View source for Non-linear oscillations

Oscillations in physical systems described by non-linear systems of ordinary differential equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n0671601.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n0671602.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n0671603.png" /> contains terms of at least the second degree in the components of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n0671604.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n0671605.png" /> is a vector function of the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n0671606.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n0671607.png" /> is a small parameter (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n0671608.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n0671609.png" />). Possible generalizations are connected with the discussion of discontinuous systems, actions with discontinuous characteristic (for example, of hysteresis type), delay and random actions, integro-differential and differential-operator equations, oscillating systems with distributed parameters described by partial differential equations, and also with the use of methods of optimal control of non-linear oscillating systems. The basic general problems of non-linear oscillations are: the search for equilibrium positions, for stationary regimes (in particular, for periodic motions, auto-oscillations), and the investigation of their stability, and problems of synchronization and stabilization of non-linear oscillations.

Strictly speaking, all physical systems are non-linear. One of the most characteristic features of non-linear oscillations is the violation of the principle of superposition of oscillations: The result of every action in the presence of a second turns out to be different from the case when the second action is absent. Quasi-linear systems are systems (1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716010.png" />. A basic method for studying them is the method of the small parameter (cf. [[Small parameter, method of the|Small parameter, method of the]]). Above all there is the Poincaré–Lindstedt method for determining periodic solutions of quasi-linear systems that are analytic in the parameter for sufficiently small values of it, either in the form of power series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716011.png" /> (see [[#References|[1]]], Chapt. IX) or in the form of power series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716012.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716013.png" /> added to the initial values of the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716014.png" /> (see [[#References|[1]]], Chapt. III). For the subsequent development of this method see, for example, [[#References|[2]]]–[[#References|[4]]].

Another method of a small parameter is that of [[Averaging|averaging]]. At the same time, in the study of quasi-linear systems new methods emerged: asymptotic methods (see [[#References|[5]]], [[#References|[6]]]), the method of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716016.png" />-functions (see [[#References|[7]]]), which is based on the fundamental results of A.M. Lyapunov and N.G. Chetaev, and others.

Essentially non-linear system are those in which the small parameter prescribed in advance is absent (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716018.png" /> in (1)). For the Lyapunov system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716020.png" /></td> </tr></table>

and where there are no multiple roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716021.png" /> among the eigen values of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716022.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716024.png" /> is an analytic vector function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716025.png" /> the expansion of which begins with terms of at least second order, and there is an analytic first integral of special form. Lyapunov (see [[#References|[8]]], Sect. 42) has proposed a method of search for periodic solutions in the form of a series in powers of an arbitrary constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716026.png" /> (for which one can use the initial value of one of the two critical variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716027.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716028.png" />).

For systems close to Lyapunov systems,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716029.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716031.png" /> are as in (2) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716032.png" /> is an analytic vector function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716033.png" /> and a small parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716034.png" />, continuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716035.png" />-periodic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716036.png" />, also a method has been proposed for determining periodic solutions (see [[#References|[4]]], Chapt. VIII). Systems of Lyapunov type (2) in which the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716037.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716038.png" /> zero eigen values with simple elementary divisors, a pair of purely imaginary eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716039.png" /> and no multiple eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716040.png" />, and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716041.png" /> is as in (2), may be reduced to Lyapunov systems (see [[#References|[9]]], Chapt. IV. 2). A study has also been made of non-linear oscillation in Lyapunov systems and in so-called Lyapunov systems with damping, and the general problem of energy transfer in them has been solved (see [[#References|[9]]], Chapts. I, III, IV).

Suppose that an essentially non-linear [[Autonomous system|autonomous system]] has been reduced to the Jordan form of its linear part:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716043.png" /></td> </tr></table>

where the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716044.png" /> has by assumption at least one non-zero component, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716045.png" /> are zero or one according whether non-simple elementary divisors of the matrix of the linear part are absent or present, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716046.png" /> are coefficients, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716047.png" /> is the set of values of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716048.png" /> with integer components such that:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716049.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716050.png" /></td> </tr></table>

Then there exists a normalizing transformation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

reducing (3) to the [[Normal form|normal form]] of differential equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716052.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716053.png" /></td> </tr></table>

such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716054.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716055.png" />. Thus, the normal form (5) contains only resonance terms, that is, the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716056.png" /> in (5) can be non-zero only for those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716057.png" /> for which the resonance equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716058.png" /></td> </tr></table>

which plays an essential role in the theory of oscillations, holds. The convergence and divergence of the normalizing transformation (4) has been investigated (see , Part I, Chapts. II, III); the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067160/n06716059.png" /> have been computed (by means of their symmetrization) (see [[#References|[9]]], Sect. 5.3). In a number of problems on non-linear oscillations of essentially non-linear autonomous systems, the method of normal forms has proved effective (see , [[#References|[9]]], Chapts. VI–VIII).

Of the other methods for studying essentially non-linear systems the method of point mappings (see [[#References|[2]]], [[#References|[11]]]), the stroboscopic method [[#References|[12]]] and methods of functional analysis [[#References|[13]]] have been applied.

