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Register a new userDynamics on unbounded domains; co-solutions and inheritance of stability
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We consider the dynamics of semiflows of patterns on unbounded domains that are equivariant under a noncompact group action. We exploit the unbounded nature of the domain in a setting where there is a strong `global norm and a weak `local norm. Relative equilibria whose group orbits are closed manifolds for a compact group action need not be closed in a noncompact setting; the closure of a group orbit of a solution can contain `co-solutions. The main result of the paper is to show that co-solutions inherit stability in the sense that co-solutions of a Lyapunov stable pattern are also stable (but in a weaker sense). This means that the existence of a single group orbit of stable relative equilibria may force the existence of quite distinct group orbits of relative equilibria, and these are also stable. This is in contrast to the case for finite dimensional dynamical systems where group orbits of relative equilibria are typically isolated.
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This thesis is concerned with the asymptotic behavior of solutions of stochastic $p$-Laplace equations driven by non-autonomous forcing on $mathbb{R}^n$. Two cases are studied, with additive and multiplicative noise respectively. Estimates on the tails of solutions are used to overcome the non-compactness of Sobolev embeddings on unbounded domains, and prove asymptotic compactness of solution operators in $L^2(mathbb{R}^n)$. Using this result we prove the existence and uniqueness of random attractors in each case. Additionally, we show the upper semicontinuity of the attractor for the multiplicative noise case as the intensity of the noise approaches zero.
In the study of dynamical systems on networks/graphs, a key theme is how the network topology influences stability for steady states or synchronized states. Ideally, one would like to derive conditions for stability or instability that instead of microscopic details of the individual nodes/vertices rather make the influence of the network coupling topology visible. The master stability function is an important such tool to achieve this goal. Here we generalize the master stability approach to hypergraphs. A hypergraph coupling structure is important as it allows us to take into account arbitrary higher-order interactions between nodes. As for instance in the theory of coupled map lattices, we study Laplace type interaction structures in detail. Since the spectral theory of Laplacians on hypergraphs is richer than on graphs, we see the possibility of new dynamical phenomena. More generally, our arguments provide a blueprint for how to generalize dynamical structures and results from graphs to hypergraphs.
We analyse the stability of periodic, travelling-wave solutions to the Kawahara equation and some of its generalizations. We determine the parameter regime for which these solutions can exhibit resonance. By examining perturbations of small-amplitude solutions, we show that generalised resonance is a mechanism for high-frequency instabilities. We derive a quadratic equation which fully determines the stability region for these solutions. Focussing on perturbations of the small-amplitude solutions, we obtain asymptotic results for how their instabilities develop and grow. Numerical computation is used to confirm these asymptotic results and illustrate regimes where our asymptotic analysis does not apply.
In this work we analyze PT-symmetric double-well potentials based on a two-mode picture. We reduce the problem into a PT-symmetric dimer and illustrate that the latter has effectively two fundamental bifurcations, a pitchfork (symmetry-breaking bifurcation) and a saddle-center one, which is the nonlinear analog of the PT-phase-transition. It is shown that the symmetry breaking leads to ghost states (amounting to growth or decay); although these states are not true solutions of the original continuum problem, the systems dynamics closely follows them, at least in its metastable evolution. Past the second bifurcation, there are no longer states of the original continuum system. Nevertheless, the solutions can be analytically continued to yield a new pair of branches, which is also identified and dynamically examined. Our explicit analytical results for the dimer are directly compared to the full continuum problem, yielding a good agreement.
We show how to solve hyperbolic equations numerically on unbounded domains by compactification, thereby avoiding the introduction of an artificial outer boundary. The essential ingredient is a suitable transformation of the time coordinate in combination with spatial compactification. We construct a new layer method based on this idea, called the hyperboloidal layer. The method is demonstrated on numerical tests including the one dimensional Maxwell equations using finite differences and the three dimensional wave equation with and without nonlinear source terms using spectral techniques.
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