Critical points with asymmetric schemes

IOURNAL OF MATHEMATICAL Critical ANALYSIS Points AND APPLICATIONS with SVERDLOVE of Mathematics, Notre Dame, Submitted 452463 Asymmetric RONALD Department 88, (1982) Schemes * University of Notre Indiana 46556 Dame, by K. L. Cooke If P,(x, y) and QJx, y) are real homogeneous polynomials of degree n with no common real linear factors, then the system i = P,, 9 = Q, has a unique critical point at the origin. If the origin is a non-rotation point, the global structure of the system in rhe plane is a sequence of sectors separated by integral rays. which is symmetric with respect to the origin. Consider the problem of realizing a similar global picture for which the sequence of sectors is not symmetric. by polynomial vector fields of minimal degree. The schemes are characterized for which such a realization is possible and examples are constructed. For the homogeneous case. a method is given for finding P, and Q, from the scheme which is simpler than the original method of Forster. 1. INTRODUCTION The general critical point of an analytic plane vector field either is a rotation point or has a local scheme consisting of a finite sequence of sectors of parabolic, hyperbolic, and elliptic types IS]. For certain homogeneous polynomial fields, there is a unique critical point at the origin, and it is possible not only to determine the local structure near the origin, but also the global structure of the solutions. Such systems were first analyzed by Forster could occur for homogeneous [319 who determined what structures polynomials of a given degree, and gave an algorithm for constructing examples of all possible types. For these systems, the global structure consists of a finite number of unbounded sectors separated by integral rays. However, not every local structure can occur in a homogeneous system. Homogeneity implies symmetry or anti-symmetry of the phase portrait with respect to the origin, depending on the parity of the degree, so the local scheme decomposes into two identical “half-schemes.” Suppose a local scheme is given which does not satisfy this symmetry property. Is it still possible to find a polynomial vector field whose global structure consists of a unique critical point at the origin and a finite number * Present address: RCA David Sarnoff Research 452 0022-247X/82/080452-12$02.00/0 Copyright Q 1982 by Academic Press. Inc. All rights of reproduction in any form reserved. Center. Princeton. N. J. 08540. CRITICAL POINTS WITH ASYMMETRIC SCHEMES 453 of unbounded sectors following the prescribed sequence of types, separated by integral rays? A related construction has been carried out by Gil’derman [3], who constructed continuous, piecewise-linear vector fields of such global types, and then extended the method to higher even dimensions [4]. We shall show that it is not always possible to achieve homogeneous-like, asymmetric global structures using polynomials. We give some necessary and some sufficient conditions for the existence of such polynomials and show how to find them when they do exist. As a special case, we give a method much easier than Forster’s for the symmetric (homogeneous) case. The principal technique used here is that of “blowing up” the origin. We do not include much discussion of the method itself, since it has been thoroughly treated in many other places, in particular by Singer and Schecter in [ 71. They have also considered the related problem of constructing polynomial fields which have a prescribed local scheme at the origin. In the appendix of [6], they give an algorithm for such a construction which works for an arbitrary (asymmetric) local scheme. They are able to do this by adding more sectors, and so more critical directions, to the scheme in a way that does not change the local topological type, but does change the global type. In particular, any elliptic sector in a local scheme can be assumed to have parabolic sectors adjacent to it on both sides. But globally, an elliptic sector may be separated from an adjacent non-parabolic sector by a single parabolic orbit. Such a parabolic orbit is a global separatrix, but not a local one. Thus, if we want to preserve the global separatrix structure, we cannot add parabolic sectors to the scheme as is done in [ 61. and so we cannot realize certain global types which are locally realizable. Furthermore, the addition of sectors to the scheme may increase the degrees of the resulting polynomials. 