of “spires” or “streaks” in the local singular values on the tori. These are analogous to the gaps in cantori of two-dimensional twist maps.