Continuous Semiflow Research Articles

The Central Limit Theorem for Markov Processes that are Exponentially Ergodic in the Bounded-Lipschitz Norm

In this paper, we establish a version of the central limit theorem for Markov–Feller continuous time processes (with a Polish state space) that are exponentially ergodic in the bounded-Lipschitz distance and enjoy a continuous form of the Foster–Lyapunov condition. As an example, we verify the assumptions of our main result for a specific piecewise-deterministic Markov process, whose deterministic component evolves according to continuous semiflows, switched randomly at the jump times of a Poisson process.

On two notions of expansiveness for continuous semiflows

We study two notions of expansiveness for continuous semiflows: expansiveness in the sense of Alves, Carvalho and Siqueira (2017), and an adaptation of positive expansiveness in the sense of Artigue (2014). We prove that if X is a metric space and ϕ is an expansive semiflow on X according to the first definition, then the semiflow ϕ is trivial and the space X is uniformly discrete. In particular, if X is compact then it is finite. With respect to the second definition, we prove that if X is a compact metric space and ϕ is a positive expansive semiflow on it, then X is a union of at most finitely many closed orbits, unbranched tails and isolated singularities.

Upper metric mean dimensions for impulsive semi-flows

In this paper we introduce three definitions of upper metric mean dimensions for semi-flows. We prove that the three definitions coincide for continuous semi-flows. Then we conclude that any impulsive semi-flow, under certain conditions, is semi-conjugate to a continuous semi-flow with same upper metric mean dimensions. As a corollary, a variational principle is obtained for impulsive semi-flows.

Delay Differential Equations with Differentiable Solution Operators on Open Domains in C((-∞, 0], Rn) and Processes for Volterra Integro-Differential Equations

For autonomous delay differential equations x'(t)=f(xt){x'(t)=f(x_t)} we construct a continuous semiflow of continuously differentiable solution operators x0xt{x_0 \to x_t}, t0{t \le 0}, on open subsets of the Frechet space C((-,0],Rn){C((-\infty, 0], R^n)}. For nonautonomous equations this yields a continuous process of differentiable solution operators. As an application, we obtain processes which incorporate all solutions of Volterra integro-differential equations x'(t)=∫0tk(t,s)h(x(s))ds{x'(t)={\int_0}^t k(t,s) h(x(s)) ds}.

A continuous semiflow on a space of Lipschitz functions for a differential equation with state-dependent delay from cell biology

We analyze a system of differential equations with state-dependent delay (SD-DDE) from cell biology, in which the delay is implicitly defined as the time when the solution of an ODE, parametrized by the SD-DDE state, meets a threshold. We show that the system is well-posed and that the solutions define a continuous semiflow on a state space of Lipschitz functions. Moreover we establish for an associated system a convex and compact set that is invariant under the time-t-map for a finite time. It is known that, due to the state dependence of the delay, necessary and sufficient conditions for well-posedness can be related to functionals being almost locally Lipschitz, which roughly means locally Lipschitz on the restriction of the domain to Lipschitz functions, and our methodology involves such conditions. To achieve transparency and wider applicability, we elaborate a general class of two component functional differential equation systems, that contains the SD-DDE from cell biology and formulate our results also for this class.

On absolute continuity of invariant measures associated with a piecewise-deterministic Markov process with random switching between flows

We are concerned with the absolute continuity of stationary distributions corresponding to some piecewise deterministic Markov process, being typically encountered in biological models. The process under investigation involves a deterministic motion punctuated by random jumps, occurring at the jump times of a Poisson process. The post-jump locations are obtained via random transformations of the pre-jump states. Between the jumps, the motion is governed by continuous semiflows, which are switched directly after the jumps. The main goal of this paper is to provide a set of verifiable conditions implying that any invariant distribution of the process under consideration that corresponds to an ergodic invariant measure of the Markov chain given by its post-jump locations has a density with respect to the Lebesgue measure.

Towards a Koopman theory for dynamical systems on completely regular spaces

The Koopman linearization of measure-preserving systems or topological dynamical systems on compact spaces has proven to be extremely useful. In this article, we look at dynamics given by continuous semiflows on completely regular spaces, which arise naturally from solutions of PDEs. We introduce Koopman semigroups for these semiflows on spaces of bounded continuous functions. As a first step we study their continuity properties as well as their infinitesimal generators. We then characterize them algebraically (via derivations) and lattice theoretically (via Kato's equality). Finally, we demonstrate-using the example of attractors-how this Koopman approach can be used to examine properties of dynamical systems. This article is part of the theme issue 'Semigroup applications everywhere'.