2. THE SYMMETRIC CASE Analyzing the symmetric case first will indicate how we will need to proceed when we come to the asymmetric case. Given a symmetric sequence of sectors, we wish to construct homogeneouspolynomials, P, and Q,,, of minimal degree n, having no common linear factors, satisfying the condition xQ, - yP, f 0, and such that the system 1= P,(x,4’), i = e,cx,Y) has the prescribed global structure. Forster’s method [2] for doing this involves changing to polar coordinates and decomposing the functions into partial fractions. Here we construct the field in blown-up form and then work backwards to find the desired functions P, and Q,. 454 RONALD SVERDLOVE To simplify the calculations, we shall always assume that the vertical direction is not specified as a critical direction. A vertical ray can always be obtained afterwards by a rotation of coordinates. Therefore, we shall always blow up the origin by making the change of coordinates: (x, JJ) + (x, A), where ,J= y/x. The vector field can then be extended analytically to the entire XA plane [7]. In the x,4 plane, system (I) becomes, after scaling by XI-n 9 1 = xP,( 1, n>, X=Q,(l,n)-nP”(l,n). (2) Each critical line, B = 8,) in the xy plane becomes a horizontal line, A= 1, = tan S,, in the x2 plane, consisting of two half-line orbits meeting at a critical point on the J axis. There is a one-to-one correspondence between critical points on the 1 axis and critical directions for the original system. We classify the critical lines in two ways. If the flow on a line is topologically equivalent to the one-dimensional system, i = fx, we call the line linear, while if the flow is equivalent to i = *x2, we call the line quadratic. In the blown-up picture, the corresponding horizontal lines are similarly called linear or quadratic. LEMMA 1. A scheme is symmetric blowup has all critical lines linear. if and only if the corresponding ProoJ In a homogeneous system either all directions are linear or all are quadratic, when n is odd or even, respectively. Since in the blowup of a system in which the lowest-degree terms are even, all orbits in the left halfplane have their directions reversed (because of the scale factor XI-“), the blowup of any homogeneous (symmetric) system has all of its critical lines linear. Conversely, if all the lines are linear, the continuity of the field on the 1 axis together with the linearity forces every sector to be of the same type as the opposite one, so the scheme is symmetric. In a neighborhood of the origin in each sector of the xq plane, 8 has constant sign. A critical ray is said to be equistable if 8 has opposite signs in the two sectors it separates, and semi-stable if 6 has the same signs in them. In the homogeneous case, this definition reduces to Forster’s definition of rays of unchanging (equi-stable) or changing (semi-stable) behavior. The difference is that for inhomogeneous systems, there may be no trajectories in a given sector which approach a given one of its boundary rays as r approaches 0 or 00, which would be required for the application of Forster’s definition. For example, in the saddle-node system i=x*,J;= y, the first and fourth quadrants are parabolic sectors in which the orbits are all asymptotically horizontal as t + *co, except for the y axis itself. Thus, none of the orbits in these sectors approach the vertical boundary rays. In general, two opposite rays may have different stability types, but in the homogeneous CRITICAL POINTS WITH ASYMMETRIC SCHEMES 455 case this does not occur, so we may speak of the stability type of a critical direction. We will see shortly that even for asymmetric schemes, it will be necessary to restrict our considerations to such cases. THEOREM 1. Let a symmetric scheme be given with critical lines I,, 2,..., I, in the xy plane corresponding to angles 0,) 8, ,..., 8,, -n/2 < 8. < 42, jar all i. Set g. = 1 if 1. is equi-stable and g. = 2 if 1. is semi-stable Let n = (Cf=, gi) - l.