Averaging method in systems with impulses

Mitropol’skii, Yu. A.; Самойленко, А. М.; Parestyuk, N. A.

doi:10.1007/bf01056851

Cited by 4 publications

(12 citation statements)

References 2 publications

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“…Theorem 2 [6]. In order that solutions of system (2) These definitions are well known. The definitions for the pulse system (2) do not differ from corresponding dx definitions for the system --= A (t)x.…”

Section: Stability Of Linear Systems Of Differential Equations With Pmentioning

confidence: 96%

“…Later, Yu. 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).…”

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confidence: 99%

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Stability of solutions of pulsed systems

Perestyuk

Chernikova²

1997

Ukr Math J

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We present the principai results in the theory of stability of pulse differential equations obtained by mathematicians of the Kiev scientific school of nonlinear mechanics. We also present some results of foreign authors.In the last I5-20 years, the theory of differential equations with pulse influence has been extensively developed. Pulse differential equations appear in various problems of nonlinear mechanics; pulse differential equations are equations that provide an adequate description for real processes in systems subjected to the action of instantaneous perturbations. The theory of pulse differential equations is basically founded and developed by scientists from the Kiev school of nonlinear mechanics which is associated with the names of such prominent scientists as N. M. Ka-ylov, N. N. Bogolyubov, and Yu. A. Mitropol'skii. As early as in 1937, by using the averaging method, oscillations of a pendulum under pulsed influence were studied in [1]. Later, Yu. 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.It turned out that the method of integral manifolds, which was developed in the works of Yu. A. Mitropol'skii and his disciples for various classes of differential equations (see, e.g., [ 13]), is quite efficient for the investigation of stability of systems under pulsed influence.Below, we present results which, from our point of view, describe the contemporary state of the theory of stability of systems with pulsed influence.Let the following system of differential equations with pulsed influence be given dx --= f(t, x), t 7~ zi(x), dt

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Section: Stability Of Linear Systems Of Differential Equations With Pmentioning

confidence: 96%

mentioning

confidence: 99%

Stability of solutions of pulsed systems

Perestyuk

Chernikova²

1997

Ukr Math J

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“…For the case I(k) = I= const ~ R, the dynamical system (14), (15) was comprehensively studied in [6], where one can find explicit expressions for the solutions of problem (14), (15) and the description of the behavior of its phase trajectories. In particular, it was established that all solutions of problem (14), (15) are periodic, although, generally speaking, with different periods.…”

Section: Pct/)mentioning

confidence: 99%

“…As in the case of constant pulsed influence [6], in the general case 1 = I(5:), all trajectories of the dynamical system (14), (15) whose initial data belong to the domain { (x, 5:) : 5:2 + r < o32xg} C R x R are periodic with period T= 2n/(.o and are not affected by pulses. If, at a certain time, the phase point of system (14), (15) …”

Section: Pct/)mentioning

confidence: 99%

On periodic solutions of linear differential equations with pulsed influence

Самойленко

Elgondyev

1997

Ukr Math J

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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...

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“…This method was generalized to various classes of differential equations [1][2][3][4][5], including quasilinear systems with pulse action [6]. Conditions for the existence of an integral manifold for a single-frequency pulse oscillation system were established in [7]. In [8], this result was generalized to the case of multifrequency resonance systems.…”

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confidence: 99%

Stability of the integral manifold of an oscillation system with slowly varying frequencies and pulse action

Petryshyn

Dudnyts’kyi

2004

Nonlinear Oscill

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517.928Using uniform estimates for oscillation integrals and sums, we prove theorems on asymptotic and conditional asymptotic stability of the integral manifold of a multifrequency system of ordinary differential equations with pulse action at fixed times.The method of integral manifolds in the theory of differential equations is an efficient mathematical apparatus because the problem of the determination of solutions of these equations becomes considerably simpler if they belong to a manifold of lower dimension than the dimension of the phase space. This method was generalized to various classes of differential equations [1][2][3][4][5], including quasilinear systems with pulse action [6]. Conditions for the existence of an integral manifold for a single-frequency pulse oscillation system were established in [7]. In [8], this result was generalized to the case of multifrequency resonance systems.In the present paper, using uniform estimates of oscillation integrals and sums, we study the problem of the stability of the integral manifold constructed in [8] for a pulse oscillation system with slowly varying frequencies.Consider the following nonlinear oscillation system of ordinary differential equations with slowly varying frequencies and pulse action at fixed times t ν = ε −1 τ ν : dx dτ = a(x, τ ) +ã(x, ϕ, τ ) + εA(x, ϕ, τ, ε),θ is a positive number, D is a bounded domain, and the right-hand sides of the equalities in (1) belong to certain classes of functions smooth and 2π-periodic in each coordinate ϕ s , s = 1, m, of the vector ϕ. Without loss of generality, we can assume that the averages with respect to ϕ in the cube of periods of the functionsã andp are identically equal to zero. Let

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Averaging method in systems with impulses

Cited by 4 publications

References 2 publications

Stability of solutions of pulsed systems

Stability of solutions of pulsed systems

On periodic solutions of linear differential equations with pulsed influence

Stability of the integral manifold of an oscillation system with slowly varying frequencies and pulse action

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