__Periodic Orbits and Their Bifurcations__-
Periodic orbits are fundamental objects within dynamical systems. They represent cyclic processes in dynamical models of biological systems. Attracting periodic orbits can be found by simulation; i.e., solving initial value problems for long times. However, simulation alone is insufficient to obtain all of the information that we would like to have about periodic orbits. There are many circumstances in which it is desirable to have methods that compute periodic orbits directly. The algorithms used by such methods are examples of boundary value solvers. In addition to solving for individual periodic orbits, the methods enable one to determine local stability information, compute sensitivity information that describes how the orbit changes with perturbations of parameters, identify parameter values at which bifurcations occur and track families of periodic orbits as parameters vary. Our contributions in this area have centered on the use of a technique called automatic differentiation in the solution of the boundary value problems. Automatic differentiation enables the computation of derivatives of functions without the truncation errors that arise from the use of finite differences. This facilitates the implementation of methods of very high order that are conceptually simple and utilize coarse meshes in discretizing the periodic orbits. Another aspect of our research on periodic orbits has been to generate rigorous proofs of the qualitative properties of numerically computed dynamical systems.

__Dynamical Systems with Multiple Time Scales__Multiple time scales present challenges in the simulation of dynamical systems. Resolving the dynamics of fast time scales while computing system behavior for long times on slower time scales requires special i

*mplicit*algorithms that make assumptions about the character of the fast time dynamics. There are also qualitative features of the dynamics in systems with multiple time scales that do not appear in systems with single time scales. My research is directed at extending the qualitative theory of dynamical systems to apply to systems with multiple time scales. Much of this work began as a collaborative effort with Kathleen Hoffman and Warren Weckesser. We studied the classical forced van der Pol system intensively. The pheomenon of chaos in dissipative dynamical systems was first discovered and analyzed in this example by Cartwright and Littlewood during the 1940's and 1950's. Our work and that of my former student Radu Haiduc simplifies and extends the results of Cartwright and Littlewood. Haiduc's thesis establishes parameter values for which the forced van der Pol equation has a chaotic invariant set and is structurally stable. In joint workwith Martin Wechselberger and Lai-Sang Young, I explored the occurrence of chaotic attractors in concrete vector fields. We developed a general theory relating "Henon-like" attractors of two dimensional diffeomorphisms to three dimensional flows with two slow and one fast variable, and then illustrated the theory with a modification of the forced van der Pol system.

Mathieu Desroches, Bernd Krauskopf, Christian Kuehn, Hinke Osinga, Martin Wechselberger and I have systematically brought multiple time scale methods to bear on understanding mixed mode oscillations. There is no precise definition of mixed mode oscillations, but they have been identified in systems of chemical reactors, neural oscillations and in other application domains. We give examples of slow-fast systems where the interactions of slow and fast time scales leads to small oscillations that are not apparent in the dynamics of either the slow or the fast time scales individually. This work analyzes several example more extensively than previous studies and includes a comprehensive list of references about mixed mode oscillations.

__Invariant Manifolds__Computation of invariant manifolds of dynamical systems with dimension larger than one has proved to be a challenging problem. There are two sources of the difficulty in computing these manifolds. First, they have an intricate geometric structure with folds and spiraling structures. Second there is "geometric stiffness" reflected in trajectories that evolve with very different speeds. Thus, attempts to compute the manifolds by solving initial value problems for a few trajectories tend to produce poor results because the trajectories don't spread uniformly, and where they do spread they diverge rapidly enough to leave large gaps in the manifolds. Alex Vladimirsky and I have used "ordered upwinding" methods from the numerical analysis of partial differential equations to develop and implement efficient methods for computing these manifolds. The methods as we have developed them are low order, but very fast. Several research groups have introduced different approaches to the computation of invariant two dimensional stable and unstable manifolds of equilibrium points of vector fields. Krauskopf et al. give a survey of these methods using computation of the unstable manifold of the origin in the Lorenz systems as a benchmark problem.

Invariant manifolds also play a key role in systems with multiple time scales. Christian Kuehn and I developed an algorithm for computing slow manifolds of saddle-type. These manifolds are the subject of the Exchange Lemma for slow-fast systems, and they are an important component of classes of homoclinic orbits of systems with multiple time scales. Homoclinic orbits arise as traveling wave profiles of partial differential equations. Traveling waves of the FitzHugh-Nagumo model for action potentials have been investigated intensively. Our algorithm yields improved calculations of these periodic orbits. This work also highlights how tangential intersections of invariant manifolds underlie bifurcations that create both homoclinic orbits and mixed mode oscillations.

__El Niño and the Southern Oscillation__El Niño is a phenomenon in the equatorial Pacific ocean. In "normal" conditions, there is a gradient of sea surface temperature with much warmer waters in the western Pacific and colder surface water off the coast of South America. Surface winds in these equatorial latitudes blow from east to west in contrast to prevailing westerlies in mid latitudes that are driven by the coriolos force accompanying the earth's rotation. Every few years, this pattern of winds and waves changes: the winds diminish and surface waters of the eastern Pacific warm. This is El Niño. Its strength and timing is highly irregular, with exceptionally strong events ocurring on a decadal time scale. The strong event affect weather around the globe, including in the US. Atmospheric scientists have invested increasing effort in monitoring and forecasting this Southern Oscillation. Models that have been developed ranges from detailed coupled ocean-atmosphere climate models, to small systems of ordinary or delayed differential equation models that attempt to capture the large scale phenomena underlying El Niño.

Andrew Roberts and I, together with Henk Djikstra, Axel Timmermann, Esther Widiasih and Chris Jones, invesstigated a three dimensional vector field formulated by Fei Fei Jin in the 1990's as one of the smallest models of the Southern Oscillation. Its phase space variables represent the sea surface temperatures in the eastern and western equatorial Pacific and the depth of the thermocline separating cold, deep water form warmer surface waters. I joined Roberts, Jones and Widiasih in applying multiple time scale methods to analyze the dynamics of the Jin model. Though the model was not formulated as a slow-fast sytem with multiple time scales, we observed that it displays dynamics as if it were. We found oscillations within the model that resemble strong El Niño events. Furthermore, there are parameter regimes in which the timing of these events is highly unpredictable. This suggests that the irregularity of the strong El Niño events may be an intrinsic feature of the large scale dynamics of the Pacific ocean.

__Lines of Curvature on Embedded Surfaces__The differential geometry of smooth surfaces embedded in three space is a classical mathematical subject. At each point of a surface, there are orthogonal directions along which normal slices of the surface have the largest and smallest curvature. These principal directions form fields on the surface that are similar to vector fields, but they are not orientable. Lines of curves are characterized by having tangents in the principal directions. They are analogous to the trajectories of a vector field and together comprise the principal foliations of the surface, analogous to the phase portrait of a vector field. Incontrast to vector fields, there are few examples of surfaces whose principal foliations have been determined. The example displayed in most differential geometry texts are the principal foliations of a triaxial ellipsoid which were found by Monge already in the eighteenth century.

One of the highlights of dynamical systems theory is the characterization of structurally stable vector fields on compact surfaces. The geometric perspective of the theory was initiated by Poincaré and further developed by Andronov and Pontryagin in the 1930s. The Poincaré-Bendixson Theorem immplies that the limit set of all trajectories of a vector field on the two sphere is either a periodic orbit or contains an equilibrium point. Structurally stable vector fields on the two spheres are characterized byhaving hyperbolic equilibrium points and periodic orbits and no trajectories that connect saddle points. Sotomayor and Gutierrez developed an analogous theory for principal foliations. The singular points are umbiics where the curvatures of all normal slices are the same. Generic umbiilic points were classified by Darboux over a century ago: the three types are now called stars, monstars and lemons.

Software for visualizing the phase portraits of two dimensional vector fields is plentiful. Software for visualizing principal foliations is not. I have begun to address this gap, and the results are surprising. Sotomayor and Gutierrez realized that the Poincaré-Bendixson Theorem fails for compact convex surfaces and gave examples of surfaces with dense lines of curvature. They posed the problem of determining the principal foliations of perturbations of the triaxial ellipsoid in their lecture notes. I have solved this exercise, demonstrating that these principal foliations can be "lifted" to vector fields on the two torus which have no equilibrium points. Return maps of these flows are diffeomorphisms of the circle, a subject that has been studied extensively -- again beginning with Poincaré. He defined a rotation number for such diffeomorphisms, and Denjoy proved that smooth diffeomorphisms with irrational rotation number have dense trajectories. Seminal work by Herman and Yoccoz established that irrational rotation numbers occur for positive measure sets of parameters in one parameter families of circle diffeomorphisms. Applying this theory to principal foliations, I have characterized the principal foliations for perturbations of triaxial ellipsoids. Many have dense lines of curvature.

Last Update: February 10, 2020