A permutation characterization of Sturm attractors of Hamiltonian type

We consider Neumann boundary value problems of the form $u_t = u_{xx} + f$ on the interval $0 \leq x \leq \pi$ for dissipative nonlinearities $f = f (u)$. A permutation characterization for the global attractors of the semiflows generated by these equations is well known, even in the general case $f = f (x, u, u_x )$. We present a permutation characterization for the global attractors in the restrictive class of nonlinearities $f = f (u)$. In this class the stationary solutions of the parabolic equation satisfy the second order ODE $v^{\prime\prime} + f (v) = 0$ and we obtain the permutation characterization from a characterization of the set of $2\pi$-periodic orbits of this planar Hamiltonian system. Our results are based on a diligent discussion of this mere pendulum equation

Design of Sturm global attractors 1: Meanders with three noses, and reversibility

We systematically explore a simple class of global attractors, called Sturm due to nodal properties, for the semilinear scalar parabolic partial differential equation (PDE) u(t) = u(xx) + f(x, u, u(x)) on the unit interval 0 < x < 1, under Neumann boundary conditions. This models the interplay of reaction, advection, and diffusion. Our classification is based on the Sturm meanders, which arise from a shooting approach to the ordinary differential equation boundary value problem of equilibrium solutions u(t) = 0. Specifically, we address meanders with only three "noses," each of which is innermost to a nested family of upper or lower meander arcs. The Chafee-Infante paradigm, with cubic nonlinearity f = f(u), features just two noses. Our results on the gradient-like global PDE dynamics include a precise description of the connection graphs. The edges denote PDE heteroclinic orbits v(1) (sic) v(2) between equilibrium vertices v(1), v(2) of adjacent Morse index. The global attractor turns out to be a ball of dimension d, given as the closure of the unstable manifold W-u(O) of the unique equilibrium with maximal Morse index d. Surprisingly, for parabolic PDEs based on irreversible diffusion, the connection graph indicates time reversibility on the (d - 1)-sphere boundary of the global attractor

Rainbow meanders and Cartesian billiards

nuloIn this paper we relate several objects from quite diverse areas of mathematics. Closed meanders are the configurations which arise when one or several disjoint closed Jordan curves in the plane intersect the horizontal axis transversely. The question of their connectivity also arises when evaluating traces in Temperley-Lieb algebras. The variant of open meanders is closely related to the detailed dynamics of Sturm global attractors, i.e. the global attractors of parabolic PDEs in one space dimension; see the groundbreaking work of Fusco and Rocha [FuRo91]. Cartesian billiards have their corners located on the integer Cartesian grid with corner angles of ±90 degrees. Billiard paths are at angles of ±45 degrees with the boundaries and reflect at half-integer coordinates. We indicate and explore some close connections between these seemingly quite different objects

Dynamics of Patterns

Patterns and nonlinear waves arise in many applications. Mathematical descriptions and analyses draw from a variety of fields such as partial differential equations of various types, differential and difference equations on networks and lattices, multi-particle systems, time-delayed systems, and numerical analysis. This workshop brought together researchers from these diverse areas to bridge existing gaps and to facilitate interaction

Classification of global and blow-up sign-changing solutions of a semilinear heat equation in the subcritical fujita range:Second-order diffusion

AbstractIt is well known from the seminal paper by Fujita [22] for 1 p0there exists a class of sufficiently "small" global in time solutions. This fundamental result from the 1960-70s (see also [39] for related contributions), was a cornerstone of further active blow-up research. Nowadays, similar Fujita-type critical exponents p0, as important characteristics of stability, unstability, and blow-up of solutions, have been calculated for various nonlinear PDEs. The above blow-up conclusion does not include solutions of changing sign, so some of them may remain global even for p ≤ p0. Our goal is a thorough description of blow-up and global in time oscillatory solutions in the subcritical range in (0.1) on the basis of various analytic methods including nonlinear capacity, variational, category, fibering, and invariant manifold techniques. Two countable sets of global solutions of changing sign are shown to exist. Most of them are not radially symmetric in any dimension N ≥ 2 (previously, only radial such solutions in ℝNor in the unit ball B1⊂ℝNwere mostly studied). A countable sequence of critical exponents, at which the whole set of global solutions changes its structure, is detected:, l = 0, 1, 2, ... .. See [47, 48] for earlier interesting contributions on sign changing solutions

Enharmonic motion: Towards the global dynamics of negative delayed feedback

In this thesis, we establish a new method for describing the qualitative dynamics of the so-called Hopf-Smale attractors in scalar delay differential equations with symmetric negative delayed feedback. The dynamics of Hopf-Smale attractors are robust under regular perturbations. Qualitatively, the attractor consists of an equilibrium, periodic orbits, and connections between them. We describe the mechanism that produces the periodic orbits and show how their formation creates new connecting orbits via sequences of Hopf bifurcations. As a result, we obtain an enumeration of all the phase diagrams, that is, the directed graphs encoding the equilibrium and periodic orbits as vertices and the connections as edges. In particular, we have obtained a prototype, the so-called enharmonic oscillator, that realizes all Hopf-Smale phase diagrams. Besides describing the Hopf-Smale attractors, our method also sheds insight into the formation process of certain global attractors with positive delayed feedback.In dieser Arbeit wird eine neue Methode zur Beschreibung der qualitativen Dynamik der sogenannten Hopf-Smale-Attraktoren in skalaren retardierten Differentialgleichung mit symmetrischer negativer verzögerter Rückkopplung entwickelt. Die Dynamik von Hopf-Smale-Attraktoren ist robust gegenüber regelmäßigen Störungen. Qualitativ besteht der Attraktor aus einem Gleichgewicht, periodischen Orbits und Orbits zwischen diesen. Wir beschreiben den Mechanismus, der die periodischen Orbits erzeugt und zeigen, wie dieser neue verbindende Orbits über Sequenzen von Hopf-Bifurkationen erzeugt. Als Ergebnis erhalten wir eine Aufzählung aller Phasendiagramme, d.h. der gerichteten Graphen, die die Gleichgewichts- und periodischen Bahnen als Knoten und die Verbindungen als Kanten kodieren. Insbesondere haben wir einen Prototyp, den sogenannten enharmonischen Oszillator, gefunden, der alle Hopf-Smale-Phasendiagramme verwirklicht. Neben der Beschreibung der Hopf-Smale-Attraktoren gibt unsere Methode auch Aufschluss über den Entstehungsprozess bestimmter globaler Attraktoren mit positiver verzögerter Rückkopplung