A Desch–Schappacher perturbation theorem for bi‐continuous semigroups

AbstractWe prove a Desch–Schappacher type perturbation theorem for one‐parameter semigroups on Banach spaces which are not strongly continuous for the norm, but possess a weaker continuity property. In this paper we chose to work in the framework of bi‐continuous semigroups. This choice has the advantage that we can treat in a unified manner two important classes of semigroups: implemented semigroups on the Banach algebra of bounded, linear operators on a Banach space E, and semigroups on the space of bounded and continuous functions over a Polish space induced by jointly continuous semiflows. For both of these classes we present an application of our abstract perturbation theorem.

A Differential Equation with a State-Dependent Queueing Delay

We consider a differential equation with a state-dependent delay motivated by a queueing process. The time delay is determined by an algebraic equation involving the length of the queue for which a discontinuous differential equation holds. The new type of state-dependent delay raises some problems that are studied in this paper. We formulate an appropriate framework to handle the system, and show that the solutions define a Lipschitz continuous semiflow in the phase space. The second main result guarantees the existence of slowly oscillating periodic solutions.

Solutions with dense short segments from regular delays

We construct a delay functionaldU with values in (0,r) and find a positive number h<r such that the negative feedback equationx′(t)=−x(t−dU(xt,r)), with the segmentxt,r:[−r,0]→R given by xt,r(s)=x(t+s), has a solution whose short segmentsxt,h=xt,r|[−h,0] are dense in an open subset of the space C1([−h,0],R). The domain U of dU is open in C1([−r,0],R), and the delay differential equation defines a continuous semiflow of continuously differentiable solution operators on the solution manifold{ϕ∈U:ϕ′(0)=−ϕ(−dU(ϕ))}. The result implies a kind of chaotic solution behaviour which is not confined to a thin Cantor dust.

Embeddability of pairs of weighted quasi-arithmetic means into a semiflow

We determine the form of all semiflows of pairs of weighted quasi-arithmetic means, those over positive dyadic numbers as well as the continuous ones. Then the iterability of such pairs is characterized: necessary and sufficient conditions for a given pair of weighted quasi-arithmetic means to be embeddable into a continuous semiflow are given. In particular, it turns out that surprisingly the existence of a square iterative root in the class of such pairs implies the embeddability.

Topological entropy for discontinuous semiflows

We study two variations of Bowen's definitions of topological entropy based on separated and spanning sets which can be applied to the study of discontinuous semiflows on compact metric spaces. We prove that these definitions reduce to Bowen's ones in the case of continuous semiflows. As a second result, we prove that our entropies give a lower bound for the τ-entropy defined by Alves, Carvalho and Vásquez (2015). Finally, we prove that for impulsive semiflows satisfying certain regularity condition, there exists a continuous semiflow defined on another compact metric space which is related to the first one by a semiconjugation, and whose topological entropy equals our extended notion of topological entropy by using separated sets for the original semiflow.

Decay of correlation implies strongly mixing property

ABSTRACTThis paper proves that a continuous transformation of a probability measure space is mixing in the sense of statistics if and only if it exhibits strongly mixing property, which implies topological mixing property and Li-Yorke sensitivity. Moreover, some analogical results for continuous semi-flows are obtained, improving the main results in Xu, Lin, and Ruan [2004. “Decay of Correlations Implies Chaos in the Sense of Devaney.” Chaos, Solitons and Fractals 22: 305–310] and Li [2015. “A Note on Decay of Correlation Implies Chaos in the Sense of Devaney.” Applied Mathematical Modelling 39: 6705–6710].

H s -class construction of an almost invariant slow subspace for the Klein–Gordon equation in the non-relativistic limit

We consider the linear Klein–Gordon equation in one spatial dimension with periodic boundary conditions in the non-relativistic limit where ε = ℏ2/(mc2) tends to zero. It is classical that the equation is well posed, for example, in the sense of possessing a continuous semiflow into spaces Hs+1 × Hs for wave function and momentum, respectively. In this paper, we iteratively construct a family of bounded operators FslowN:Hs+1→Hs whose graphs are O(εN)-invariant subspaces under the Klein–Gordon evolution for O(1) times. Contrary to a naive asymptotic series, there is no “loss of derivatives” in the iterative step; i.e., the Sobolev index s can be chosen independent of N. This is achieved by solving an operator Sylvester equation at each step of the construction.

