Narrow Results

We develop causality theory for upper semi-continuous distributions of cones over manifolds generalizing results from mathematical relativity in two directions: non-round cones and non-regular differentiability assumptions. We prove the validity of most results of the regular Lorentzian causality theory including: causal ladder, Fermat’s principle, notable singularity theorems in their causal formulation, Avez–Seifert theorem, characterizations of stable causality and global hyperbolicity by means of (smooth) time functions. For instance, we give the first proof for these structures of the equivalence between stable causality, $K$-causality and existence of a time function. The result implies that closed cone structures that admit continuous increasing functions also admit smooth ones. We also study proper cone structures, the fiber bundle analog of proper cones. For them, we obtain most results on domains of dependence. Moreover, we prove that horismos and Cauchy horizons are generated by lightlike geodesics, the latter being defined through the achronality property. Causal geodesics and steep temporal functions are obtained with a powerful product trick. The paper also contains a study of Lorentz–Minkowski spaces under very weak regularity conditions. Finally, we introduce the concepts of stable distance and stable spacetime solving two well-known problems (a) the characterization of Lorentzian manifolds embeddable in Minkowski spacetime, they turn out to be the stable spacetimes, (b) the proof that topology, order and distance (with a formula à la Connes) can be represented by the smooth steep temporal functions. The paper is self-contained, in fact we do not use any advanced result from mathematical relativity.

This paper proposes an adaptive design of nonlinear feedback controllers for a class of complex nonlinear applications that have ill-defined mathematical models due to the effects of uncertainties and external disturbances. The design is aimed at estimating the uncertain parameters of the system while using a feedforward-like technique to cancel the effect of disturbances and unwanted nonlinearities. This is being achieved using a combination of state feedback and Lyapunov-based techniques that can guarantee the asymptotic stability of the closed loop system. The controller is synthesized such that it will follow the performance of a reference model via satisfying a certain criterion. The control law is demonstrated to be easily achievable for applications that can be modeled by low-order dynamics, e.g., industrial processes (level, flow, pressure, etc.) and some automotive applications (active suspension). The key factor in the design is arriving at the best parameter update law that guarantees both stability and satisfactory transient performance. The application of the proposed controller is extended to higher-order systems via proposing a low-order nonlinear model that is capable of encapsulating the dominant dynamics of the system without using linearization techniques. Trade-offs between stability and performance are carefully studied along with comparisons with a nonmodel-based PID controller to highlight the superiority of the proposed design. A simulated nonlinear Duffing oscillator and a continuous stirred tank reactor are used to exemplify the suggested technique. Finally, a conclusion is submitted with comments regarding feasibility of the controller along with its advantages and limitations.

This paper is devoted to the synchronization of a dynamical system defined by two different coupling versions of two identical piecewise linear bimodal maps. We consider both local and global studies, using different tools as natural transversal Lyapunov exponent, Lyapunov functions, eigenvalues and eigenvectors and numerical simulations. We obtain theoretical results for the existence of synchronization on coupling parameter range. We characterize the synchronization manifold as an attractor and measure the synchronization speed. In one coupling version, we give a necessary and sufficient condition for the synchronization. We study the basins of synchronization and show that, depending upon the type of coupling, they can have very different shapes and are not necessarily constituted by the whole phase space; in some cases, they can be riddled.

In this paper, the global exponential attractive sets of the T chaotic system are studied. The method for constructing Lyapunov functions that are applied to the former chaotic systems [Liao, 2004; Liao et al., 2008; Wang et al., 2011; Zhang et al., 2011a; Zhang et al., 2011b; Yu & Liao, 2008; Li et al., 2009; Yu et al., 2009] does not work for the T chaotic system. We overcome this difficulty by introducing a cross term xy. We get a perfect result that contains the case for the parameter values considered covering the most interesting case of the chaotic attractor system.

We explore Itô stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier–Stokes equation and stochastic Ginsburg–Landau equation.

We investigate the discretization of nonautonomous cocycle delay differential equations near uniform pullback attractors of arbitrary shape and show that for sufficiently small step sizes a Runge–Kutta discretization admits a discrete uniform pullback attractor located close to that of the original system.

We establish an integral test describing the exact cut-off between recurrence and transience for normally reflected Brownian motion in certain unbounded domains in a class of warped product manifolds. Besides extending a previous result by Pinsky, who treated the case in which the ambient space is flat, our result recovers the classical test for the standard Brownian motion in model spaces. Moreover, it allows us to discuss the recurrence/transience dichotomy for certain generalized tube domains around totally geodesic submanifolds in hyperbolic space.

In this paper, the stability of impulsive functional differential equations with infinite delays are investigated. By using Lyapunov functions and the Razumikhin technique, a new theorem on the uniform asymptotic stability and global asymptotic stability for such differential equations is obtained. An example is given to illustrate the feasibility of the result.

