On Takens' Last Problem: tangencies and time averages near heteroclinic networks

We obtain a structurally stable family of smooth ordinary differential equations exhibiting heteroclinic tangencies for a dense subset of parameters. We use this to find vector fields $C^2$-close to an element of the family exhibiting a tangency, for which the set of solutions with historic behaviour contains an open set. This provides an affirmative answer to Taken's Last Problem (F. Takens (2008) Nonlinearity, 21(3) T33--T36). A limited solution with historic behaviour is one for which the time averages do not converge as time goes to infinity. Takens' problem asks for dynamical systems where historic behaviour occurs persistently for initial conditions in a set with positive Lebesgue measure. The family appears in the unfolding of a degenerate differential equation whose flow has an asymptotically stable heteroclinic cycle involving two-dimensional connections of non-trivial periodic solutions. We show that the degenerate problem also has historic behaviour, since for an open set of initial conditions starting near the cycle, the time averages approach the boundary of a polygon whose vertices depend on the centres of gravity of the periodic solutions and their Floquet multipliers. We illustrate our results with an explicit example where historic behaviour arises $C^2$-close of a $\textbf{SO(2)}$-equivariant vector field

Book of Abstracts

USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud São Carlos, Brasil. 3-7 february 2014

Bifurcations from codimension-one D4m-equivariant homoclinic cycles

Das Thema dieser Arbeit ist eine detaillierte Beschreibung der Dynamik in der Nähe von D4m-symmetrischen relativen homoklinen Zykeln mit Hilfe von Lins Methode. Die homoklinen Zykel haben die Kodimension-1, d.h. wir beobachten ihre generische Entfaltung innerhalb einer einparametrigen Familie. Sie bestehen aus mehreren Trajektorien, die sowohl für positive als auch negative Zeit derselben hyperbolischen Gleichgewichtslage zustreben (Homokline Trajektorien) und die alle durch die von einer endlichen Gruppe induzierten Symmetrie voneinander abhängig sind. Wir nehmen reelle führende Eigenwerte und homokline Trajektorien an, die sich der Gleichgewichtslage entlang führender Richtungen nähern. Die Homoklinen befinden sich in flussinvarianten Unterräumen. Insbesondere für solche homoklinen Zykel in Differentialgleichungen mit Dk-Symmetrie (Dk ist die Symmetriegruppe eines regelmäßigen k-Ecks in der Ebene), bei denen k ein Vielfaches von 4 ist, stehen einige dieser flussinvarianten Unterräume senkrecht zueinander. Dies impliziert das Verschwinden der typischerweise auftretenden Terme führender exponentieller Konvergenzordnung in einigen der aus Lins Methode gewonnenen Bestimmungsgleichungen. Um eine genaue Beschreibung der nichtwandernden Dynamik eines solchen homoklinen Zykels zu geben, d.h. eine Beschreibung der Lösungen, die in der Umgebung des Zykels sowohl im Phasen- als auch im Parameterraum verbleiben, sind weitere Informationen über die Restterme in den Bestimmungsgleichungen erforderlich. In dieser Arbeit stellen wir eine verfeinerte Darstellung der Restterme in den Bestimmungsgleichungen vor und identifizieren zwei weitere Terme mit nächsthöheren exponentiellen Konvergenzraten. Darauf aufbauend diskutieren wir die Lösbarkeit der resultierenden Bestimmungsgleichungen für homokline Zykel in R4. Dabei sind zwei Fälle zu unterscheiden, die vom Größenverhältnis der beiden neuen Terme abhängen. In einem Fall beobachten wir einen endlichen Subshift. Im anderen Fall erweist sich die Analysis als schwieriger, so dass wir die Untersuchung auf periodische Lösungen beschränken.The topic of this thesis is a detailed description of the dynamics near D4m-symmetric relative homoclinic cycles by using Lin’s method. The homoclinic cycles have codimension-one, that is we observe the generic unfolding within a one- parameter family. They consist of several trajectories that are homoclinic to a hyperbolic equilibrium and which are all related to each other by means of the symmetry induced by a finite group. We assume real leading eigenvalues and connecting trajectories that approach the equilibrium along leading directions. The homoclinics are situated in flow-invariant subspaces. Especially for such homoclinic cycles in differential equations with Dk-symmetry (Dk is the symmetry group of a regular k-gon in the plane) where k is a multiple of 4 some of these flow-invariant subspaces are perpendicular to each other. This implies the vanishing of the typically appearing leading order terms in some of the determination equations gained from Lin’s method. In order to give a precise description of the nonwandering dynamics of such a homoclinic cycle, that is a description of the solutions that remain in the neighbourhood of the cycle both in phase and parameter space, further information about the residual terms in the determination equations are needed. In this thesis we present a more sophisticated representation of the residual terms in the determination equations and identify two further terms of next leading exponential rates. Based on this we discuss the solvability of the resulting determination equations for homoclinic cycles in R4. Thereby two cases must be distinguished, depending on the size ratio of the two new terms. In one case we observe subshifts of finite type. In the other case the analysis turns out to be more difficile so we restrict the investigation to periodic solutions. Beyond that we show how vector fields in R4 containing a homoclinic cycle with Dk-symmetry can be constructed. Those can be used for numerical investigations. One of these examples we consider numerically to verify some of the analytic results

