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University of Colorado

1. Fox, Adam Merritt. Destruction of Invariant Tori in Volume-Preserving Maps.

Degree: PhD, Applied Mathematics, 2013, University of Colorado

Invariant rotational tori play an important role in the dynamics of volume-preserving maps. When integrable, all orbits lie on these tori and KAM theory guarantees the persistence of some tori upon perturbation. When these tori have codimension-one they act as boundaries to transport, and therefore play a prominent role in the global stability of the system. 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. This dissertation begins by extending Greene's residue criterion to three-dimensional, reversible, volume-preserving maps. The application of Greene's residue criterion requires the repeated computation of periodic orbits, which is costly if the system is nonreversible. We describe a quasi-Newton, Fourier-based scheme to numerically compute the conjugacy of a torus and demonstrate how the growth of the Sobolev norm or singular values of this conjugacy can be used to predict criticality. We will then use this method to study both reversible and nonreversible volume-preserving maps in two and three dimensions. The near-critical conjugacies, and the gaps that form within them, will be explored in the context of Aubry-Mather and Anti-Integrability theory, when applicable. This dissertation will conclude by exploring the locally and globally most robust tori in area-preserving maps. Advisors/Committee Members: James D. Meiss, Juan Restrepo, Keith Julien, Elizabeth Bradley, James Curry.

Subjects/Keywords: KAM theory; Greene's residue criterion; near-critical conjugacies; Applied Mathematics

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APA (6th Edition):

Fox, A. M. (2013). Destruction of Invariant Tori in Volume-Preserving Maps. (Doctoral Dissertation). University of Colorado. Retrieved from

Chicago Manual of Style (16th Edition):

Fox, Adam Merritt. “Destruction of Invariant Tori in Volume-Preserving Maps.” 2013. Doctoral Dissertation, University of Colorado. Accessed January 23, 2021.

MLA Handbook (7th Edition):

Fox, Adam Merritt. “Destruction of Invariant Tori in Volume-Preserving Maps.” 2013. Web. 23 Jan 2021.


Fox AM. Destruction of Invariant Tori in Volume-Preserving Maps. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2021 Jan 23]. Available from:

Council of Science Editors:

Fox AM. Destruction of Invariant Tori in Volume-Preserving Maps. [Doctoral Dissertation]. University of Colorado; 2013. Available from: