Regular, quasiperiodic motion is ubiquitous in dynamical systems with sufficient symmetry. A prominent example occurs in the Hamiltonian or symplectic case, where these "invariant tori" persist---even for nearly-integrable motion, as is explained by "KAM theory." The destruction of tori in the two-dimensional case is explained by Aubry-Mather theory and renormalization results. However, a concomitant understanding of the destruction of tori upon perturbation in higher dimensions has proved elusive. In this proposal, the implications of integrability, due to symmetries and invariants, of volume-preserving dynamics will be investigated. The loss of integrability under perturbation will be studied by a combination of analytical (Aubry's anti-integrable limit, Fourier series) and numerical (invariant manifold and continuation) techniques. Tori are both created and destroyed by bifurcations, and a study of the normal forms for codimension-one and two bifurcations of fixed points will lead to classification possible phenomena. Transport will be investigated numerically with the goal of developing analytical measures of flux and transport distributions. In a second project, the PI will investigate bifurcations in nonsmooth systems appropriate to the modeling of chemical reactions, the systematic simplification of these systems by center manifold reduction, as well as the study of transport caused by weak coupling of chaotic motion to regular motion.
Conservative dynamical models are used in designing particle accelerators, obtaining rates for simple chemical reactions, calculating confinement times in plasma fusion devices, understanding the spectra of highly excited atomic systems, and designing efficient spacecraft trajectories. Dynamics in such systems is often chaotic and prediction of individual trajectories is difficult; nevertheless, chaos can be profitably utilized, for example, to improve efficiency of spacecraft trajectories, by judiciously applying small course corrections, or to enhance the lifetimes of particles in confinement devices and the rates of chemical reactions. Volume-preserving dynamics models the flow of incompressible fluids and magnetic fields and a quantitative understanding of chaos in these systems is crucial for the development of efficient mixing in microscale bioreactors as well as of predictive planetary scale weather models. Most of our current theoretical understanding is limited to the two-dimensional case that is appropriate for flows in rapidly rotating or thin layers of fluid. While this has been useful in the understanding of such phenomena as the trapping of nutrients in gulf stream rings, the formation of the ozone hole and the creation of vortex-induced mixing in sinuous tubes, even in these systems, three-dimensional, chaos-induced transport needs to be understood. The PI seeks to develop analytical and computational methods for the study of regular and chaotic volume-preserving motion both to contribute broadly to our fundamental understanding of the richness of the behavior of low-dimensional deterministic evolution, and, to relate it to mixing and transport.
The motion of a drop of dye in stirred fluid can often exhibit surprising complexity, even when the fluid itself is moving in a regular way. An understanding of transport and the ultimate mixing of such a droplet has many applications, ranging from–on a global scale–the transport of heat and phytoplankton in oceanic currents, to–on an industrial scale–the design of micro-mixers for granular mixing in pharmaceuticals, to finally–on a picoliter scale–chemical analysis using "labs-on-a-chip". A distinguishing feature of most of these systems is that the flows are not turbulent; moreover, since they are subsonic (thus incompressible) the dynamics is volume preserving. The complex folding and stretching in these systems is an example of chaos, and–as was emphasized in pioneering work of Arnold and Aref–occurs even in laminar flows. Volume-preserving flows and maps are applicable not only to the motion of passive particles in fluids, but also to such areas as the motion of charged particles in plasma fusion devices and the quasiperiodically forced dynamics of comets. A major goal of the PI's NSF-funded project "Chaos and Bifurcations in Volume-Preserving Dynamics" was to develop a fundamental understanding of the onset and control of this chaotic behavior by studying generic models to elucidate the range of possible phenomena, by developing new tools, and by contributing to design principles for efficient mixers. A primary goal was to extend tools familiar from the study of symplectic maps: many were generalized, and a number of new tools were developed. Concepts of particular interest were integrability, bifurcation, resonance, twist, and transition to chaos. Twenty-three journal articles were published by the PI with his collaborators and students during this grant. Fundamental results include the development of "standard" models representing the dynamics near a generic saddle-center bifurcation and that near resonance. A new route for the onset of chaos was discovered; it leads to the destruction of regular orbits (invariant circles) by the creation of bubbles visually similar to those seen in the collision of smoke-rings. The investigation of the onset of chaos through the destruction of transport barriers (invariant tori) led to the discovery of a power-law decay in "stickiness" as the barriers disintegrate, and the extension of a famous conjecture of John Greene using the stability of nearby periodic orbits to this volume-preserving case. The grant supported several graduate students: David Simpson (PhD 2008)---currently a faculty member at Massey University in New Zealand, Brock Mosovsky (PhD 2012)--currently a Fulbright Scholar in the Netherlands, and Adam Fox (PhD anticipated 2013). In addition the PI was a co-adviser for Zachary Alexander (PhD 2012) with Professor Bradley of computer science, for Ted Galanthay (PhD anticipated 2013) with Professor Flaxman in Biology, and for Seksun Sirisubtawee with Professor Sivaselvan of Civil Engineering on dynamics projects. Research from this proposal has been used to provide examples and motivation in the PI’s dynamics classes at CU Boulder: APPM 3310, 5470 and 7100. This work is being incorporated into a textbook to complement the PI's "Differential Dynamical Systems" (SIAM, 2007). The PI is an editor for the Maps section of Scholarpedia's Encyclopedia of Dynamical Systems, as well as the new Encyclopedia of Applied Mathematics . The PI has been a mentor for a number of undergraduate research projects, also supported by the Department’s NSF funded MCTP grant. For example, one project studied mode locking and synchronization in coupled oscillators such as electrical circuits or pendulums. The undergraduates presented their research at a fall MCTP conference in Boulder, and the Spring "Front Range SIAM" meeting in Denver. Their project report is online at the University's website. Many of these students have extended their study after the BS degree by going to top graduate schools in STEM disciplines. During the summer of 2010, the PI was mentor for a high school student. This student began a study of the validity of statistical mechanics methods using an ideal-gas-like model called the self-consistent Fermi map. He presented his work at the local science fair, and was one of two Colorado students to become a semifinalist in the Siemens competition. He is currently in graduate school at MIT.