Continuous semi-flows with the almost average shadowing property

In this article, we introduce the notion of almost average shadowing property (ALASP) for continuous semi-flows and study its properties. We give a class of continuous semi-flows possessing the ALASP. We also investigate the relationship of the ALASP with various known dynamical properties for continuous semi-flows, for instance, topological transitivity, weak mixing, strong ergodicity, equicontinuity, sensitivity, Li-Yorke chaos and Takens and Ruelle chaos.

Multi-sensitivity, syndetical sensitivity and the asymptotic average- shadowing property for continuous semi-flows

Multi-sensitivity, syndetical sensitivity and the asymptotic average- shadowing property for continuous semi-flows

Measures under the flat norm as ordered normed vector space

The space of real Borel measures \(\mathcal {M}(S)\) on a metric space S under the flat norm (dual bounded Lipschitz norm), ordered by the cone \(\mathcal {M}_+(S)\) of nonnegative measures, is considered from an ordered normed vector space perspective in order to apply the well-developed theory of this area. The flat norm is considered in place of the variation norm because subsets of \(\mathcal {M}_+(S)\) are compact and semiflows on \(\mathcal {M}_+(S)\) are continuous under much weaker conditions. In turn, the flat norm offers new challenges because \(\mathcal {M}(S)\) is rarely complete and \(\mathcal {M}_+(S)\) is only complete if S is complete. As illustrations serve the eigenvalue problem for bounded additive and order-preserving homogeneous maps on \(\mathcal {M}_+(S)\) and continuous semiflows. Both topics prepare for a dynamical systems theory on \(\mathcal {M}_+(S)\).

Generalized solutions to nonlinear Fokker–Planck equations

The nonlinear Fokker–Planck equationut+divx(D(x,u)u)−Δβ(u)=0 in (0,∞)×Rd;u(0,x)=u0(x), where β:R→R is monotonically increasing and continuous, is treated as a nonlinear Cauchy problem of accretive type in the space L1(Rd). Two distinct cases: D(x,u)≡b(u) and that of linear drift term D(x,u)≡D(x) are treated separately. In the first case, one proves the existence of an entropy generalized solution u=u(t,x) in sense of S. Kružkov [11] and, in the second, one proves the existence of a generalized solution under additional hypothesis β,β−1∈Lip(R). In both cases, the generalized solution u=u(t,u0) defines a continuous semiflow u0→u(t,u0) of contractions in L1(Rd).

Measure-valued branching processes associated with Neumann nonlinear semiflows

We construct a measure-valued branching Markov process associated with a nonlinear boundary value problem, where the boundary condition has a nonlinear pseudo monotone branching mechanism term −β, which includes the case β(u)=−um, with 0<m<1. The process is then used in the probabilistic representation of the solution of the parabolic problem associated with a nonlinear Neumann boundary value problem. In this way the classical association of the superprocesses to the Dirichlet boundary value problems also holds for the nonlinear Neumann boundary value problems. It turns out that the obtained branching process behaves on the measures carried by the given open set like the linear continuous semiflow, induced by the reflected Brownian motion, while the branching occurs on the measures having non-zero traces on the boundary of the open set, with the behavior of the (−β)-superprocess, having as spatial motion the process on the boundary associated to the reflected Brownian motion.

Prolongations, Suspensions and Telescopes

Autonomous differential equations induced by continuous vector fields usually appear in non-smooth mechanics and other scientific contexts. For these type of equations, given an initial condition, one has existence theorems but, in general, the uniqueness of the solution can not be ensured. For continuous vector fields, the equation solutions do not generally present a continuous flow structure; one particular but interesting case, occurs when under some initial conditions one can ensure existence of solutions and uniqueness in forward time obtaining in this case continuous semi-flows. The discretization and return Poincare techniques induce the corresponding discrete flows and semi-flows and some inverse methods as the suspension can construct a flow from a discrete flow or semi-flow. The objective of this work is to give categorical models for the diverse phase spaces of continuous and discrete semi-flows and flows and for the relations between these different phase spaces. We also introduce some new constructions such as the prolongation of continuous and discrete semi-flows and the telescopic functors. We consider small Top-categories (weakly enriched over the category Top of topological spaces) and we take as categorical models of the solutions of these differential equations some categories of continuous functors from a small Top-category to the category of topological spaces. Moreover, the processes of discretizations, suspensions, prolongations, et cetera are described in terms of adjoint functors. The main contributions of this paper are the construction of a tensor product associated to a functor between small Top-categories and the interpretation of prolongations, suspensions and telescopes as particular cases of this general tensor product. In general, the paper is focused on the establishment of links between category theory and dynamical systems more than on the study of differential equations using some categorical terminology.