Exponentially asymptotically stable dynamical systems

In this paper, we give some sufficient conditions which guarantee that the zero solution of the differential equation with two delayed terms x(t) = a(t)x(t-p(t)) - b(t)x(t-r(t))is uniformly stable     

Asymptotic and exponential stability of certain third-order non-linear delayed differential equations: Frequency domain method

We use the frequency domain method to prove that the zero solution of certain third order nonlinear delayed differential equations is asymptotically stable, (when there is no forcing term). We also prove the existence of a bounded solution which is exponentially stable, (when there is a bounded forcing term). The situation for which the non-linear term is delayed is also proved.     

Study the stability of solutions of functional differential equations via fixed points

In this paper, we study the stability properties of solutions of a class of functional differential equations with variable delay. By using the fixed point theory under an exponentially weighted metric, we obtain some interesting sufficient conditions ensuring that the zero solution of the equations is stable and asymptotically stable.     

Stability of differential equations with piecewise constant arguments of generalized type

In this paper we continue to consider differential equations with piecewise constant argument of generalized type (EPCAG) [M.U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal. TMA 66 (2007) 367–383]. A deviating function of a new form is introduced. The linear and quasilinear systems are under discussion. The structure of the sets of solutions is specified. Necessary and sufficient conditions for stability of the zero solution are obtained. Our approach can be fruitfully applied to the investigation of stability, oscillations, controllability and many other problems of EPCAG. Some of the results were announced at The International Conference on Hybrid Systems and Applications, University of Louisiana, Lafayette, 2006.     

Families of Liapunov-Krasovskiį functionals and stability for functional differential equations

This paper discusses the stability of solutions of nonautonomous functional differential equations with infinite delay with respect to a parr of admissible phase spaces of Hale and Kato. A one-parameter family of Liapunov-Krasovskiį functional, together with some additional analysis, is used to prove new sufficient conditions of asymptotic and uniform asymptotic stability for such equations. It is also shown that the so-called Razumikhin condition is unessential when families of Liapunov-Krasovskiį functionals are used. Entrata in Redazione il 25 settembre 1997. Invited address at the Second Marrakesh International Conference on Differential Equations, Marrakesh, Morocco, June 1995.     

Existence of periodic solutions for a system of delay differential equations

In this paper we mainly study the existence of periodic solutions for a system of delay differential equations representing a simple two-neuron network model of Hopfield type with time-delayed connections between the neurons. We first examine the local stability of the trivial solution, propose some sufficient conditions for the uniqueness of equilibria and then apply the Poincaré-Bendixson theorem for monotone cyclic feedback delayed systems to establish the existence of periodic solutions. In addition, a sufficient condition that ensures the trivial solution to be globally exponentially stable is also given. Numerical examples are provided to support the theoretical analysis.     

Permanence and stability of a predator–prey system with stage structure for predator

A stage-structured predator–prey system with delays for prey and predator, respectively, is proposed and analyzed. Mathematical analysis of the model equations with regard to boundedness of solutions, permanence and stability are analyzed. Some sufficient conditions which guarantee the permanence of the system and the global asymptotic stability of the boundary and positive equilibrium, respectively, are obtained.     

Dynamics and stability of impulsive hybrid setvalued integro-differential equations with delay

We study the dynamics and stability theory for impulsive hybrid set integro-differential equations with delay. Sufficient conditions for the stability of the null solution of impulsive hybrid set integro-differential equations with delay are presented.     

Almost periodic solutions of recurrent neural networks with continuously distributed delays

In this paper, a class of recurrent neural networks with continuously distributed delays is discussed. Without resorting to the theory of exponential dichotomy, several new sufficient conditions are obtained ensuring the existence of an almost periodic solution for this model based on a special functional and analysis technique. Moreover, by constructing suitable Lyapunov functions, the attractivity and exponential stability of the almost periodic solution are also considered for the system. The results obtained are helpful to design globally stable almost periodic oscillatory neural networks. A numerical example is given to show the feasibility of the results obtained.     

Synchronization of the unified chaotic system

This paper studies the synchronization problem of the unified chaotic system. Three different methods, linear feedback method, nonlinear feedback method and impulsive control method are used to control synchronization of the unified chaotic systems. Based on the Lyapunov stability theory and impulsive control method, the conditions of synchronization are discussed, and they are also proved theoretically. Numerical simulations show the effectiveness of the three different methods.     

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.     

Stability criteria for impulsive systems on time scales

In this paper we study stability of impulsive system on time scales. By using comparison method, Lyapunov function and analysis method, the asymptotic stability criteria for system with impulses at fixed times and impulses at variable times on time scales are obtained, respectively. An example is presented to illustrate the efficiency of proposed result.     