In this paper, we investigate the dynamical behavior of a class of periodic SVEIR epidemic model. Since the nonautonomous phenomenon often occurs as cyclic pattern, our model is then a periodic time-dependent system. It follows from persistence theory that the basic reproduction number is the threshold parameter above which the disease is uniformly persistent and below which disease-free periodic solution is globally asymptotically stable. The threshold dynamics extends the classic results for the corresponding autonomous model. Furthermore, we show that eradication policy on the basis of the basic reproduction number of the autonomous system may overestimate the infectious risk when the disease follows periodic behavior. The according simulation results are also given.

In this paper, we study the existence, uniqueness and stability of memristor-based synchronous switching neural networks with time delays. Several criteria of exponential stability are given by introducing multiple Lyapunov functions. In comparison with the existing publications on simplice memristive neural networks or switching neural networks, we consider a system with a series of switchings, these switchings are assumed to be synchronous with memristive switching mechanism. Moreover, the proposed stability conditions are straightforward and convenient and can reflect the impact of time delay on the stability. Two examples are also presented to illustrate the effectiveness of the theoretical results.

In this paper, a class of stochastic Lotka–Volterra system with feedback controls is considered. The purpose is to establish some criteria to ensure the system is globally dissipative in the mean square. By constructing suitable Lyapunov functions as well as combining with Jensen inequality and It$\widehat{\mathrm{\text{o}}}$ formula, the sufficient conditions are established and they are expressed in terms of the feasibility to a couple linear matrix inequalities (LMIs). Finally, the main results are illustrated by examples.

This paper is devoted to the study of the stability of a CD$4^{+}$ T cell viral infection model with diffusion. First, we discuss the well-posedness of the model and the existence of endemic equilibrium. Second, by analyzing the roots of the characteristic equation, we establish the local stability of the virus-free equilibrium. Furthermore, by constructing suitable Lyapunov functions, we show that the virus-free equilibrium is globally asymptotically stable if the threshold value ${{ℛ}}_0\le 1$; the endemic equilibrium is globally asymptotically stable if ${{ℛ}}_0>1$ and $d{u}^{\ast }-\delta w^{\ast }\ge 0$. Finally, we give an application and numerical simulations to illustrate the main results.

In this paper, a mathematical analysis of the global dynamics of a viral infection model in vivo is carried out. We study the dynamics of a hepatitis C virus (HCV) model, under therapy, that considers both extracellular and intracellular levels of infection. At present, most mathematical modeling of viral kinetics after treatment only addresses the process of infection of a cell by the virus and the release of virions by the cell, while the processes taking place inside the cell are not included. We prove that the solutions of the new model with positive initial values are positive, exist globally in time and are bounded. The model has two virus-free steady states. They are distinguished by the fact that viral RNA is absent inside the cells in the first state and present inside the cells in the second. There are basic reproduction numbers associated to each of these steady states. If the basic reproduction number of the first steady state is less than one, then that state is asymptotically stable. If the basic reproduction number of the first steady state is greater than one and that of the second less than one, then the second steady state is asymptotically stable. If both basic reproduction numbers are greater than one, then we obtain various conclusions which depend on different restrictions on the parameters of the model. Under increasingly strong assumptions, we prove that there is at least one positive steady state (infected equilibrium), that there is a unique positive steady state and that the positive steady state is stable. We also give a condition under which every positive solution converges to a positive steady state. This is proved by methods of Li and Muldowney. Finally, we illustrate the theoretical results by numerical simulations.

The gradient flow of a Morse function on a smooth closed manifold generates, under suitable transversality assumptions, the Morse–Smale–Witten complex. The associated Morse homology is an invariant for the manifold, and equals the singular homology, which yields the classical Morse relations. In this paper we define Morse–Conley–Floer homology, which is an analogous homology theory for isolated invariant sets of smooth, not necessarily gradient-like, flows. We prove invariance properties of the Morse–Conley–Floer homology, and show how it gives rise to the Morse–Conley relations.

In this paper we give a pedagogical account of the PDE approach to invariant and Gibbs measures in finite and infinite dimensions. As an application we describe some recent new results on the classical problem whether “invariance implies Gibbsian” and illustrate how they apply to a well-studied lattice model from statistical mechanics with non-compact single spin spaces.

In this paper we review some recent results on stability of multilinear switched systems, under arbitrary switchings. An open problems is stated

In this paper we announce a new framework for a rigorous stability analysis of sliding-mode controllers. We give unrestrictive conditions under which such feedback controllers are robustly stabilizing. These conditions make allowance for large disturbance signals, for modeling, actuator and observation measurement errors, and also for the effects of digital implementation of the control. The proposed stability analysis techniques involve two Lyapunov-type functions. The first is associated with passage to the sliding surface in finite time; the second, with convergence to the state associated with the desired equilibrium point. Application of the techniques is illustrated with reference to higher-order linear systems in control canonical form.