We explore a definition of uniformity on noncompact manifolds that does not require a Riemannian metric, but is equivalent to bounded gemetry. These are unfinished research notes (and will likely never be published), but since they were cited by other people, I'm submitting this here for easier referencing.

We establish the existence of chimera states, simultaneously supporting synchronous and asynchronous dynamics, in a network of two symmetrically linked star subnetworks of identical oscillators with shear and Kuramoto-Sakaguchi coupling. We show that the chimera states may be metastable or asymptotically stable. If the intra-star coupling strength is of order e, the chimera states persist on time scales at least of order 1/e in general, and on time-scales at least of order 1/e (2) if the intra-star coupling is of Kuramoto-Sakaguchi type. If the intra-star coupling configuration is sparse, the chimeras are asymptotically stable. The analysis relies on a combination of dimensional reduction using a Mobius symmetry group and techniques from averaging theory and normal hyperbolicity.

We study global properties of the global (center-)stable manifold of a normally attracting invariant manifold (NAIM), the special case of a normally hyperbolic invariant manifold (NHIM) with empty unstable bundle. We restrict our attention to continuous-time dynamical systems, or flows. We show that the global stable foliation of a NAIM has the structure of a topological disk bundle, and that similar statements hold for inflowing NAIMs and for general compact NHIMs. Furthermore, the global stable foliation has a C-k disk bundle structure if the local stable foliation is assumed C-k. We then show that the dynamics restricted to the stable manifold of a compact inflowing NAIM are globally topologically conjugate to the linearized transverse dynamics at the NAIM. Moreover, we give conditions ensuring the existence of a global C-k linearizing conjugacy. We also prove a C-k global linearization result for inflowing NAIMs; we believe that even the local version of this result is new, and may be useful in applications to slow-fast systems. We illustrate the theory by giving applications to geometric singular perturbation theory in the case of an attracting critical manifold: we show that the domain of the Fenichel normal form can be extended to the entire global stable manifold, and under additional nonresonance assumptions we derive a smooth global linear normal form.

We review the role of dual pairs in mechanics and use them to derive particle-like solutions to regularized incompressible fluid systems. In our case we have a dual pair resulting from the action of diffeomorphisms on point particles (essentially by moving the points). We then augment our dual pair by considering the action of diffeomorphisms on Taylor series, also known as jets . The augmented weak dual pairs induce a hierarchy of particle-like solutions and conservation laws with particles carrying a copy of a jet group. We call these augmented particles jetlets . The jet groups serve as finite-dimensional models of the diffeomorphism group itself, and so the jetlet particles serve as a finite-dimensional model of the self-similarity exhibited by ideal incompressible fluids. The conservation law associated to jetlet solutions is shown to be a shadow of Kelvin’s circulation theorem. Finally, we study the dynamics of infinite time particle mergers. We prove that two merging particles at the zeroth level in the hierarchy yield dynamics which asymptotically approach that of a single particle in the first level in the hierarchy. This merging behavior is then verified numerically as well as the exchange of angular momentum which must occur during a near collision of two particles. The resulting particle-like solutions suggest a new class of meshless methods which work in dimensions n ≥ 2 and which exhibit a shadow of Kelvin’s circulation theorem. More broadly, this provides one of the first finite-dimensional models of self-similarity in ideal fluids.

The classical question whether nonholonomic dynamics is realized as limit of friction forces was first posed by Carathéodory. It is known that, indeed, when friction forces are scaled to infinity, then nonholonomic dynamics is obtained as a singular limit. Our results are twofold. First, we formulate the problem in a differential geometric context. Using modern geometric singular perturbation theory in our proof, we then obtain a sharp statement on the convergence of solutions on infinite time intervals. Secondly, we set up an explicit scheme to approximate systems with large friction by a perturbation of the nonholonomic dynamics. The theory is illustrated in detail by studying analytically and numerically the Chaplygin sleigh as an example. This approximation scheme offers a reduction in dimension and has potential use in applications.