‘Then n is the smallest’numbe~such that there exists-a system of the form (1) with the prescribed structure, unlessall sectors are elliptic, or all are parabolic and n is even, in which case n + 1 is the smallest such number. Prooj We shall prove this, as Forster proves his similar result, by explicitly constructing the system and showing that certain polynomials to be found must have at least n roots and therefore must be of degree of at least n. The vertical isoclines of (2), aside from the A axis, are the horizontal lines along which P,( 1, n) is identically 0. These correspond exactly to the vertical isoclines in the XJJ plane, which are lines through the origin. Let e, h, and p denote the number of sectors of elliptic, hyperbolic, and parabolic type in one half of the given symmetric scheme. Then the number of branches of the vertical isocline is v = e + h f 1, where the - occurs if the sectors containing the y axis are hyperbolic, and the + otherwise. Let li = tan Bi, i = l,..., k, and let {pji), j = I,..., v, be a set of 1 values chosen in the appropriate intervals, so that 1= pi will be a vertical isocline. LEMMA 2. v - n (mod 2) so we can choose a polynomial f (A) which has a simple root at each ,uj and no other real roots. Proof. First eliminate all parabolic sectors from the half-scheme. In this case, we have k = e + h. The number of changes from E to H or H to E in the half-scheme is then equal to the number ss of semi-stable directions. But the former number is always even because of the symmetry, so es, the number of equi-stable directions satisfies es E k (mod 2) since es + ss = k. Therefore, with no parabolic sectors, es E e + h (mod 2). We now construct an arbitrary half-scheme by inserting parabolic sectors one at a time. If we put a P between an E and an H, es is either unchanged or increased by 2. If we put a P between two E’s or two H’s, es is unchanged. We can easily check that in all the other cases, where a P is placed between two sectors, at least one of which is also a P, es is unchanged or increased by 2. At the end of this process, we have the original, prescribed scheme, which must still satisfy es E e + h (mod 2), since e + h has not changed, while es can only have changed by a multiple of 2. Now, n = 2ss+ es - 1, so n s es- 1 (mod 2). But v E e + h - 1 (mod 2). Therefore, n s v (mod 2), and the lemma is proved. 456 RONALD SVERDLOVE We shall now define a polynomial f(A) in terms of which P, can be constructed. Let m = max[O, (n - u)/2]. Then we can set f(A) = (A -p ,) ... (A -,u,)(A’ + 1)“. The degree offis n except when u > n. This only happens when all sectors are elliptic, so u = e + 1, while n = e - 1. Such a structure cannot be achieved by a polynomial of degree n, but requires one of degree n + 1. Thus, we will have the first exceptional case of the theorem. For the second exception, we note that if all sectors are parabolic, then all critical lines must be linear which, together with Lemma 1, implies that n must be odd. Thus, we can achieve an all-parabolic structure with n + 1 = 2j critical lines by taking all lines equi-stable, but to get n = 2j - 1 critical lines, we must take one of them to be semi-stable. In general, to get the minimum n, in any sequence of consecutive parabolic sectors, we try to make as many of the bounding lines as possible equi-stable. In this special case. it does not quite work to do that. We can now set P,(x, y) = fx”f(~/x). The sign is chosen according to the desired direction of the flow on the vertical isoclines. To find Q,, we observe that the critical directions correspond to horizontal isoclines in the .YAplane. Thus, C&(1, A) - AP,(l, A) must have as its roots exactly the Ats. Since the sign of 1 in a right half-strip is the same as the sign of 8 in the corresponding sector of the xy plane, Q, - AP, should have single roots corresponding to equi-stable directions, and double roots corresponding to semi-stable directions. Therefore, Q, - AP,,cannot have degree less than n, which shows that n is .the minimal degree. So let R, +,(A) = f nf=, (A - Ai)Ri,where the sign is chosen to be the same as the sign of P,. Then set Q,(x, 4’) = [YP,<X, Y) -xn+’ Rn+,(y/x)]/x. Since f and R,+! were both chosen to be manic, the y”’ ’ terms cancel and the expression in brackets is divisible by x, so Q, has degree n. Since system (2) has the structure of the blowup of the original prescribed system, system (1) has the prescribed structure. This completes the proof of the theorem. 