Nonlinear Hybrid Continuous/Discrete-Time Models

It is observed, [280], that there are at least three scientific communities, which contribute to theory of hybrid systems: computer science, modeling and simulation, systems and control communities. The results of our book are about hybrid systems, which are obtained through modeling activity, and they are initiated in analysis of systems which operate in different modes, continuous and discrete. That is, we consider a narrow class of hybrid systems. Let us describe it more precisely. Denote by \( \mathbb{N},\;\mathbb{Z} \) and \( \mathbb{R} \) the sets of all natural numbers, integers and real numbers, respectively. The state variable in our book is finite dimensional, \( x \in \mathbb{R}^{n} ,\;n \in \mathbb{N}. \) Denote by x(t) the position of the state variable, where \( t \in \mathbb{R} \) is the time argument. Suppose that there is a discrete set of moments θk, \( k \in \mathbb{Z}, \) which one calls, switching moments [279]. We shall call also θk, an event time [280]. If t is in the continuous part, then x(t) satisfies a differential equation, otherwise the value of jumps of x is evaluated by jumps equation [20]. With each event time we associate a switch and a jump. That is, at each event time an event occurs, such that variables x and t jump, and the right-hand-side of the differential equation and the jumps equation switch at the event time. If one specify a hybrid system, then events and switching moments have to be determined. If events are externally induced, then the switching and jumps are controlled, otherwise they are autonomous [280]. In chapters 2–8 we analyze hybrid systems without jumps equations. In the last two chapters jumps equations are considered to model blood pressure distribution and biological integrate-and-fire oscillators.

In this chapter we start investigation with the most simple linear systems of differential equations with piecewise constant argument. Then the analysis will be extended to quasilinear systems. Existence-uniqueness of solutions, the linear space of solutions, fundamental matrix, stability problems are under discussion.

The theory of integral manifolds founded by H. Poincaré and A. M. Lyapunov [215, 267] became a very powerful instrument for investigating problems of the qualitative theory of differential equations. Over the past several decades, many researchers have been studying the methods of reducing high dimensional problems to low dimensional ones. If we discuss this problem for long-time dynamics of differential equations, we should consider the Reduction Principle [264, 265]. For a brief history of the principle, the reader is referred to the papers [199, 219, 264]. As it is well known that the principle was utilized in the center manifold theory, as well as in the theory of inertial manifolds [68, 118, 155]. On the other hand, it is natural that the exploration of the properties and neighborhoods of manifolds is one of the most interesting problems of the theory of differential equations [59, 68, 74, 153, 180, 248, 268]. One should not be surprised that integral manifolds and the reduction principle are among the major subjects of investigation for specific types of differential and difference equations [39,51,73,75,100,118,149,150,200,249–251,268,294]. The main novelty of this chapter is to extend the principle to differential equations with piecewise constant arguments.

The problem of the existence of periodic solutions is one of the most interesting topics for applications. The method of small parameter is introduced by Poincaré [267] to investigate the problem, and it has been developed by many authors (see, for example, [218, 273], and the references cited therein). This method remains as one of the most effective methods for this problem and it is important that the results obtained in this field can be extended to the bifurcation theory [48, 233]. In this chapter, we investigate the existence and stability of periodic solutions of quasilinear system with piecewise constant an argument and a small parameter in noncritical and critical cases. Theorems on continuous dependence of solutions with respect to initial conditions and parameters, and an analogue of the Gronwall-Bellman lemma are also proved. Examples illustrating the obtained results are constructed as well. In the first section of this chapter we consider a simpler problem of a non-critical case with the delay argument-function β (t) In the last section critical case is considered as well as the argument-function γ(t) of the mixed, advanced-delayed, type is used for the system.

In this section we develop the Lyapunov’s second method for stability of differential equations with piecewise constant argument of generalized type by employing the Razumikhin technique [150, 270]. Sufficient conditions are established for stability, uniform stability and uniform asymptotic stability of the trivial solution of such equations. We also provide appropriate examples to illustrate our results.

In previous chapters, the differential equations with piecewise constant argument of generalized type (differential equations with piecewise constant arguments) of the form \( {\frac{dx(t)}{dt}} = f(t,x(t),\;x(\beta (t))), \) (6.1) are considered, where β (t) = θ _i if θ _i  ≤ t < θ _i+1, i are integers, is an identification function, θ _i is a strictly increasing sequence of real numbers.

This chapter presents existence and stability of almost periodic solutions of the following system \( {\frac{dx(t)}{dt}} = A(t)x(t) + f(t,x(\theta_{\upsilon (t) - p1} ),x(\theta_{\upsilon (t) - p2} ), \ldots ,x(\theta_{\upsilon (t) - pm} )), \) (7.1) where \( x \in \mathbb{R}^{n} ,\;t \in \mathbb{R}, \) υ(t) = 1 if θ _i  ≤ t < θ _i+1, i = …,-2,-1,0,1,2,…, is an identification function, θi is a strictly ordered sequence of real numbers, unbounded on the left and on the right, pj, j = 1,2,…,m, are fixed integers, and the linear homogeneous system associated with (7.1) satisfies exponential dichotomy. The problem of the existence is studied without any sign condition on deviations of the argument.

In this chapter, the method of Lyapunov functions for differential equations with piecewise constant argument considered in Chapter 5 is applied to a model of recurrent neural networks (RNNs). The model includes both advanced and delayed arguments. We obtain sufficient conditions for global exponential stability of the equilibrium point. The feasibility of the results are illustrated by examples with numerical simulations.

In paper [262] C. Peskin develops the integrate-and-fire model of the cardiac pacemaker [181] to a population of identical pulse-coupled oscillators. Thus, it was proposed to consider a model of cardiac pacemaker, where signal of fire arises not from an outside stimuli, but in the population of cells itself. Then, the well known conjectures of selfsynchronization were formulated. Solution of these conjectures for identical oscillators [216, 262], and a wide discussion of the subject stimulated mathematicians as well as biologists for the intensive investigations in the field [60, 109, 122, 188, 222, 285, 296, 298, 299]. It is natural that the problem has been considered in the more general form. In [216] the method of phase diagrams effectively is used to discuss the models. In the paper [24] we suggest a special map, which helped us to solve the synchronization problem for nonidentical oscillators. A version of the model is considered such that perturbations can be evaluated still to save the synchronization.