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. [1]–[3]).

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 [6].)

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 [7] by constructing asymptotic approximations, for $ \lambda \rightarrow \infty $, for the amplitude

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

and for the period (see also [8])

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

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

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 [9]. 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 [10]). 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 [11], [12], [9]).

Periodic solutions of the type of relaxation oscillations of non-autonomous systems of ordinary differential equations have also been studied (see e.g. [13]).

References

[1]	A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Dover, reprint (1987) (Translated from Russian) MR0925417 Zbl 0188.56304
[2]	N.S. Landa, "Auto-oscillations in systems with a finite number of degrees of freedom" , Moscow (1980) (In Russian) MR0600107
[3]	Yu.M. Romanovskii, N.V. Stepanova, D.S. Chernavskii, "Mathematical modelling in biophysics" , Moscow (1975) (In Russian)
[4]	B. van der Pol, Phil. Mag. Ser. 7 , 2 : 11 (1926) pp. 978–992
[5]	N.A. Zheleztsov, L.V. Rodygin, Dokl. Akad. Nauk SSSR , 81 : 3 (1951) pp. 391–394
[6]	D.V. Anosov, "Limit cycles of systems of differential equations with small parameters in front of the highest derivatives" Mat. Sb. , 50 : 3 (1960) pp. 299–334 (In Russian) MR140772
[7]	A.A. Doronitsyn, "Asymptotic solution of van der Pol's equation" Prikl. Mat. i Mekh. , 11 : 3 (1947) pp. 313–328 (In Russian) (English abstract)
[8]	M.I. Zharov, E.F. Mishchenko, N.Kh. Rozov, "On some special functions and constants arising in the theory of relaxation oscillations" Soviet Math. Dokl. , 24 : 3 (1981) pp. 672–675 Dokl. Akad. Nauk SSSR , 261 : 6 (1981) pp. 1292–1296 Zbl 0504.34031
[9]	E.F. Mishchenko, N.Kh. Rozov, "Differential equations with small parameters and relaxation oscillations" , Plenum (1980) (Translated from Russian) MR0750298 Zbl 0482.34004
[10]	N.Kh. Rozov, "Asymptotic computation of solutions of systems of second-order differential equations close to discontinuous periodic solutions" Soviet Math. Dokl. , 3 : 4 (1962) pp. 932–934 Dokl. Akad. Nauk SSSR , 145 : 1 (1962) pp. 38–40 Zbl 0135.31002
[11]	L.S. Pontryagin, "Asymptotic behaviour of solutions of systems of differential equations with a small parameter in front of the highest order derivatives" Izv. Akad. Nauk SSSR Ser. Mat. , 21 : 5 (1957) pp. 605–626 (In Russian)
[12]	E.F. Mishchenko, "Asymptotic calculation of periodic solutions of differential equations with small parameters in front of the derivatives" Izv. Akad. Nauk SSSR Ser. Mat. , 21 : 5 (1957) pp. 627–654 (In Russian)
[13]	M. Levi, "Qualitative analysis of the periodically forced relaxation oscillations" , Amer. Math. Soc. (1981) MR0617687 Zbl 0448.34032

Comments

For weakly-forced and weakly-coupled relaxation oscillations one can derive formulas for synchronization and phase shift between the oscillators, see [a1]. In this reference also a survey of the Western literature on relaxation oscillations is given. Moreover, a method is presented to analyze chaotic relaxation oscillations.

When by a Hopf bifurcation a relaxation oscillation branches off from a stable equilibrium, a typical transition is observed known as the "canardcanard" . It was first analyzed mathematically with non-standard analysis, see [a2].

The study of perturbed equations

$$ \tag{a1 } \dot{x} = f( x, y) ,\ \ \dot{y} = \epsilon g ( x, y) $$

and the corresponding constrained differential equation

$$ \tag{a2 } \dot{x} = f( x, y) ,\ \ g ( x, y) = 0 $$

and their solutions in relation to each other belongs to the theory of singular perturbations; cf. also Differential equations with small parameter and Boundary-layer theory. A selection of references dealing with singular perturbations is [a3]–[a5]. Some early papers dealing with the interrelations of solutions of (a1), (a2) are [a6]–[a8]. To discuss the relations between solutions of (a1), (a2) one needs an appropriate concept of solution for constrained differential equations like (a2). For this see [a9], [a10].

A brief, explicit discussion of relaxation oscillations for the case of the van der Pol equation (complete with figure) can be found in [a11], pages 34-35, see also [a12], sections 14.2 and 14.3.

The phrase "relaxation oscillation" was introduced by B. van der Pol in 1926.

References

[a1]	J. Grasman, "Asymptotic methods for relaxation oscillations and applications" , Springer (1987) MR0884527 Zbl 0627.34037
[a2]	J.L. Callot, F. Diener, M. Diener, "Le problème de la "chasse au canard" " C.R. Acad. Sci. Paris , A286 (1987) pp. 1059–1061 MR0500368 Zbl 0419.34039
[a3]	K.W. Chang, F.A. Howes, "Nonlinear singular perturbation phenomena: theory and application" , Springer (1984) MR0764395
[a4]	W. Eckhaus, "Asymptotic analysis of singular perturbations" , North-Holland (1979) MR0553107 Zbl 0421.34057
[a5]	R.E. O'Malley, "Introduction to singular perturbations" , Acad. Press (1974) Zbl 0287.34062
[a6]	N. Levinson, "Perturbations of discontinuous solutions of nonlinear systems of differential equations" Acta Math. , 82 (1950) pp. 71–106 MR0022010
[a7]	N.R. Lebovitz, R. Schaar, "Exchange of stabilities in autonomous systems" Studies Appl. Math. , 54 (1975) pp. 229–260 MR0477319 MR0477320 Zbl 0312.34040
[a8]	J. Levin, N. Levinson, "Singular perturbations of nonlinear systems of differential equations and an associated boundary layer equation" J. Rat. Mech. Anal. , 3 (1954) pp. 247–270 MR61241
[a9]	F. Takens, "Constrained equations: a study of implicit differential equations and their discontinuous solutions" P. Hilton (ed.) , Structural Stability, the Theory of Catastrophes, and Applications in the Sciences , Springer (1976) pp. 143–234 MR0478236 Zbl 0386.34003
[a10]	S.S. Sastry, C.A. Desoer, P.P. Varaiya, "Jump behaviour of circuits and systems" IEEE Trans. Circuits and Systems (1980)
[a11]	A.H. Nayfeh, "Perturbation methods" , Wiley (1973) MR0404788 Zbl 0265.35002
[a12]	M.I. Rabinovich, D.I. Trubetskov, "Oscillations and waves in linear and nonlinear systems" , Kluwer (1989) (Translated from Russian) MR1037263 Zbl 0712.70001