Dynamic behaviors of a delay differential equation model of plankton allelopathy

In this paper, we consider a modified delay differential equation model of the growth of n-species of plankton having competitive and allelopathic effects on each other. We first obtain the sufficient conditions which guarantee the permanence of the system. As a corollary, for periodic case, we obtain a set of delay-dependent condition which ensures the existence of at least one positive periodic solution of the system. After that, by means of a suitable Lyapunov functional, sufficient conditions are derived for the global attractivity of the system. For the two-dimensional case, under some suitable assumptions, we prove that one of the components will be driven to extinction while the other will stabilize at a certain solution of a logistic equation. Examples show the feasibility of the main results.     

On the stability and boundedness of solutions of nonlinear vector differential equations of third order

In this paper, by using Liapunov’s second method, we establish some new results for stability and boundedness of solutions of nonlinear vector differential equations of third order. By constructing a Liapunov function, sufficient conditions for stability and boundedness of solutions of equations considered are obtained. Concerning to the subject, some explanatory examples are also given. Our results improve and include a result existing in the literature.     

Novel delay-dependent asymptotical stability of neutral systems with nonlinear perturbations

This paper address the asymptotical stability of neutral systems with nonlinear perturbations. Some novel delay-dependent asymptotical stability criteria are formulated in terms of linear matrix inequalities (LMIs). The resulting delay-dependent stability criteria are less conservative than the previous ones, owing to the introduction of free-weighting matrices, based on a class of novel augment Lyapunov functionals. Numerical examples are given to demonstrate that the derived conditions are much less conservative than those given in the literature.     

Problems of the partial stability and detectability of dynamical systems

The conditions under which uniform stability (uniform asymptotic stability) with respect to a part of the variables of the zero equilibrium position of a non-linear non-stationary system of ordinary differential equations signifies uniform stability (uniform asymptotic stability) of this equilibrium position with respect the other, larger part of the variables, which include an additional group of coordinates of the phase vector, are established. These conditions include the condition for uniform asymptotic stability of the zero equilibrium position of the “reduced” subsystem of the original system with respect to the additional group of variables. Since within the conditions obtained the stability with respect to the remaining unmeasured coordinates of the phase vector remains undetermined or is investigated additionally, partial zero-detectability of the original system occurs in this case, and the conditions obtained supplement the series of known results from partial stability theory. The application of the results obtained to problems of the partial stabilization of non-linear controlled systems, particularly to the problem of stabilizing an asymmetric rigid body relative to an assigned direction in an inertial space, is considered. The partial detectability of linear systems with constant coefficients is also investigated.     

Dynamic Hopf bifurcations generated by nonlinear terms

We consider the so-called delayed loss of stability phenomenon for singularly perturbed systems of differential equations in case that the associated autonomous system with a scalar parameter undergoes the Hopf bifurcation at the zero equilibrium point. It is assumed that the linearization of the associated system is independent of the parameter and the next terms in the expansion of the right-hand parts at zero are positive homogeneous of order α>1. Simple formulas are presented to estimate the asymptotic delay for the delayed loss of stability phenomenon. More precisely, we suggest sufficient conditions which ensure that zeros of a simple function ψ defined by the positive homogeneous nonlinear terms are the Hopf bifurcation points of the associated system, the sign of ψ at other points determines stability of the zero equilibrium, and the asymptotic delay equals the distance between the bifurcation point and a zero of some primitive of ψ.     

Stability analysis for neutral systems with mixed delays

Relationships between system states contained in the neutral equation are used to address the delay-dependent stability of a neutral system with time-varying state delay. Using linear matrix inequalities, we present a new asymptotic stability criterion, and a new robust stability criterion, for neutral systems with mixed delays. Since the criteria take into account the sizes of the neutral delay, discrete delay and the derivative of discrete delay, they are less conservative than those produced by previous approaches. Numerical examples are presented to demonstrate that these criteria are indeed more effective.     

Asymptotic stability and instability with respect to part of variables for solutions to impulsive systems

We obtain sufficient conditions for asymptotic stability with respect to part of variables for the zero solution to an impulsive system with the fixed moments of impulse effects.     

Differential equations with state-dependent piecewise constant argument

A new class of differential equations with state-dependent piecewise constant argument is introduced. It is an extension of systems with piecewise constant argument. Fundamental theoretical results for the equations—the existence and uniqueness of solutions, the existence of periodic solutions, and the stability of the zero solution—are obtained. Appropriate examples are constructed.