Monotone Skew-product Semiflows Research Articles

The exponential ordering for nonautonomous delay systems with applications to compartmental Nicholson systems

The exponential ordering is exploited in the context of nonautonomous delay systems, inducing monotone skew-product semiflows under less restrictive conditions than usual. Some dynamical concepts linked to the order, such as semiequilibria, are considered for the exponential ordering, with implications for the determination of the presence of uniform persistence or the existence of global attractors. Also, some important conclusions on the long-term dynamics and attraction are obtained for monotone and sublinear delay systems for this ordering. The results are then applied to almost periodic Nicholson systems and new conditions are given for the existence of a unique almost periodic positive solution which asymptotically attracts every other positive solution.

Almost automorphy of minimal sets for C1${C}^{1}$‐smooth strongly monotone skew‐product semiflows on Banach spaces

We focus on the presence of almost automorphy in strongly monotone skew-product semiflows on Banach spaces. Under the $C^1$-smoothness assumption, it is shown that any linearly stable minimal set must be almost automorphic. This extends the celebrated result of Shen and Yi [Mem. Amer. Math. Soc. 136(1998), No. 647] for the classical $C^{1,\alpha}$-smooth systems. Based on this, one can reduce the regularity of the almost periodically forced differential equations and obtain the almost automorphic phenomena in a wider range.

Monotone skew-Product Semiflows for Carathéodory Differential Equations and Applications

The first part of the paper is devoted to studying the continuous dependence of the solutions of Carath\'eodory constant delay differential equations where the vector fields satisfy classical cooperative conditions. As a consequence, when the set of considered vector fields is invariant with respect to the time-translation map, the continuity of the respective induced skew-product semiflows is obtained. These results are important for the study of the long-term behavior of the trajectories. In particular, the construction of semicontinuous semiequilibria and equilibria is extended to the context of ordinary and delay Carath\'eodory differential equations. Under appropriate assumptions of sublinearity, the existence of a unique continuous equilibrium, whose graph coincides with the pullback attractor for the evolution processes, is shown. The conditions under which such a solution is the forward attractor of the considered problem are outlined. Two examples of application of the developed tools are also provided.

Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems

We determine sufficient conditions for uniform and strict persistence in the case of skew-product semiflows generated by solutions of non-autonomous families of cooperative systems of ODEs or delay FDEs in terms of the principal spectrums of some associated linear skew-product semiflows which admit a continuous separation. Our conditions are also necessary in the linear case. We apply our results to a noncooperative almost periodic Nicholson system with a patch structure, whose persistence turns out to be equivalent to the persistence of the linearized system along the null solution.

Group actions on monotone skew-product semiflows with applications

We discuss a general framework of monotone skew-product semiflows under a connected group action. In a prior work, a compact connected group G -action has been considered on a strongly monotone skew-product semiflow. Here we relax the strong monotonicity and compactness requirements, and establish a theory concerning symmetry or monotonicity properties of uniformly stable 1-cover minimal sets. We then apply this theory to show rotational symmetry of certain stable entire solutions for a class of nonautonomous reaction-diffusion equations on \mathbb R^n , as well as monotonicity of stable traveling waves of some nonlinear diffusion equations in time-recurrent structures including almost periodicity and almost automorphy.

Topological Dynamics for Monotone Skew-Product Semiflows with Applications

Classical techniques of topological dynamics are used to prove a flow extension result for linearly stable minimal sets in monotone and differentiable skew-product semiflows. Moreover, motivated by the field of delay equations, a new version of the concept of continuous separation is introduced and studied in an abstract setting. The application of these results to the skew-product semiflows defined by almost periodic ordinary differential equations, delay equations and parabolic partial differential equations permits us to extend previous results guaranteeing the presence of almost automorphic minimal sets.

Phase-translation group actions on strongly monotone skew-product semiflows

We establish a convergence property for pseudo-bounded forward orbits of strongly monotone skew-product semiflows with invariant phasetranslation group actions. The results are then applied to obtain global convergence of certain chemical reaction networks whose associated systems in reaction coordinates are monotone, as well as the dynamics of certain reactiondiffusion systems in time-recurrent structure including periodicity, almost periodicity and almost automorphy.

Asymptotic behaviour of comparable skew-product semiflows with applications

The (almost) 1-cover lifting property of omega-limit sets is established for nonmonotone skew-product semiflows, which are comparable to uniformly stable eventually strongly monotone skew-product semiflows. These results are then applied to study the asymptotic behavior of solutions to the nonmonotone comparable systems of ODEs, reaction-diffusion systems, differential systems with time delays and semilinear parabolic equations.

A note on equilibrium of eventually strongly monotone skew-product semiflows

In this paper, we show that for an eventually strongly monotone skew-product semiflow τ, the strict ordering on Ec (the set consisting of continuous equilibria of τ) implies the strong one.

Asymptotic symmetry in strongly monotone skew-product semiflows with applications

For strongly monotone skew-product semiflows on which a compact connected group acts, it is shown that any stable minimal set is residually symmetric and any uniformly stable trajectory is asymptotically symmetric. These results are then applied to study the spatio-temporal asymptotics of stable solutions of reaction–diffusion equations on a symmetric domain in time-recurrent structures including almost periodicity. In particular, the 1-covering property of omega-limit sets is established for uniformly stable bounded solutions of reaction–diffusion equations on a ball.

Exponential Ordering for Nonautonomous Neutral Functional Differential Equations

We study monotone skew-product semiflows generated by families of nonautonomous neutral functional differential equations with infinite delay and stable D-operator, when the exponential ordering is considered. Under adequate hypotheses of stability for the order on bounded sets, we show that the omega-limit sets are copies of the base to explain the long-term behavior of the trajectories. The application to the study of the amount of material within the compartments of a neutral compartmental system with infinite delay shows the improvement with respect to the standard ordering.

Strictly ordered minimal subsets of a class of convex monotone skew-product semiflows

We study the topological and ergodic structure of a class of convex and monotone skew-product semiflows. We assume the existence of two strongly ordered minimal subsets K 1, K 2 and we obtain an ergodic representation of their upper Lyapunov exponents. In the case of null upper Lyapunov exponents, we obtain a lamination into minimal subsets of an intermediate region where the restriction of the semiflow is affine. In the hyperbolic case, we deduce the long-time behaviour of every trajectory ordered with K 2. Some examples of skew-product semiflows generated by non-autonomous differential equations and satisfying the assumptions of monotonicity and convexity are also presented.