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...

The local limit theorem describes how the density of a sum of random variables follows the normal curve. However the local limit theorem is often seen as a curiosity of no particular importance when compared with the central limit theorem. Nevertheless the local limit theorem came first and is in fact associated with the foundation of probability theory by Blaise Pascal and Pierre de Fer...

We state and prove a Local Stable Manifold Theorem (Theorem 4.1) for non-linear stochastic differential systems with finite memory (viz. stochastic functional differential equations (sfde’s)). We introduce the notion of hyperbolicity for stationary trajectories of sfde’s. We then establish the existence of smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary traject...

We study the behavior of perturbations of small nonlinear Dirac standing waves. We assume that the linear Dirac operator of reference H = Dm + V has only two double eigenvalues and that degeneracies are due to a symmetry of H (theorem of Kramers). In this case, we can build a small 4-dimensional manifold of stationary solutions tangent to the first eigenspace of H . Then we assume that a resona...

In this paper, we establish a local stable manifold theorem near a hyperbolic equilibrium point for planar fractional differential equations. The construction of this stable manifold is based on the associated Lyapunov-Perron operator. An example is provided to illustrate the result.

In this paper, we deduce a predator–prey model with discrete time in the interior of R+2 using new method to study its local dynamics and Neimark–Sacker bifurcation. Compared continuous models, ones have many unique properties that help understand changing patterns biological populations from completely perspective. The existence stability three equilibria are analyzed, formation conditions bif...

We prove that any Kähler manifold admitting a flat complex conformal connection is a Bochner-Kähler manifold with special scalar distribution and zero geometric constants. Applying the local structural theorem for such manifolds we obtain a complete description of the Kähler manifolds under consideration.

This paper is a survey of recent results on the dynamics of Stochastic Burgers equation (SBE) and two-dimensional Stochastic Navier–Stokes Equations (SNSE) driven by affine linear noise. Both classes of stochastic partial differential equations are commonly used in modeling fluid dynamics phenomena. For both the SBE and the SNSE, we establish the local stable manifold theorem for hyperbolic sta...

In this paper, a stage-structured predator-prey system with time delays is considered, where the time delays are regarded as bifurcation parameters. By analyzing the corresponding characteristic equation, the local stability of a positive equilibrium is investigated. Moreover, we show that Hopf bifurcations occurs when time delay crosses some critical values. By using the normal form method and...

The dynamics of a system of two semiconductor lasers, which are delay coupled via a passive relay within the synchronization manifold, are investigated. Depending on the coupling parameters, the system exhibits synchronized Hopf bifurcation and the stability switches as the delay varies. Employing the center manifold theorem and normal form method, an algorithm is derived for determining the Ho...