3. THE ASYMMETRIC CASE: FORM OF SOLUTIONS To realize an asymmetric structure, we must obviously use terms of more than one degree in our polynomials. It follows directly from Forster’s main result in [2] (see [ 1]), that if P, and Q, satisfy the conditions given at the beginning of the last section, then the system i = P,(x, Y) + J-(x, Y), it = Q,<x,Y) + g(x, ~1, (3) where f and g contain arbitrary higher-order terms, is locally topologically equivalent to (2) at the origin. Thus, to obtain an asymmetric structure, we CRITICAL POINTS WITH ASYMMETRIC SCHEMES 457 will have to violate one of those conditions: P, and Q, will have common factors. Since, according to Lemma 1, symmetric schemesare characterized by having all critical lines in the blowup linear, we will see that for the asymmetric case, P, and Q, have common factors corresponding exactly to the quadratic lines in the blowup. Thus, the topological equivalence theorem will,not be violated. First, we give a necessary condition for realizability. THEOREM 2. if a schemeis realizable with all critical directions distinct, then for every opposite pair of critical orbits, both must have the same stability type. ProoJ Suppose~9~is equi-stable and Bi + x is semi-stable. Let S, (0 < 13~) and S, (0 > ei) be the sectors adjacent to Bi and let S, and S, have the same relationship to the ray Bi + rr. We assumethat near 0, 8 > 0 in S,, and 4 < 0 in S,, S,, and S,. The other cases will follow similarly. In the blown-up picture, consider the segmentsof the 1 axis adjacent to the point Ai = tan Bi. Near this point in the right half-plane i > 0 for ,l < li and x < 0 for 1 > ki, by the conditions on 8 in S, and S,. Thus, the segmentsof the I axis are contained in the stable manifold of the point (0, Ai). However, the conditions on 8 in S, and S,, and consequently on i in the left half-plane require the same segmentsto be contained in a center mainfold. Since i is continuous, this is a contradiction. Remark 1. As mentioned above, this necessary condition is automatically satisfied in the symmetric case since each of the rays of a pair separatessectors of the same two types. Remark 2. The parity of the degree of the blowup (degree of lowestorder terms) does not have any effect on this argument. Remark 3. This is a local theorem and does not depend on an assumption that the global separatrices are straight lines. Remark 4. If a structure were specified which did not satisfy the condition of the theorem, it could not be synthesized from a single blowup, but would require more than one, since it would have to have at least two global separatrices approaching the origin in the same (not opposite) direction. At least one such separatrix, of course, could not be a straight line. Remark 5. In [6], Singer and Schecter restrict their schemesto allow parabolic sectors to be specified only between two hyperbolic sectors. They then suggest the following way to modify a scheme. Separate any adjacent pair of the form EH or HE by a P. For each P in the original scheme,add another P next to it, so the sequence HPH becomes HPPH. What this process does is it eliminates all semi-stable directions, so the resulting 458 RONALD SVERDLOVE scheme satisfies the condition of the theorem. Note that the second type of change, P becoming PP, does not affect the global topological type, so we will consider such a change, or the reverse, permissible, but not the first type. Suppose now that we are given an asymmetric scheme satisfying the condition of Theorem 2 and we want to realize it by a polynomial vector field. The first step is to determine the minimum degree, n, for the lowest order terms. This is done exactly as in the symmetric case. The g:s are still well defined by the assumption on the scheme. Since opposite sectors may no longer be of the same type, we can no longer use straight lines for the vertical isoclines, but rather we must use the branches of a rational function. Consider a system of the form i= PAX* r>+p,+,(x,Y ), .? = Q,k Y) + Q,, ,(x, Y). (4) In the x1 plane (4) becomes, again after scaling by xl-“. Then the vertical isoclines, aside from the A axis, satisfy the equation x = R(A) = -P,(L A)/Pn+ ,(I, A>. 0 rice the parity of n is known, the entire structure of the flow in the XL plane can be drawn, from which the positions of the vertical isoclines can be determined. Each sector of elliptic or hyperbolic type will have one such branch, while a parabolic sector will have two or zero if the opposite sector is or is not parabolic, respectively. It is clear that a parabolic sector must contain an even number of branches of the vertical isocline. Since we are going to construct the vertical isodine as the graph of a rational function, there must be at least one branch between every two critical lines. Therefore, the smallest number of branches that can be contained in a parabolic sector opposite another parabolic sector is two. The graph of the function R(A) has as horizontal asymptotes all linear critical lines and single lines in opposite pairs of sectors consisting of one E and one H. Its vertical asymptote is the A axis. It crosses the A