In this paper a numerical method for the detection and computation of degenerate Hopf bifurcation points is presented. The degeneracies are classi ed and de ning equations characterizing each of the equivalence classes are constructed by means of a generalized Liapunov-Schmidt reduction. The numerical computation of the sign of the rst Liapunov coe cient which determines the stability of the bi...

In this paper, we first compute the multiple non-trivial solutions of the Schrödinger equation on a square, by using the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory, combinedwith Legendre pseudospectralmethods. Then, starting from the non-trivial solution branches of the corresponding nonlinear problem, we further obtain the whole positive solution branch with D4 symmetr...

The classical reduction techniques of bifurcation theory, Liapunov–Schmidt reduction and centre manifold reduction, are investigated where symmetry is present. The symmetry is given by the action of a finite or continuous group. The symmetry is exploited systematically by using the algebraic structure of the module of equivariant polynomial tuples. We generalize the concept of SAGBI-bases to mo...

We study solutions of the stationary Cahn-Hilliard equation in a bounded smooth domain which have a spike in the interior. We show that a large class of interior points (the “nondegenerate peak” points) have the following property: there exist such solutions whose spike lies close to a given nondegenerate peak point. Our construction uses among others the methods of viscosity solution, weak con...

A numerical tool for the detection of degenerated symmetry breaking bifurcation points is presented. The degeneracies are classified and numerically processed on 1-D restrictions of the bifurcation equation. The test functions that characterise each of the equivalence classes are constructed by means of an equivariant numerical version of the Liapunov-Schmidt reduction. The classification suppl...

We study a crime hotspot model suggested by Short-Bertozzi-Brantingham [18]. The aim of this work is to establish rigorously the formation of hotspots in this model representing concentrations of criminal activity. More precisely, for the onedimensional system, we rigorously prove the existence of steady states with multiple spikes of the following types: (i) Multiple spikes of arbitrary number...

In this paper, we review analytical methods for a rigorous study of the existence and stability of stationary, multiple spots for reaction-diffusion systems. We will consider two classes of reaction-diffusion systems: activator-inhibitor systems (such as the Gierer-Meinhardt system) and activator-substrate systems (such as the Gray-Scott system or the Schnakenberg model). The main ideas are pre...

We consider the linear stability and structural stability of non-ground state traveling waves of a pair of coupled nonlinear Schrr odinger equations (CNLS) which describe the evolution of co-propagating polarized pulses in the presence of birefringence. Viewing the CNLS equations as a Hamiltonian perturbation of the Manakov equations, we nd parameter regimes in which there are two stable famili...