We prove the existence of integral (stable, unstable, center) manifolds of admissible classes for the solutions to the semilinear integral equation $u\left(t\right)=U(t,s)u\left(s\right)+{\int }_s^{t}U(t,\xi )f(\xi ,u\left(\xi \right))d\xi$ when the evolution family ${\left(U(t,s)\right)}_{t\ge s}$ has an exponential trichotomy on a half-line or on the whole line, and the nonlinear forcing term f satisfies the (local or global) φ-Lipschitz conditions, i.e., ||f(t,x)-f(t,y)|| ≤ φ(t)||x-y|| where φ(t) belongs to some classes of admissible function spaces. These manifolds are formed by trajectories of the solutions belonging...

Hopf bifurcation, dynamics at infinity and robust modified function projective synchronization (RMFPS) problem for Sprott E system with quadratic perturbation were studied in this paper. By using the method of projection for center manifold computation, the subcritical and the supercritical Hopf bifurcation were analyzed and obtained. Then, in accordance with the Poincare compactification of polynomial vector field in $R^3$, the dynamical behaviors at infinity were described completely. Moreover, a...

Let $\gamma$ be an integral solution of an analytic real vector field $\xi$ defined in a neighbordhood of $0\in {ℝ}^3$. Suppose that $\gamma$ has a single limit point, $\omega \left(\gamma \right)=\left\{0\right\}$. We say that $\gamma$ is non oscillating if, for any analytic surface $H$, either $\gamma$ is contained in $H$ or $\gamma$ cuts $H$ only finitely many times. In this paper we give a sufficient condition for $\gamma$ to be non oscillating. It is established in terms of the existence of “generalized iterated tangents”, i.e. the existence of a single limit point for any transform property for...

In this paper we examine some features of the global dynamics of the four-dimensional system created by Lou, Ruggeri and Ma in 2007 which describes the behavior of the AIDS-related cancer dynamic model in vivo. We give upper and lower ultimate bounds for concentrations of cell populations and the free HIV-1 involved in this model. We show for this dynamics that there is a positively invariant polytope and we find a few surfaces containing omega-limit sets for positive half trajectories in the positive...

We prove the existence and conditional stability of periodic solutions to semilinear evolution equations of the form u̇ = A(t)u + g(t,u(t)), where the operator-valued function t ↦ A(t) is 1-periodic, and the operator g(t,x) is 1-periodic with respect to t for each fixed x and satisfies the φ-Lipschitz condition ||g(t,x₁) - g(t,x₂)|| ≤ φ(t)||x₁-x₂|| for φ(t) being a real and positive function which belongs to an admissible function space. We then apply the results to study the existence, uniqueness...