Coexistence The special case when all the independent solutions of a linear, T -periodic ODE are T -periodic. Hill’s equation A second order ODE of the form ẍ + p(t)x = 0, with p(t) T -periodic. Instability pockets Finite domains, usually intersections of instability tongues, where the trivial solution of linear, T -periodic ODEs is unstable. Instability tongues Domains in parameter space where...

The Γ-convergence theory shows that under certain conditions the diblock copolymer equation has spot and ring solutions. We determine the asymptotic properties of the critical eigenvalues of these solutions in order to understand their stability. In two dimensions a threshold exists for the stability of the spot solution. It is is stable if the sample size is small and unstable if the sample si...

We construct center-stable and center-unstable manifolds, as well as stable and unstable manifolds, for the nonlinear Klein–Gordon equation with a focusing energy subcritical nonlinearity, associated with a family of solitary waves which is generated from any radial stationary solution by the action of all Lorentz transforms and spatial translations. The construction is based on the graph trans...

We consider the mass-critical focusing nonlinear Schrödinger equation in the presence of an external potential, when the nonlinearity is inhomogeneous. We show that if the inhomogeneous factor in front of the nonlinearity is sufficiently flat at a critical point, then there exists a solution which blows up in finite time with the maximal (unstable) rate at this point. In the case where the crit...

The reaction–diffusion Holling–Tanner predator–prey model with Neumann boundary condition is considered. We perform a detailed stability and Hopf bifurcation analysis and derive conditions for determining the direction of bifurcation and the stability of the bifurcating periodic solution. For partial differential equation (PDE), we consider the Turing instability of the equilibrium solutions an...

to study the newly mentioned significance of the obstructive lesion morphology in coronary artery disease (cad), a study has been designed to compare this feature between patients having stable angina (sa) or unstable angina (ua) who underwent coronary angiography in tehran heart institute. study objectives were detection of the angina producing artery (ap a) and its morphologic lesion type, de...

A class of damped wave equations with superlinear source term is considered. It is shown that every global solution is uniformly bounded in the natural phase space. Global existence of solutions with initial data in the potential well is obtained. Finally, not only finite time blow up for solutions starting in the unstable set is proved, but also high energy initial data for which the solution ...

We study quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains. Our main results concern the asymptotic behavior of the solutions in the vicinity of an equilibrium. The local center, center–stable, and center–unstable manifolds are constructed and their dynamical properties are established using nonautonomous cuto...