Ed2 + (U2-1)du + u = 0; dt!= Xi(xi, PERTURBATIONS OF DISCONTINUOUS SOLUTIONS OF NON- plane could be exploited).

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1 214 MA TIIEMAA ICS: N. LEVINSON PROC. N^. A. S. PERTURBATIONS OF DISCONTINUOUS SOLUTIONS OF NON- LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS BY NORMAN LEVINSON MASSACHUSETTS INSTITUTE OF TECHNOLOGY Communicated May 3, 1947 Theorems about solutions of the system!= Xi(xi, P...,xn,t, e), i = 1,..., n,(1 where the Xi are regular functions of e for small 1, are classical. More recently studies of cases of (1) subject to the various conditions that allow one or several Xi to become large as e > 0 have been studied.1 In all the studies just referred to it is assumed that the system (1) has' a solution with a continuous derivative in case E = 0; sufficient conditions are given for this also to be the case for e $ 0. There are cases of considerable interest where; when e = 0, the system (1) has only a discontinuous "solution." It is known as a practical matter that in these cases when e $ 0 but small the system may have a continuous solution which approaches the discontinuous one as e - 0. Practical applications of this have been made by the Russian school of non-linear mechanics (Mandlestam, Chaikin, Lochakow). A rigorous treatment of this case appears so far to have been given2 only for a special relaxation oscillation problem involving two unknown functions (so that the simple topology of the phase plane could be exploited). Here the following system of arbitrary order is considered: dx, du f- + +zi=1, 2,...,p ni d2u du e- 2 + g- +h 0. (2) In (2), fs, 4,, g and h are all functions of xi,...x,, u, and t which we shall denote hereafter by fj(x, u, t), etc. We shall first show that (2) is general enough to include certain cases of obvious interest: Example 1. The van der Pol equation, with a change of time scale, becomes Example 2. Ed2 + (U2-1)du + u = 0; 2 The Rayleigh type equation, (X + (X2-1)X + X + X3 = 0,

2 VOL. 33, 1947 MA THEMA TICS: N. LEVINSON 2152 is not included in (2). dx However if we differentiate it and set * = u we get =U, which is of the form (2); Exdmple 3. Consider the system3 2 + (3U2 1)- + U +'3X2U = =Hi(x, w, t),i= 1,...,n, dw Ct= G(x, w, t). In case the right members above are not linear in w, this system can be brought to the form (2) simply by differentiating the last equation only with respect to t. In case the right members are linear in w, we can introduce u defined by u w and the system assumes the form dxi d-u -= f(x, t)d- + (x, t), d2u du = g(x, t)- + dyt2 h(x, k(t) t), which is a special case of (2). To formulate our result precisely we consider along with (2) where e > 0, the degenerate system dy= f(yp vi t) dv + Oj(y v, t), i l1 2,...n. dv g(y, v, t)w- + h(y, v, 't)=0, (3) which is (2) with e = 0. We shall regard a solution of (3) 'as a curve Co in En+2, the n + 2 dimensional space (y, v, t). We shall assume that there is an open continuum D in En+2 which contains the solution Co and in which fi, 4, g and h are continuous and have continuous first order partial derivatives with respect to yj, v and t. Definition. The n + 1 functions yi(t), v(t), denoted briefly by the n + 1 dimensional vector Y(t), is said to be a solution of (2) the degenerate system (3) in the interval a _ t _,B if:

3 216 MA THEMA TICS: N. LE VINSON PROC. N. A. S. (1.1) Y(t) is continuous and possesses a continuous derivative except possibly at a finite number of interior points, rj, of the closed interval (a, d) and except at the rj, Y(t) = (yi(t), v(t)) satisfies (3). (1.2) Except at the points rj, g(y(t), v(t), t) $ 0. (1.3) Both Y(rj - 0) and Y(rj + 0) exist, and we denote them by Y- and Y+, respectively. Similarly we define y-, v-, etc. Byfi- we mean fi(y-, V-, Tj), etc. -(1.4) For each j, g- = 0; (1.5) For eachj, g+ $ 0; (1.6) Denoting by y(v, rj) the solution, Yi, of dy, -= f2(y' V, Tr) dv for which y(v-, Tj) = y- and y(v+, Tj) = y+, let v+ f g(y(v, rj), v, Tj)dv = 0. v_ (1.7) There exists no vo interior to the interval (v-, v+) such that the integral in (1.6) vanishes if v+ is replaced by vo in the upper limit of the integral. (1.8) Let I denote -)g - + E flbg -fi. Replacing y by y-, v by v- and t by Ti, c-) i layi I becomes I-. We assume that (I-)(h-) > O. (1.9) For small a > 0, g(x(qr - 6), VQ7-j - 6), Tj - 6) > 0. The curve Co in En+2 is determined by Y(t) for t X Ti and by y(v, Tj) where v goes from v- to v+ when t = rj. Thus Co is continuous, and Co possesses a tangent except at the several points Y- and Y+. The above rather complicated definition of a discontinuous solution of (1.3) is justified by the following restilt. THEOREM I. Let the degenerate system (3) have a solution Co, for a < t < A. Let X(t) = (x(t), u(t)) be a solution of (2) for E > 0 such that n JX(a) - Y(a)j = Elxi(a) - yi(a)i + lu(a) - v(a)i _ 51 du(a) dv(a) <62 _

