Discrete Geometric Singular Perturbation Theory

We propose a mathematical formalism for discrete multi-scale dynamical systems induced by maps which parallels the established geometric singular perturbation theory for continuous-time fast-slow systems. We identify limiting maps corresponding to both 'fast' and 'slow' iteration under the map. A notion of normal hyperbolicity is defined by a spectral gap requirement for the multipliers of the fast limiting map along a critical fixed-point manifold $S$. We provide a set of Fenichel-like perturbation theorems by reformulating pre-existing results so that they apply near compact, normally hyperbolic submanifolds of $S$. The persistence of the critical manifold $S$, local stable/unstable manifolds $W^{s/u}_{loc}(S)$ and foliations of $W^{s/u}_{loc}(S)$ by stable/unstable fibers is described in detail. The practical utility of the resulting discrete geometric singular perturbation theory (DGSPT) is demonstrated in applications. First, we use DGSPT to identify singular geometry corresponding to excitability, relaxation, chaotic and non-chaotic bursting in a map-based neural model. Second, we derive results which relate the geometry and dynamics of fast-slow ODEs with non-trivial time-scale separation and their Euler-discretized counterpart. Finally, we show that fast-slow ODE systems with fast rotation give rise to fast-slow Poincar\'e maps, the geometry and dynamics of which can be described in detail using DGSPT.Comment: Updated to include minor corrections made during the review process (no major changes

Geometric blow-up of a dynamic Turing instability in the Swift-Hohenberg equation

We present a rigorous analysis of the slow passage through a Turing bifurcation in the Swift-Hohenberg equation using a novel approach based on geometric blow-up. We show that the formally derived multiple scales ansatz which is known from classical modulation theory can be adapted for use in the fast-slow setting, by reformulating it as a blow-up transformation. This leads to dynamically simpler modulation equations posed in the blown-up space, via a formal procedure which directly extends the established approach to the time-dependent setting. The modulation equations take the form of non-autonomous Ginzburg-Landau equations, which can be analysed within the blow-up. The asymptotics of solutions in weighted Sobelev spaces are given in two different cases: (i) A symmetric case featuring a delayed loss of stability, and (ii) A second case in which the symmetry is broken by a source term. In order to characterise the dynamics of the Swift-Hohenberg equation itself we derive rigorous estimates on the error of the dynamic modulation approximation. These estimates are obtained by bounding weak solutions to an evolution equation for the error which is also posed in the blown-up space. Using the error estimates obtained, we are able to infer the asymptotics of a large class of solutions to the dynamic Swift-Hohenberg equation. We provide rigorous asymptotics for solutions in both cases (i) and (ii). We also prove the existence of the delayed loss of stability in the symmetric case (i), and provide a lower bound for the delay time.Comment: 69 pages. A notational misprint in equation (17) has been correcte

Singularly Perturbed Boundary-Focus Bifurcations

We consider smooth systems limiting as $\epsilon \to 0$ to piecewise-smooth (PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable local normal form, we study the dynamics for the smooth system with $0 < \epsilon \ll 1$ using a combination of geometric singular perturbation theory and blow-up. We show that the type of BF bifurcation in the PWS system determines the bifurcation structure for the smooth system within an $\epsilon-$dependent domain which shrinks to zero as $\epsilon \to 0$, identifying a supercritical Andronov-Hopf bifurcation in one case, and a supercritical Bogdanov-Takens bifurcation in two other cases. We also show that PWS cycles associated with BF bifurcations persist as relaxation cycles in the smooth system, and prove existence of a family of stable limit cycles which connects the relaxation cycles to regular cycles within the $\epsilon-$dependent domain described above. Our results are applied to models for Gause predator-prey interaction and mechanical oscillation subject to friction

