# Relaxation oscillation

A periodic process in which slow smooth change of the state of an object over a finite interval of time is alternated with rapid irregular change of the state during an infinitely short time. Such oscillatory processes are observed in many real mechanical, radiotechnical, biological, etc., objects (see e.g. ).

The mathematical models describing relaxation oscillations are autonomous systems (cf. Autonomous system) of ordinary differential equations with a small parameter in front of some of the derivatives:

$$\tag{1 } \epsilon \dot{x} = f( x, y),\ \ \dot{y} = g( x, y),\ \ \dot{ {}} = \frac{d}{dt} ,$$

$$x \in \mathbf R ^ {k} ,\ y \in \mathbf R ^ {m} ,\ 0 < \epsilon \ll 1 .$$

A periodic solution with respect to the time $t$ of such a system is called a relaxation oscillation. The traditional example of a system with one degree of freedom and having relaxation oscillations is the van der Pol equation

$$\tag{2 } \frac{d ^ {2} x }{d \tau ^ {2} } - \lambda ( 1- x ^ {2} ) \frac{d x }{d \tau } + x = 0$$

for large positive values of the parameter $\lambda$( from this point of view the value $\lambda = 10$ can be considered as large). Putting

$$y = \int\limits _ { 0 } ^ { x } ( x ^ {2} - 1) dx + \frac{1} \lambda \frac{dx}{d \tau } ,\ \ t = \frac \tau \lambda ,\ \ \epsilon = \frac{1}{\lambda ^ {2} } ,$$

equation (2) is reduced to a system of the form (1):

$$\epsilon \dot{x} = y - \frac{x ^ {3} }{3} + x,\ \ \dot{y} = - x.$$

The problem of the existence and number of relaxation oscillations in a system (1) is solved in terms of a degenerate system

$$\tag{3 } f( x, y) = 0,\ \ \dot{y} = g( x, y),$$

which is a hybrid system of equations. The trajectories of the system (3) in the phase space $\mathbf R ^ {k} \times \mathbf R ^ {m}$ are naturally treated as limits of the phase trajectories of the non-degenerate system (1) as $\epsilon \rightarrow 0$. In particular, the trajectory of a relaxation oscillation of the system (1), as $\epsilon \rightarrow 0$, tends towards a closed trajectory of the system (3) that consists of alternating sections of two types: sections lying on $f( x, y) = 0$ and satisfying $\dot{y} = g( x, y)$ and "jumps" from one point of $f( x, y) = 0$ to another. Each of these jumps starts at a break point, i.e. at a point where

$$f( x, y) = 0,\ \ \mathop{\rm det} \left \| \frac{\partial f }{\partial x } \right \| = 0$$

and lies in a plane parallel to $\mathbf R ^ {k}$. The solution of the system (3) corresponding to such a closed trajectory is called a discontinuous periodic solution. Consequently, the relaxation oscillation of the system (1) is often called the periodic solution close to the discontinuous one, or even simply the discontinuous oscillation. (The system (3) may have a closed trajectory entirely lying on the surface $f( x, y) = 0$ and not passing through a break point. In this case (1) has a closed trajectory near to it, but the periodic solution of (1) corresponding to it will not be a relaxation oscillation; see .)

An important question is the asymptotic (for $\epsilon \rightarrow 0$) calculation of the phase trajectory of the relaxation oscillation of the system (1), and the establishment of asymptotic formulas for the characteristics of this oscillation — its period, amplitude, etc. The trajectory of the relaxation oscillation of the van der Pol equation (2) has been calculated by A.D. Dorodnitsyn  by constructing asymptotic approximations, for $\lambda \rightarrow \infty$, for the amplitude

$$a = 2 + 0.77937 \lambda ^ {-} 4/3 - \frac{16}{27} \frac{ \mathop{\rm ln} \lambda }{ \lambda ^ {2} } - 0.8762 \lambda ^ {-} 2 + O( \lambda ^ {-} 8/3 ),$$

$$T = 1.613706 \lambda + 7.01432 \lambda ^ {-} 1/3 - \frac{2}{3} \frac{ \mathop{\rm ln} \lambda } \lambda +$$

$$- 1.3233 \lambda ^ {-} 1 + O( \lambda ^ {-} 5/3 ).$$

If the system (1) is of the second order (i.e. for $k= m= 1$) with break points in general position, the problem of the asymptotic calculation of the relaxation oscillation has been solved completely . In particular, the structure of the asymptotic expansion, for $\epsilon \rightarrow 0$, of the period of the relaxation oscillation has been clarified:

$$T = T _ {0} + \sum _ { n= } 2 ^ \infty \epsilon ^ {n/3} \sum _ {\nu = 0 } ^ { { } \chi ( n- 2) } K _ {n, \nu } \mathop{\rm ln} ^ \nu \frac{1} \epsilon ,$$

where

$$\chi ( n) = \frac{n}{3} + \frac{2 \sqrt 3 }{9} \mathop{\rm tan} \frac{\pi n }{3} ,\ \ n \in \mathbf Z ;$$

and the $K _ {n, \nu }$ are coefficients which are effectively calculable directly from the functions $f( x, y)$ and $g( x, y)$( see ). In a general system (1) of arbitrary order, the results of L.S. Pontryagin and E.F. Mishchenko (1983) have not been surpassed: they calculated the asymptotics of the relaxation oscillation correctly up to $O( \epsilon )$( see , , ).

Periodic solutions of the type of relaxation oscillations of non-autonomous systems of ordinary differential equations have also been studied (see e.g. ).

How to Cite This Entry:
Relaxation oscillation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Relaxation_oscillation&oldid=48503
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article