Linear Semiflows Research Articles

Control Semiflows, Chain Controllability, and the Selgrade Decomposition for Linear Delay Systems

A continuous semiflow is introduced for linear control systems with delays in the states and controls and bounded control range. The state includes the control functions. It is proved that there exists a unique chain control set which corresponds to the chain recurrent set of the semiflow. The semiflow can be lifted to a linear semiflow on an infinite dimensional vector bundle with chain transitive base flow. A decomposition into exponentially separated subbundles is provided by a recent generalization of Selgrade’s theorem.

Relationships among Various Chaos for Linear Semiflows Indexed with Complex Sectors

In this paper, we investigate the relationships among point transitivity, topological transitivity, Li–Yorke chaos, and the existence of irregular vectors for a linear semiflow {Tt}t∈Δ indexed with a complex sector. We reveal the equivalence between topological transitivity and point transitivity for a linear semiflow {Tt}t∈Δ, especially in case the range of some operator Tt,t∈Δ is not dense. We also prove that Li–Yorke chaos is equivalent to the existence of a semi-irregular vector and that point transitivity is stronger than the existence of an irregular vector for any linear semiflow Ttt∈Δ. At last, unlike the conclusion for traditional linear dynamical systems, we show that there exists a Li–Yorke chaotic C0-semigroup Ttt∈Δ without irregular vectors. The results and proof methods presented in this paper demonstrate the differences in the dynamical behavior between linear semiflows {Tt}t∈Δ and traditional linear systems with the acting semigroup S=Z+ and S=R+.

On transitive and chaotic dynamics of linear semiflows

In this paper, we study the underlying relationship between transitivity and chaos for linear semiflows, which provides a better understanding of chaotic dynamics in topological spaces. We show that syndetically transitive linear semiflows with abelian acting semigroups are thickly syndetically-transitive and weakly mixing. As a result, Devaney chaos implies the weak mixing for linear semiflows with abelian acting semigroups. Besides, we give the characterizations for multi-dimensional Li-Yorke chaotic linear semiflows and show that if a linear semiflow (S,X) on a Polish space satisfies one of the following conditions: (i) S is discrete; (ii) dimX=∞; (iii) S is abelian, then Devaney chaos implies multi-dimensional Li-Yorke chaos. This answers a latest question posed in [12] (Question 2.19) to some extent. We also show that any topologically transitive linear semiflow (S,X) on a Polish space X with a translation-invariant metric appears to be densely Li-Yorke chaotic if IntX(Bx)=∅ for any compact subset B of S and any transitive point x∈X. Moreover, if the acting semigroup S is abelian, we present a weaker condition for the densely multi-dimensional Li-Yorke chaos.

The Selgrade Decomposition for Linear Semiflows on Banach Spaces

We extend Selgrade's Theorem, Morse spectrum, and related concepts to the setting of linear skew product semiflows on a separable Banach bundle. We recover a characterization, well-known in the finite-dimensional setting, of exponentially separated subbundles as attractor-repeller pairs for the associated semiflow on the projective bundle.

Dispersive semiflows on fiber bundles

This paper studies dispersiveness of semiflows on fiber bundles. The main result says that a right invariant semiflow on a fiber bundle is dispersive on the base space if and only if there is no almost periodic point and the semiflow is dispersive on the total space. A special result states that linear semiflows on vector bundles are not dispersive.

An ergodic theorem in the qualitative behavior of (non)linear semiflows

Take σ to be a continuous semiflow on the locally compact metric space Θ, and let {A(θ)}θ∈Θ be a family of (possibly unbounded) densely defined closed operators on the Banach space X.We show that the solutions of the variational equation u̇=A(σ(θ,t))u,θ∈Θ,t∈R+ have a uniform exponential decay if and only if there exist p∈[1,∞) and a sequence of positive real numbers (γn)n∈N with γn→∞ satisfying limn→∞γnn∫0n‖Φ(θ,τ)x‖pdτ=0uniformly w.r.t. θ∈Θ, for all x∈X, where (θ,t)→Φ(θ,t) is a strongly continuous cocycle solving (in the classical sense) the above variational equation.Thus, we generalize some known results obtained by Datko and Pazy.

Some Vector Borne Diseases with Structured Host Populations: Extinction and Spatial Spread

We derive from a structured population model a system of delay differential equations describing the interaction of five subpopulations, namely susceptible and infected adult and juvenile reservoirs and infected adult vectors, for a vector borne disease with particular reference to West Nile virus, and we also incorporate spatial movements by considering the analogue reaction diffusion systems with nonlocal delayed terms. Specific conditions for the disease eradication and sharp conditions for the local stability of the disease-free equilibrium are obtained using comparison techniques coupled with the spectral theory of monotone linear semi flows. A formal calculation of the minimal wave speed for the traveling waves is given and compared with field observation data.