Geometric Blow-up for Folded Limit Cycle Manifolds in Three Time-Scale Systems

$\textit{Geometric singular perturbation theory}$ provides a powerful mathematical framework for the analysis of 'stationary' multiple time-scale systems which possess a $\textit{critical manifold}$, i.e. a smooth manifold of steady states for the limiting fast subsystem, particularly when combined with a method of desingularization known as $\textit{blow-up}$. The theory for 'oscillatory' multiple time-scale systems which possess a limit cycle manifold instead of (or in addition to) a critical manifold is less developed, particularly in the non-normally hyperbolic regime. We show that the blow-up method can be applied to analyse the global oscillatory transition near a regular folded limit cycle manifold in a class of three time-scale systems with two small parameters. The systems considered behave like oscillatory systems as the smallest perturbation parameter tends to zero, and stationary systems as both perturbation parameters tend to zero. The additional time-scale structure is crucial for the applicability of the blow-up method, which cannot be applied directly to the two time-scale counterpart of the problem. Our methods allow us to describe the asymptotics and strong contractivity of all solutions which traverse a neighbourhood of the global singularity. Our results cover a range of different cases with respect to the relative time-scale of the angular dynamics and the parameter drift

Beyond Slow-Fast: Relaxation Oscillations in Singularly Perturbed Non-Smooth Systems

Sharp dynamical transitions are ubiquitous in nature, arising in fluid flow, earthquake faulting and transistor oscillations. In studying such systems, many authors appeal to piecewise-smooth (PWS) approximations. In cases where an understanding of the transitionary dynamics is required, however, greater insight can often be gained by studying singular perturbations of PWS systems. In recent decades, singular perturbation theory has developed significantly in the context slow-fast systems, via the development geometric singular perturbation theory (GSPT), and a method for geometric desingularisation known as blow-up. In this thesis, GSPT, PWS theory and blow-up methods are combined to study sharp transitions beyond the scope of slow-fast systems. We focus on mechanical and electrical relaxation oscillators as case studies. First, we consider a class of ‘two-stroke’ oscillators, which can be recast as slow-fast systems after a state-dependent desingularisation. Our analysis demonstrates the power and simplicity of a coordinate-independent GSPT, which is adapted to handle non-trivial timescale separations. We then investigate a relationship between PWS systems, GSPT and systems with exponential nonlinearity, via an analysis of electrical oscillators which may be considered as ‘prototypes’ for a second class of relaxation oscillations. Using recent advances which allow for the application of blow-up methods to essential singularities, we derive the detailed asymptotic information required to prove existence of the relaxation oscillations. Finally, we study the onset of relaxation oscillations in smooth perturbations of systems which undergo a PWS bifurcation known as boundary-focus bifurcation. We show that the PWS dynamics qualitatively determines the bifurcation structure for the corresponding smooth transition, but cannot describe the dynamical changes associated with the transition from regular to relaxation-type oscillations, which requires GSPT and blow-up

Singularly Perturbed Boundary-Equilibrium Bifurcations

Boundary equilibria bifurcation (BEB) arises in piecewise-smooth systems when an equilibrium collides with a discontinuity set under parameter variation. Singularly perturbed BEB refers to a bifurcation arising in singular perturbation problems which limit as some $\epsilon \to 0$ to piecewise-smooth (PWS) systems which undergo a BEB. This work completes a classification for codimension-1 singularly perturbed BEB in the plane initiated by the present authors in [19], using a combination of tools from PWS theory, geometric singular perturbation theory (GSPT) and a method of geometric desingularization known as blow-up. After deriving a local normal form capable of generating all 12 singularly perturbed BEBs, we describe the unfolding in each case. Detailed quantitative results on saddle-node, Andronov-Hopf, homoclinic and codimension-2 Bogdanov-Takens bifurcations involved in the unfoldings and classification are presented. Each bifurcation is singular in the sense that it occurs within a domain which shrinks to zero as $\epsilon \to 0$ at a rate determined by the rate at which the system loses smoothness. Detailed asymptotics for a distinguished homoclinic connection which forms the boundary between two singularly perturbed BEBs in parameter space are also given. Finally, we describe the explosive onset of oscillations arising in the unfolding of a particular singularly perturbed boundary-node (BN) bifurcation. We prove the existence of the oscillations as perturbations of PWS cycles, and derive a growth rate which is polynomial in $\epsilon$ and dependent on the rate at which the system loses smoothness. For all the results presented herein, corresponding results for regularized PWS systems are obtained via the limit $\epsilon \to 0$

Jelbart's diary from the "Maudheim" expedition 1949-52

DAG-19