4 VOL. 33, 1947 MA THEMA TICS: N. LE VINSON 217 If E, 51 and b2 are sufficiently small, X(t) is a solution of (2) over a. t. (. Moreover as e, 51 and 52 > 0, the curve representing X(t) in E.+2 approaches Co. In particular X(,) Y(#). Also Thus we see that the existence of a solution, even though discontinuous, of the degenerate system (3) implies the existence of nearby solutions of the system (2). Let us denote the initial values of one of the n + 1 coordinates of X(a) and Y(a) by a. Then we can regard the values of X and Y at t = (3as functions of, and a. We have THEOREM II. As e, 51and 52. 0, OX(#, a) b Y(3, a) bja bja Also as e, 61, 52 0, a du(13, a) a dv(3, a) aa aa du Moreover denoting the value of at a by c, we have as e ->0 bx(o, C) 0, a du(#, c) 0. bc ' bc In case the degenerate system (3) has a periodic solution and the Jacobian associated with the determination of this solution by varying initial conditions is distinct from zero, then it follows from Theorem II that the corresponding Jacobian for (2) also will not vanish and therefore (2) will also have a periodic solution.4 The treatment can be generalized readily to the case where not one but several or even all the equations of -(2) assume the same form as the last (n + 1st) equation and moreover where the termsf, d, g and h depend on e (but approach finite limiting values as e 0). A further generalization is the following system: dx~ in duj d-u = t(,t e ) -'f +i U, t, e)=0, i = 1,...m, E 2 + E gs,(x, u, t, e) d + h1(x, u,t, e) O,. Let us denote the square matrix (gij(x, u, t, e))

5 218 PHYSICS: G. P6L YA PROC. N. A. S. by G and its determinant by IGf. We assume that the degenerate system ( t O) 0 has a solution which can be continued up to t = r where r is such that for the solution in question IG(x(T), u(t), r, )[= 0. Part of our sufficient condition for the perturbed system to have a solution is the requirement that the characteristic equation - gjgi(xr), U((r), T, 0) -6 j = 0 have X = 0 as a simple root. 1 M. Nagumo, Uber das Verhalten der Integrals von Xy" + f(x, y, y', X) = 0 fur X 0. Proc. Phys. Math. Soc. Japan, 21, (1939). I. M. Volk, A Generalization of the Method of Small Parameter in the Theory of Non-Linear Oscillations of Non- Autonomous Systems. C. R. (Dokiady) A cad. Sci. U.S.S.R., 51, (1946). Volk considers (1) where the Xi are ineromorphic functions of e for small e and periodic in t. K. 0. Friedrichs and W. R. Wasow, Singular Perturbations of Non-Linear Oscillations. Duke Math. Jour., 13, (1946). Here the Xi are not functions of t and for i < n - 1 are not functions of e. Xn contains E in the form of a factor 1/E. 2 D. A. Flanders and J. J. Stoker, The Limit Case of Relaxation Oscillations, Studies in the Linear Vibration Theory, New York Univ., This is the system, except that t is not necessarily excluded from the right members, which is-considered by Friedrichs and Wasow, loc. cit. 4 See Friedrichs and Wasow, and Volk, loc. cit, for continuous cases where right members do not and do, respectively, depend on t. A MINIMUM PROBLEM ABOUT THE MOTION OF A SOLID TIHROUGH A FLUID BY G. P6LYA STANFORD UNIVERSITY Communicated April 25, An incompressible frictionless fluid of uniform density p fills the whole space outside a moving solid and is at rest at infinite distance. The motion of the solid is one of pure translation. The magnitude of the velocity is U, its direction cosines with respect to a coordinate system fixed in the solid X,,u, v. The kinetic energy of the fluid is of the form T = 1/2MU2. The quantity M, called the virtual mass, depends on the direction of the velocity: Alf/p = AX2 + BM2 + Cv2 + 2A'Av + 2B'vX + 2C'Xg. A, B, C, A', B', C' are uniquely determined if the shape and size of the solid and the relative location of the co6rdinate system and the solid are given.