axis on quadratic critical lines and once in each opposite pair of sectors consisting of two E’s or two H’s. From this geometric description of the graph of R(A), we can determine the functions P, and P,, , , since they are manic polynomials and the asymptotes and A-axis crossings of the graph of R determine their roots. From the above requirements, we can see that there is one exceptional case here, similar to the symmetric case with all sectors elliptic. Suppose in the original scheme, all k = es = n + 1 critical lines are both equi-stable and quadratic. Making repeated applications of these two assumptions plus the CRITICAL POINTS WITH ASYMMETRIC SCHEMES 459 asymmetry of the scheme,we easily derive the follwing seriesof conclusions. First, Lemma 1 and the properties of blowing up imply that the degree of the constructed system must be odd. Now in the given scheme, every P sector must be opposite another P, so there must be at least one opposite EH pair. Between any E and any H in the scheme,there must be an odd number of P’s, but an even number of E’s and I-rs. Applying these facts to an opposite EH pair, we find that the number of sectors in a half-scheme is even. Thus, k = n + 1 is even, so n is odd, as it should be. The problem is that if the polynomial P, were chosen to have the properties described above, it would have to have n + 1 roots. Since the degree must be odd, the minimum degree in this case is n + 2. The function Q, can now be found exactly as in the homogeneouscase. Each equi-stable line is a single root, and each semi-stableline a double root, of Q,( I,,?) - IP,(l, A). Note that if 1= 2,. is quadratic, Ai is a root of both P,( 1, ;3) and Q,( 1,1) - AP,( 1, A), so it must be a root of Q,( 1, A). Therefore P, and Q, have common factors corresponding to such directions as asserted earlier. To ensure that the lines 13= ki are orbits of (5), we must also require vanish for each ki. However, these may all be that Q,,+,(Ll>-~P,,+,(LL> simple roots, since the term is multiplied by x and so has no effect on ,i for .Y= 0. This gives us considerable freedom in choosing Q,, , . Finally, as before, we set Pi(x, y) = xiPi( 1. J/X) for i = n, n + 1, and similarly for Qi. THEOREM 3. With the above choice of the P’s and Q’s, (4) has the prescribed structure at the origin, and the sectors are globally separated bJ the lines Ii. This construction does not guarantee that the global phase-portrait will be of Forster type, i.e., having all the sectors unbounded. Indeed, there may be other critical points. The horizontal isoclines for (5) are given by x = -(Q,(l, 1) - AP,(l, A))/(Q,+ ,(I, 1) - IP,,, ,(l, A)). Thus, the Forster-type global structure can be achieved if it is possible to choose Q, + , so that the equation Q,(L A>- AP,(l, A) PAL A) P,+,(l,A) =Q,+,(l,~>-~P,+,(1,13) has no real solutions other than the 1:s. If (6) does have real solutions for any admissible choice of Q,, ,, it may still be possible to achieve the construction by adding terms of higher orders than n + 1 to (4). However, the isoclines will then be defined by a more complicated algebraic function, the analysis of whose branches, or whose synthesis from a geometric description of its branches, is much more difficult than for rational functions. 460 RONALD SVERDLOVE 4. THE ASYMMETRIC CASE: CALCULATIONS To show that system (4), with no additional terms, sufftces at least in some cases, we calculate the simplest example. Suppose the origin is to be a saddle-node with a stable manifold along the line y = -x (J. = -1) and center manifold along the line y = x (1 = 1). Since both critical directions are equistable, we may take n = 1. ,I= 1 is a quadratic ray, so the unique 0 of P,(l,A)isat l,i.e.,P,(l,J)=,I-1. Q,(l,J)-np,(l,,I)hasroots fl,from which we easily conclude that Q,( 1, ,I) = 1 - ,I. PZ( 1, A) has a root only at -1, so set P,(l, A) = (1 t A)‘. To find Q,, we observe that Qz(l, A) AP,(l, A) must have roots at f 1, so Q2(1, A) = (,I’ - l)(c - 1) t A( 1 t 1)’ = (c t 2) ,I2 + 2;1- c for some number c. Substituting into (6), we get 1-l (1 +A)’ = (l-l)-n(n1) (A’ - l)(c - n) . An analysis of this equation shows that it has no solutions if c = -1 or c = -3. Choosing c = -1, we get Q2( 1, A) = (1 t 3L)2. Putting these functions together and changing back to xy coordinates, we get the system i, = (x - y) f (x + J’)2. i = (4’ - x) + (x t y)?, The linear terms have a common linear factor along the quadratic direction, as expected. Under the change of variables w =x + ~7. z =x - y, this becomes