Dynamics of smooth essentially strongly order-preserving semiflows with application to delay differential equations

In this paper, we introduce a class of smooth essentially strongly order-preserving semiflows and improve the limit set dichotomy for essentially strongly order-preserving semiflows. Generic convergence and stability properties of this class of smooth essentially strongly order-preserving semiflows are then developed. We also establish the generalized Krein-Rutman Theorem for a compact and eventually essentially strongly positive linear operator. By applying the main results of this paper to essentially cooperative and irreducible systems of delay differential equations, we obtain some results on generic convergence and stability, the linearized stability of an equilibrium and the existence of the most unstable manifold in these systems. The obtained results improve some corresponding ones already known.     

New generic quasi-convergence principles with applications

In this paper, essentially strongly order-preserving and conditionally set-condensing semiflows are considered. Obtained is a new type of generic quasi-convergence principles implying the existence of an open and dense set of stable quasi-convergent points when the state space is order bounded. The generic quasi-convergence principles are then applied to essentially cooperative and irreducible systems in the forms of ordinary differential equations and delay differential equations, giving some results of theoretical and practical significance.     

Convergence of a class of discrete-time semiflows with application to neutral delay differential equations

In this paper, we introduce a class of pseudo-monotone maps on ordered topological spaces. By exploiting monotonicity methods and the invariance of the omega limit set, we establish a convergence principle for discrete-time semiflows generated by the maps introduced. The convergence principle is then applied to a class of periodic neutral delay differential equations, which leads to some novel and sharper results.     

Convergence for pseudo monotone semiflows on product ordered topological spaces

In this paper, we consider a class of pseudo monotone semiflows, which only enjoy some weak monotonicity properties and are defined on product-ordered topological spaces. Under certain conditions, several convergence principles are established for each precompact orbit of such a class of semiflows to tend to an equilibrium, which improve and extend some corresponding results already known. Some applications to delay differential equations are presented.     

Hilbert's 16th problem for classical Liénard equations of even degree

Classical Liénard equations are two-dimensional vector fields, on the phase plane or on the Liénard plane, related to scalar differential equations . In this paper, we consider f to be a polynomial of degree 2l−1, with l a fixed but arbitrary natural number. The related Liénard equation is of degree 2l. We prove that the number of limit cycles of such an equation is uniformly bounded, if we restrict f to some compact set of polynomials of degree exactly 2l−1. The main problem consists in studying the large amplitude limit cycles, of which we show that there are at most l.     

Lyapunov-Razumikhin pairs in the instability problem for infinite delay equations

Some theorems on complete instability of the zero solution relative to a set for nonautonomous nonlinear equations with infinite delay are provided. The right-hand side of the equation is assumed to be defined in a fading memory space and to satisfy conditions that allow the construction of limiting equations. We use conceptions of Lyapunov-Razumikhin pairs and limiting equations to obtain new instability results, which are applicable, in particular, to autonomous, periodic and almost periodic in t delay differential equations.     

Global attractivity in monotone and subhomogeneous almost periodic systems

A global attractivity theorem is first proved for a class of skew-product semiflows. Then this result is applied to monotone and subhomogeneous almost periodic reaction-diffusion equations, ordinary differential systems and delay differential equations for their global dynamics.     

On the number of zeros of Abelian integrals for a polynomial Hamiltonian irregular at infinity

Up to now, most of the results on the tangential Hilbert 16th problem have been concerned with the Hamiltonian regular at infinity, i.e., its principal homogeneous part is a product of the pairwise different linear forms. In this paper, we study a polynomial Hamiltonian which is not regular at infinity. It is shown that the space of Abelian integral for this Hamiltonian is finitely generated as a R[h] module by several basic integrals which satisfy the Picard-Fuchs system of linear differential equations. Applying the bound meandering principle, an upper bound for the number of complex isolated zeros of Abelian integrals is obtained on a positive distance from critical locus. This result is a partial solution of tangential Hilbert 16th problem for this Hamiltonian. As a consequence, we get an upper bound of the number of limit cycles produced by the period annulus of the non-Hamiltonian integrable quadratic systems whose almost all orbits are algebraic curves of degree k+n, under polynomial perturbation of arbitrary degree.     

Stability and extensibility results for abstract skew-product semiflows

In this paper we present new stability and extensibility results for skew-product semiflows with a minimal base flow. In particular, we describe the structure of uniformly stable and uniformly asymptotically stable sets admitting backwards orbits and the structure of omega-limit sets. As an application, the occurrence of almost periodic and almost automorphic dynamics for monotone non-autonomous infinite delay functional differential equations is analyzed.     

Classification of quadratic systems admitting the existence of an algebraic limit cycle

In the paper we find a set of necessary conditions that must be satisfied by a quadratic system in order to have an algebraic limit cycle. We find a countable set of ?5 parameter families of quadratic systems such that every quadratic system with an algebraic limit cycle must, after a change of variables, belong to one of those families. We provide a classification of all the quadratic systems which can have an algebraic limit cycle based on geometrical properties of the embedding of the system in the Poincaré compactification of R². We propose names for all the classes we distinguish and we classify all known examples of quadratic systems with algebraic limit cycle. We also prove the integrability of certain classes of quadratic systems.