In this paper we illustrate the potential role which relative limit cycles may play in biolocomotion. We do this by describing, in great detail, an elementary example of reduction of a lightly dissipative system modeling crawling-type locomotion in 3D. The symmetry group SE(2) is the set of rigid transformations of the horizontal (ground) plane. Given a time-periodic perturbation, the system will admit a relative limit cycle whereupon each period is related to the previous by a fixed translation and rotation along the ground. This toy model identifies how symmetry reduction and dissipation can conspire to create robust behavior in crawling, and possibly walking, locomotion.

We propose a geometric method for quantifying the difference between parametrized curves in Euclidean space by introducing a distance function on the space of parametrized curves up to rigid transformations (rotations and translations). Given two curves, the distance between them is defined as the infimum of an energy functional which, roughly speaking, measures the extent to which the jet field of the first curve needs to be rotated to match up with the jet field of the second curve. We show that this energy functional attains a global minimum on the appropriate function space, and we derive a set of first-order ODEs for the minimizer.

We study synchronization properties of networks of coupled dynamical systems with interaction akin to diffusion. We assume that the isolated node dynamics possesses a forward invariant set on which it has a bounded Jacobian, then we characterize a class of coupling functions that allows for uniformly stable synchronization in connected complex networks-in the sense that there is an open neighbourhood of the initial conditions that is uniformly attracted towards synchronization. Moreover, this stable synchronization persists under perturbations to non-identical node dynamics. We illustrate the theory with numerical examples and conclude with a discussion on embedding these results in a more general framework of spectral dichotomies.

We construct a simple, yet rigorous mathematical model for walk- ing. Our model exhibits robust limit cycle behavior, a feature commonly ob- served in biomechanical experiments. As such, it may serve as a case study for more complex models by identifying core mechanisms of walking. The singular nature of contact makes walking a difficult problem to analyze mathematically. To overcome this, we consider a simple triple mass-spring sys- tem as a model for a walker and we regularize the ground contact. We reduce the system by translation symmetries and prove the existence of limit cycles in the reduced phase space under small periodic forcings. In the unreduced phase space, the lifted trajectories are stable and relatively periodic, wherein each period is related to the previous by an x-translation. In conclusion, we have constructed a model which is complex enough to capture behavior characteristic of walking, yet simple enough to prove rigorous results.

This chapter treats Riemannian manifolds of bounded geometry. We use bounded geometry to extend uniformity assumptions that were implicit in the compact case to a noncompact setting. We redevelop various technical tools in the context of bounded geometry on Riemannian manifolds, such as a tubular neighborhood and smooth approximation of a submanifold embedding. We also explicitly work out the relation between curvature and holonomy for later use. Finally we use these tools to show how to reduce the main persistence theorem to a setting in a trivial bundle.

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all estimates throughout the proof.

[1] N. Fenichel. “Persistence and smoothness of invariant manifolds for flows”. In: Indiana Univ. Math. J. 21 (1971/1972), pp. 193–226. [2] M. W. Hirsch, C. C. Pugh, and M. Shub. Invariant manifolds. Vol. 583. Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1977, pp. ii+149. [3] J. Eichhorn. “The Banach manifold structure of the space of metrics on noncompact manifolds”. In: Differential Geom. Appl. 1.2 (1991), pp. 89–108. Statement of the theorem

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all estimates throughout the proof.

In this Master thesis we consider 't Hooft's polygon model for 2+1D gravity. After a detailed review of the polygon model in the classical context, we discuss problems associated with its quantization and calculate the explicitly the full Poisson structure of the constraints. The calculation of the Poisson structure in chapter 6 introduces corrections and generalizations to earlier published results. We show that the full Poisson structure closes on shell, but not off shell. An attempt is made to interpret the gauge orbits generated by the constraints.

We propose to investigate reductions of dynamical systems by means of a limiting procedure modelled on the underlying physics. By investigating the asymptotics, we then try to better understand the dynamics of both the reduced and unreduced system.