On periodic solutions of linear differential equations with pulsed influence

Abstract

We study periodic solutions of ordinary linear second-order differential equations with pulsed influence at fixed and nonfixed times.Numerous engineering and other practical problems are connected with the investigation of nonlinear dynamical systems with short-term processes or under the action of external forces whose duration may be neglected in creating the relevant mathematical models [1][2][3][4]. The classical example of a problem of this sort is the model of percussion clock mechanism [5]. Similar problems appear in many other fields of science and engineering, in particular, in metallurgy (in controlling temperature in thermal and open-hearth furnaces [6]), in chemical technology [7], in the dynamics of aircrafts [8], in mathematical economics [9], in medicine and biology [10], etc.The investigation of mathematical models for these systems leads to the necessity of the analysis of discontinuous dynamical systems generated by ordinary differential equations and conditions of pulsed action [4, 1 1J.The general characteristics of the qualitative behavior of solutions of systems of ordinary differential equations with pulsed influence, the similarity and difference between the problems in this field of applied mathematics and the corresponding problems in the theory of ordinary differential equations, as well as the main results obtained in this field are presented in [4,12]. Fundamental results concerning the applicability of asymptotic methods of nonlinear mechanics to weakly nonlinear differential equations with pulsed action can be found in [ 1,[13][14][15][16].In applying asymptotic methods to the investigation of weakly nonlinear differential equations, it is very important to study the so-called nonperturbed (generating) problem obtained in the case where the value of the corresponding small parameter is equal to zero. A nonlinear dynamical system generated by a linear ordinary differential equation of the form d2x dx + + qx = 0, p,q~ R,(1) d t 2 P -dtt and some conditions of pulsed action [4] can be regarded as a nonperturbed problem of this sort appearing as a result of asymptotic integration of weakly nonlinear differential equations with pulsed action [ 17,18]. There are several types of conditions of pulse action [4], among which one most often encounters conditions specified at fixed and nonfixed instants of time. In the general case, problems with pulsed action at nonfixed times are more complicated than problems with pulsed action at fixed instants of time. Mathematically, the conditions of pulsed action mean that there are certain rules according to which moving points of the considered dynamical system equation at the instants of pulsed action instantaneously changes its path by passing from one trajectory of the dynamical system to another. At first sight, this problem seems to be quite simple, but the behavior of trajectories in a dynamical system generated by ordinary differential equations with some conditions of pulsed action can be extremely complicated precisely due to the presence of...

Weakly nonlinear boundary-value problems for operator equations with pulse influence

“…(4), (5) is everywhere solvable, and this assertion holds, for example, for pulse ordinary differential.systems [3, p. 235] and for pulse systems with lag [4]. Moreover, in the case of ordinary differential systems, relation (9) is considerably simplified [1].…”

“…The method of investigation is based on passing by methods of the Lyapunov-Schmidt type from a pulse boundary-value problem to an equivalent operator system that can be solved by iteration procedures based on the fixed-point principle.The study of the qualitative theory of differential systems with pulse influence started in [1,2] has been further developed and extended in a great number of papers. Thus, an approach to the study of pulse periodic boundaryvalue problems for ordinary differential systems proposed in [1] was successfully developed and applied in the case of weakly nonlinear pulse boundary-value problems (with Noetherian linear part) for ordinary differential systems [3] and for linear boundary-value problems for differential systems with concentrated lag [4].…”

Stability of solutions of pulsed systems

“…This is due to the fact that solutions of pulse differential equations are piecewise continuous functions, whose discontinuity points depend on the solutions. Let us formulate more precise definitions of the main notions of the stability theory for system (l) (see, e.g., [4][5][6]). …”

“…A. Mitropol'skii, A. M. Samoilenko, and N. A. Perestyuk developed the ideas of [1] and applied these ideas to a larger class of systems under pulsed influence (a detailed list of the investigated problems can be found in the survey [2]). The investigations of scientists from the Kiev school stimulated comprehensive systematic studies of pulse differential equations in many other scientific centers (see [3,4] for more details). As a separate direction of the theory of pulse differential equations, the theory of stability of pulse systems was formed.…”

“…Fundamental results, associated with the investigation of the stability of solutions of differential equations with pulsed influence, were established by A. M. Samoilenko and N. A. Perestyuk. In [3][4][5][6][7][8][9][10][11][12], the classical theory of the first and second Lyapunov methods was extended to the case of pulse systems. Later, various results in the theory of stability of pulse systems were established by foreign scientists.…”