Minimal sets in monotone and sublinear skew-product semiflows I: The general case

The dynamics of a general monotone and sublinear skew-product semiflow is analyzed, paying special attention to the long-term behavior of the strongly positive semiorbits and to the minimal sets. Four possibilities arise depending on the existence or absence of strongly positive minimal sets and bounded semiorbits, as well as on the coexistence or not of bounded and unbounded strongly positive semiorbits. Previous results are unified and extended, and scenarios which are impossible in the autonomous or periodic cases are described.     

Convergence and stability for essentially strongly order-preserving semiflows

This paper is concerned with a class of essentially strongly order-preserving semiflows, which are defined on an ordered metric space and are generalizations of strongly order-preserving semiflows. For essentially strongly order-preserving semiflows, we prove several principles, which are analogues of the nonordering principle for limit sets, the limit set dichtomy and the sequential limit set trichotomy for strongly order-preserving semiflows. Then, under certain compactness hypotheses, we obtain some results on convergence, quasiconvergence and stability in essentially strongly order-preserving semiflows. Finally, some applications are made to quasimonotone systems of delay differential equations and reaction-diffusion equations with delay, and the main advantages of our results over the classical ones are that we do not require the delicate choice of state space and the technical ignition assumption.     

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.     

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.     

Schäffer spaces and uniform exponential stability of linear skew-product semiflows

We study the exponential stability of linear skew-product semiflows on locally compact metric space with Banach fibers. Our main tool is the admissibility of a pair of the so-called Schäffer spaces. This characterization is a very general one, it includes as particular cases many interesting situations among them we can mention some results due to Clark, Datko, Latushkin, van Minh, Montgomery-Smith, Randolph, Räbiger, Schnaubelt.     

Discontinuous local semiflows for Kurzweil equations leading to LaSalle's invariance principle for differential systems with impulses at variable times

In this paper, we consider an initial value problem for a class of generalized ODEs, also known as Kurzweil equations, and we prove the existence of a local semidynamical system there. Under certain perturbation conditions, we also show that this class of generalized ODEs admits a discontinuous semiflow which we shall refer to as an impulsive semidynamical system. As a consequence, we obtain LaSalle's invariance principle for such a class of generalized ODEs. Due to the importance of LaSalle's invariance principle in studying stability of differential systems, we include an application to autonomous ordinary differential systems with impulse action at variable times.     

Almost periodic solutions for abstract impulsive differential equations

This paper studies the existence and uniqueness of exponentially stable almost periodic solutions for abstract impulsive differential equations in Banach space. The investigations are carried out by means of the fractional powers of operators. We construct an example to illustrate the feasibility of our results.     

Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I

The well-known Favard's theorem states that the linear differential equation

(1)     

Normal forms for semilinear equations with non-dense domain with applications to age structured models

Normal form theory is very important and useful in simplifying the forms of equations restricted on the center manifolds in studying nonlinear dynamical problems. In this paper, using the center manifold theorem associated with the integrated semigroup theory, we develop a normal form theory for semilinear Cauchy problems in which the linear operator is not densely defined and is not a Hille–Yosida operator and present procedures to compute the Taylor expansion and normal form of the reduced system restricted on the center manifold. We then apply the main results and computation procedures to determine the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions in a structured evolutionary epidemiological model of influenza A drift and an age structured population model.     

A Model for Evaluation of Transport Policies in Multimodal Networks with Road and Parking Capacity Constraints

This paper presents a model for evaluation of transport policies in multimodal networks with road and parking capacity constraints. The proposed model simultaneously considers choices of travelers on route, parking location and mode between auto and transit. In the proposed model, it is assumed that auto drivers make a simultaneous route and parking location choice in a user equilibrium manner, and the modal split between auto and transit follows a multinomial logit formulation. A mathematical programming model with capacity constraints on road link and parking facilities is proposed that generates optimality conditions equivalent to the requirements for multimodal network equilibrium. An augmented Lagrangian dual algorithm embedded by partial linearization approach is developed to solve the proposed model. Numerical results on two example networks are presented to illustrate the proposed methodology. The results show that the service level of transit, parking charges, road link and parking capacities, and addition of a new parking location may bring significant impacts on travelers’ behavior and network performance. In addition, transport policies may result in paradoxical phenomenon.     

Stability of periodic solutions of state-dependent delay-differential equations

We consider a class of autonomous delay-differential equations

     

Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. II

In this paper we continue the research started in a previous paper, where we proved that the linear differential equation

(1)     

Saddle-point behavior for monotone semiflows and reaction-diffusion models

The saddle-point behavior is established for monotone semiflows with weak bistability structure and then these results are applied to three reaction-diffusion systems modeling man-environment-man epidemics, single-loop positive feedback and two species competition, respectively.     

Stability of ZakianI MN recursions for linear delay differential equations

Long sequences of linear delay differential equations (DDEs) frequently occur in the design of control systems with delays using iterative-numerical methods, such as the method of inequalities. ZakianI _MN recursions for DDEs are suitable for solving this class of problems, since they are reliable and provide results to the desired accuracy, economically even if the systems are stiff. This paper investigates the numerical stability property of theI _MN recursions with respect to Barwell's concept ofP-stability. The result shows that the recursions using full gradeI _MN approximants areP-stable if, and only if,N−2≤M≤N−1.     

Existence and controllability results for impulsive partial functional differential inclusions

In this paper we prove the existence and the controllability of mild and extremal mild solutions for first-order semilinear densely defined impulsive functional differential inclusions in separable Banach spaces with local and nonlocal conditions.     

Attractivity,multistability, and bifurcation in delayed Hopfieldʼs model with non-monotonic feedback

For a system of delayed neural networks of Hopfield type, we deal with the study of global attractivity, multistability, and bifurcations. In general, we do not assume monotonicity conditions in the activation functions. For some architectures of the network and for some families of activation functions, we get optimal results on global attractivity. Our approach relies on a link between a system of functional differential equations and a finite-dimensional discrete dynamical system. For it, we introduce the notion of strong attractor for a discrete dynamical system, which is more restrictive than the usual concept of attractor when the dimension of the system is higher than one. Our principal result shows that a strong attractor of a discrete map gives a globally attractive equilibrium of a corresponding system of delay differential equations. Our abstract setting is not limited to applications in systems of neural networks; we illustrate its use in an equation with distributed delay motivated by biological models. We also obtain some results for neural systems with variable coefficients.     

Stability of nonlinear waves in a ring of neurons with delays

In this paper, we consider a ring of identical neurons with self-feedback and delays. Based on the normal form approach and the center manifold theory, we derive some formula to determine the direction of Hopf bifurcation and stability of the Hopf bifurcated synchronous periodic orbits, phase-locked oscillatory waves, standing waves, mirror-reflecting waves, and so on. In addition, under general conditions, such a network has a slowly oscillatory synchronous periodic solution which is completely characterized by a scalar delay differential equation. Despite the fact that the slowly oscillatory synchronous periodic solution of the scalar equation is stable, we show that the corresponding synchronized periodic solution is unstable if the number of the neurons is large or arbitrary even.