Qualitative methods for non-linear oscillations. The starting point here is the study of the form of the integral curves of non-linear ordinary differential equations, which was undertaken by H. Poincaré (see [[#References|[14]]]). Applications to problems of non-linear oscillations that can be described by second-order autonomous systems can be found in [[#References|[2]]] and [[#References|[15]]]. Questions of the existence of periodic solutions and their stability in the large for many-dimensional systems have been studied [[#References|[16]]]; almost-periodic non-linear oscillations are treated in [[#References|[13]]]. For applications of the theory of ordinary differential equations with a small parameter in front of some of the derivatives to problems of non-linear relaxation oscillations (cf. [[Relaxation oscillation|Relaxation oscillation]]) see [[#References|[17]]].

For other important aspects of non-linear oscillations and further references see [[Perturbation theory|Perturbation theory]] and [[Oscillations, theory of|Oscillations, theory of]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Poincaré, "Oeuvres" , '''1''' , Gauthier-Villars (1951) {{MR|1787680}} {{MR|1401792}} {{MR|1401348}} {{MR|1401791}} {{MR|0392476}} {{MR|0392475}} {{MR|1554897}} {{ZBL|0894.01021}} {{ZBL|0894.01020}} {{ZBL|0894.01019}} {{ZBL|0072.24103}} {{ZBL|0059.00104}} {{ZBL|0059.00103}} {{ZBL|0049.44102}} {{ZBL|0041.37403}} {{ZBL|0041.37402}} {{ZBL|46.0004.01}} {{ZBL|36.0022.04}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Pergamon (1966) (Translated from Russian) {{MR|0198734}} {{ZBL|0188.56304}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> B.V. Bulgakov, "Oscillations" , Moscow (1954) (In Russian) {{MR|0069347}} {{ZBL|0948.81547}} {{ZBL|0061.19313}} {{ZBL|0061.19312}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.G. Malkin, "Some problems in the theory of non-linear oscillations" , Moscow (1956) (In Russian) {{MR|81402}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> N.N. Bogolyubov, "Selected works" , '''1''' , Kiev (1969) (In Russian) {{MR|1321817}} {{MR|1228446}} {{MR|1098017}} {{MR|1098016}} {{MR|0871873}} {{MR|0804234}} {{MR|0555732}} {{MR|0414297}} {{ZBL|0226.01021}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> N.N. Bogolyubov, Yu.A. Mitropol'skii, "Asymptotic methods in the theory of non-linear oscillations" , Gordon &amp; Breach , Delhi (1961) (Translated from Russian) {{MR|0100379}} {{ZBL|0151.12201}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> G.B. Kamenkov, "Selected works" , '''1–2''' , Moscow (1971–1972) (In Russian) {{MR|0492597}} {{MR|0469662}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian) {{MR|2559078}} {{MR|1229075}} {{MR|1179692}} {{MR|1154209}} {{MR|0569069}} {{MR|0039884}} {{ZBL|0161.06303}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> V.M. Starzhinskii, "Applied methods in the theory of nonlinear oscillations" , MIR (1980) (Translated from Russian) {{MR|0587667}} {{ZBL|}} </TD></TR><TR><TD valign="top">[10a]</TD> <TD valign="top"> A.D. [A.D. Bryuno] Brjuno, "Analytical form of differential equations" ''Trans. Moscow Math. Soc.'' , '''25''' (1972) pp. 131–288 ''Trudy Moskov. Mat. Obshch.'' , '''25''' (1971) pp. 119–262</TD></TR><TR><TD valign="top">[10b]</TD> <TD valign="top"> A.D. [A.D. Bryuno] Brjuno, "The analytic form of differential equations II" ''Trans. Moscow Math. Soc.'' , '''26''' (1972) pp. 199–239 ''Trudy Moskov. Mat. Obshch.'' , '''26''' (1972) pp. 199–239</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> Yu.I. Neimark, "The method of point mappings in the theory of non-linear oscillations" , Moscow (1972) (In Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> N. Minorski, "Nonlinear oscillations" , v. Nostrand (1962) (Translated from Russian)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> M.A. Krasnosel'skii, V.Sh. Burd, Yu.S. Kolesov, "Nonlinear almost periodic oscillations" , Wiley (1973) (Translated from Russian) {{MR|344595}} {{ZBL|0287.34038}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> H. Poincaré, "Mémoire sur les courbes définies par une équation différentielle I - IV" , ''Oeuvres de H. Poincaré'' , '''1''' , Gauthier-Villars (1951) pp. Part V; nos. 74–77</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> N.V. Butenin, Yu.I. Neimark, N.A. Fufaev, "Introduction to the theory of non-linear oscillations" , Moscow (1976) (In Russian)</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> V.A. Pliss, "Nonlocal problems of the theory of oscillations" , Acad. Press (1966) (Translated from Russian) {{MR|0196199}} {{ZBL|0151.12104}} </TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> E.F. Mishchenko, N.Kh. Rozov, "Differential equations with small parameters and relaxation oscillations" , Plenum (1980) (Translated from Russian) {{MR|0750298}} {{ZBL|0482.34004}} </TD></TR></table>



====Comments====


====References====
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