the present paper deals with a toxin producing phytoplankton (tpp)-zooplankton interaction in spatial environment in thecontext of phytoplankton bloom. in the absence of diffusion the stability of the given system in terms of co-existence and hopf bifurcation has been discussed. after that tpp-zooplankton interaction is considered in spatiotemporal domain by assuming self diffusion in both popu...

We show the existence of a local stable manifold for a bidirectional discrete-time nondiffeomorphic nonlinear Hamiltonian dynamics. This is the case where zero is a closed loop eigenvalue and therefore the Hamiltonian matrix is not invertible. In addition, we show the eigenstructure and the symplectic properties of the mixed direction nonlinear Hamiltonian dynamics. We extend the Local Stable M...

We study the global behavior of (weakly) stable constant mean curvature hypersurfaces in general Riemannian manifolds. By using harmonic function theory, we prove some one-end theorems which are new even for constant mean curvature hypersurfaces in space forms. In particular, a complete oriented weakly stable minimal hypersurface in Rn+1, n ≥ 3, must have only one end. Any complete noncompact w...

In this paper we go over the basics of stable and unstable manifolds associated to the fixed points of a dynamical system. We will start with an overview of stable and unstable sets in general, and then look at some simple examples from linear systems, mostly taken from Proctor (2010). To study some non-linear examples we will use (without proof) the stable manifold theorem and will conclude wi...

Building on results obtained in [ 21 ], we prove Local Stable and Unstable Manifold Theorems for nonlinear, singular stochastic delay differential equations. The main tools are rough paths theory a semi-invertible Multiplicative Ergodic Theorem cocycles acting measurable fields of Banach spaces 20 ].

Rigorous Melnikov analysis is accomplished for Davey–Stewartson II equation under singular perturbation. Unstable fiber theorem and center-stable manifold theorem are established. The fact that the unperturbed homoclinic orbit, obtained via a Darboux transformation, is a classical solution, leads to the conclusion that only local well posedness is necessary for such a Melnikov analysis. The mai...

We prove a compactness theorem for holomorphic curves in 4-dimensional symplectizations that have embedded projections to the underlying 3-manifold. It strengthens the cylindrical case of the SFT compactness theorem [BEH03] by using intersection theory to show that degenerations of such sequences never give rise to multiple covers or nodes, so transversality is easily achieved. This has applica...

A. Katok has conjectured that a C map g : Mn → Mn, n ≥ 2, which is Hölder conjugated to an Anosov diffeomorphism is also an Anosov diffeomorphism. Using Pesin stable manifold theorem and Liao spectrum theorem, we show that under the hypothesis of such a conjecture, g is an Axiom A diffeomorphism having no cycles. Particularly, if g is Hölder conjugated to a hyperbolic toral automorphism, then g...

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...