Assessment and control of transition to turbulence in plane Couette flow

Transition to turbulence in shear flows is a puzzling problem regarding the motion of fluids flowing, for example, through the pipe (pipe flow), as in oil pipelines or blood vessels, or confined between two counter-moving walls (plane Couette flow). In this kind of flows, the initially laminar (ordered and layered) state of fluid motion is linearly stable, but turbulent (disordered and swirling) flows can also be observed if a suitable perturbation is imposed. This thesis concerns the assessment of transitional properties of such flows in the uncontrolled and controlled environments allowing for the quantitative comparisons of control strategies aimed at suppressing or trigerring transition to turbulence. Efficient finite-amplitude perturbations typically take the form of small patches of turbulence embedded in the laminar flow and called turbulent spots. Using direct numerical simulations, the nonlinear dynamics of turbulent spots, modelled as exact solutions, is investigated in the transitional regime of plane Couette flow and a detailed map of dynamics encompassing the main features found in transitional shear flows (self-sustained cycles, front propagation and spot splitting) is built. The map represents a quantitative assessment of transient dynamics of turbulent spots as a dependence of the relaminarisation time, i.e. the time it takes for a finite-amplitude perturbation, added to the laminar flow, to decay, on the Reynolds number and the width of a localised perturbation. By applying a simple passive control strategy, sinusoidal wall oscillations, the change in the spot dynamics with respect to the amplitude and frequency of the wall oscillations is assessed by the re-evaluation of the relaminarisation time for few selected localised initial conditions. Finally, a probabilistic protocol for the assessment of transition to turbulence and its control is suggested. The protocol is based on the calculation of the laminarisation probability, i.e. the probability that a random perturbation decays as a function of its energy. It is used to assess the robustness of the laminar flow to finite-amplitude perturbations in transitional plane Couette flow in a small computational domain in the absence of control and under the action of sinusoidal wall oscillations. The protocol is expected to be useful for a wide range of nonlinear systems exhibiting finite-amplitude instability

Pattern formation in the wake of external mechanisms

University of Minnesota Ph.D. dissertation. June 2016. Major: Mathematics. Advisor: Arnd Scheel. 1 computer file (PDF); xiii, 189 pages.Pattern formation in nature has intrigued humans for centuries, if not millennia. In the past few decades researchers have become interested in harnessing these processes to engineer and manufacture self-organized and self-regulated devices at various length scales. Since many natural pattern forming processes nucleate or grow from a homogeneous unstable state, they typically create defects, caused by thermal and other inherent sources of noise, which can hamper effectiveness in applications. One successful experimental method for controlling the pattern forming process is to use an external mechanism which moves through a system, transforming it from a stable state to an unstable state from which the pattern forming dynamics can take hold. In this thesis, we rigorously study partial differential equations which model how such triggering mechanisms can select and control patterns. We first use dynamical systems techniques to study the case where a spatial trigger perturbs a pattern forming freely invading front in a scalar partial differential equation. We study such perturbations for the two generic types of scalar invasion fronts, known as pulled and pushed fronts, which roughly correspond to fronts which invade either through a linear or nonlinear mechanism. Our results give the existence of perturbed fronts and provide expansions in the speed of the triggering mechanism for the wavenumber perturbation of the pattern formed. With the hope of moving towards the more complicated geometries which can arise in two spatial dimensions, where many dynamical systems methods cannot be readily applied, we also develop a functional analytic method for the study of Hopf bifurcation in the presence of continuous spectrum. Our method, while still giving computable information about the bifurcating solution, is more direct than previously proposed methods. We develop this method in the context of a triggered Cahn-Hilliard equation, in one spatial dimension, which has been used to study many triggered pattern forming systems. Furthermore, we use these abstract results to characterize an explicit example and also use our method to give a simplified proof of the bifurcation of oscillatory shock solutions in viscous conservation laws