Studying a manifold with a certain geometric structure becomes an interesting topic in differential topology when such a structure has some stability. The local stability of symplectic and contact structures, the Darboux theorem, is a fundamental property for symplectic and contact topology. There is a well-known global stability theorem for contact structures, the Gray theorem (see [3]): a one...

For a two-neuron network with self-connection and time delays, we carry out stability and bifurcation analysis. We establish that a Hopf bifurcation occurs when the total delay passes a sequence of critical values. The stability and direction of the local Hopf bifurcation are determined using the normal form method and center manifold theorem. To show that periodic solutions exist away from the...

We show the existence of a local solution to a Hamilton–Jacobi–Bellman (HJB) PDE around a critical point where the corresponding Hamiltonian ODE is not hyperbolic, i.e., it has eigenvalues on the imaginary axis. Such problems arise in nonlinear regulation, disturbance rejection, gain scheduling, and linear parameter varying control. The proof is based on an extension of the center manifold theo...

In this study, a discrete-time prey-predator model based on the Allee effect is presented. We examine parametric conditions for local asymptotic stability of fixed points model. Furthermore, with use center manifold theorem and bifurcation theory, we analyze existence directions period-doubling Neimark-Sacker bifurcations. The plots maximum Lyapunov exponents provide indications complexity chao...

We consider positive, radial and exponentially decaying steady state solutions of the general reaction-diffusion and Klein-Gordon type equations and present an explicit construction of infinite-dimensional invariant manifolds in the vicinity of these solutions. The result is a precise stable manifold theorem for the reaction-diffusion equation and a precise center-stable manifold theorem for th...

First we prove a general spectral theorem for the linear NavierStokes (NS) operator in both 2D and 3D. The spectral theorem says that the spectrum consists of only eigenvalues which lie in a parabolic region, and the eigenfunctions (and higher order eigenfunctions) form a complete basis in H (l = 0, 1, 2, · · · ). Then we prove the existence of invariant manifolds. We are also interested in a m...

‎using nehari manifold methods and mountain pass theorem‎, ‎the existence of nontrivial and radially symmetric solutions for a class of $p$-kirchhoff-type system are established.

When fluid in a rectangular tank sits upon a platform which is oscillating with sufficient amplitude, surface waves appear in the "Faraday resonance." Scientists and engineers have done bifurcation analyses which assume that there is a center manifold theory using a finite number of excited spatial modes. We establish such a center manifold theorem for Xiao-Biao Lin’s model in which potential f...

this paper investigates the dynamics and stability properties of a discrete-time lotka-volterra type system. we first analyze stability of the fixed points and the existence of local bifurcations. our analysis shows the presence of rich variety of local bifurcations, namely, stable fixed points; in which population numbers remain constant, periodic cycles; in which population numbers oscillate amo...