the standard saddle-node, ti = 2w’, i = -22. For a general structure, we must analyze Eq. (6) in more detail to determine if it can be realized by a system of the form (4). A simple algebraic manipulation shows that any 1 (other than the values on the critical lines) which satisfies (6) must satisfy We need some additional notation so that we can write down explicit forms for these functions. As above, (Ai, i = l,..., k) are the critical directions. Let E, S, L, and Q be the sets of i’s for which the li’s are, respectively, equistable, semi-stable, linear, and quadratic. Thus, E U S = L U Q = ( l,..., k). Let (pi, i= l,..., u ) (resp. {yi, i = I,..., w)) be chosen so that one ,u (resp. y) occurs between each two consecutive A’s which are linear (resp. quadratic) and oppositely oriented. Then we can write our polynomials (which are always evaluated at (1, A)) as follows: isQ i=l CRITICAL POINTS P n+I=~~ WITH tneni) ASYMMETRIC fi i=l SCHEMES 461 C1-YiJBtA) Qn= - ,QE (A- li>n (A-Ai)’+Af’n iaS Q n+lTmi$ (A-ni>c(n)+np”+l. Here, A, B, and C are manic polynomials of the appropriate degrees, each of which can be chosen to have at most one real root which, if it exists, lies outside all intervals between 2’s. Now, we have QnixPn-P,+,Q, = 1*1(A- Ai)(- n (A- ni)C(A)+ Afi (A- y.) I B(A)) i=l icQ i=l + A fi (A -,Ui) A(l)) fi (A - I’[) B(tl). i=l i=l so, R(‘)=-(Q,+,P,-P,+,Q,) fi (‘-A,) n tk-li)] i=$ ieSnQ = R,(A)-R,(A), where RI(~) R,(l) = = ieEnQ i=l r [ icEnL ieSnL i=l In all of the above, the actual values of the A’s, ,u’s, and y’s are not important as long as they are ordered correctly. Thus, we can use the choice of these values to try to obtain an appropriate R. If we can find an R of the above form which has no real roots, then we can use it to construct Q,, , so that system (4) has a unique critical point at the origin. 409/00/2- 10 462 RONALD SVERDLOVE LEMMA 3. A necessary condition for the possibility of choosing the parameters so that R has no real roots is that between any two real roots of R,(R,) there is an even number of roots of R2(R,). Proof If there is ever an odd number, then the graphs of R, and R, must intersect since the functions are continuous. COROLLARY. If (4) has no critical points other than the origin, then the schemecan have no EH pair of opposite sectors (or equivalently, v = 0), and no opposite pairs of type EE or HH, both of whose boundary) lines are equistable. Proof Suppose there were such opposite pairs of sectors. In the first case there would be exactly one root of R,, a ,u, between two consecutive L’s which are roots of R,; in the second, exactly one root of R,, a y, between two consecutive I’s which are roots of R, . Other than these cases covered by the corollary, it is hard to give a simple description of any necessary geometric conditions on the scheme. However, the conditions of the lemma can easily be checked for any given structure. Obviously, the example at the beginning of this section satisfies them. LEMMA 4. The conditions of the previous lemma are also suflcient. Proof: This is a special case of a general theorem on differences of polynomials with no real roots. The theorem is proved in [Sl using the method of the root locus, a standard tool in linear systems theory. Nonintersection of the graphs is achieved by the selecting the roots of R, and R2 to occur in pairs, the members of which are very close to each other, except for the largest and smallest which are pushed out toward *co. The pairing is possible by the evenness condition. THEOREM 4. It is possible to choose Q,, , such that (4) has a prescribed Forster-type global structure if and only if the scheme is such that the orderings and orientations of the critical lines satisfy the conditions of Lemma 3. REFERENCES I. C. COLEMAN, Equivalence of planar dynamical and differential Equafions 1 (1965), 222-233. 2. H. FORSTER, iiber das Verhalten der Integralkurven einer tialgleichung erster Ordnung in der Umgebung eines singuliires (1938), 271-320. 3. YIJ. I. GIL’DERMAN, angles, Siberian Math. Plane dynamical J. 13 (1972), systems 44-5 I. with continuous systems, J. Dr@?renfial gewlihnlichen DifferenPunktes, Math. Z. 43 straight sections linear in CRITICAL POINTS WITH ASYMMETRIC SCHEMES 463 4. Yu. I. GIL DERMAN. On dynamical systems linear in cones, Siberian Math. J. 15 (1974). 720-730. 5. P. HARTMAN, “Ordinary Differential Equations,” Philip Hartman, Baltimore, Md.. 1973. 6. S. SCHECTER AND M. F. SINGER, Singular points of planar vector fields, in “Global Theory of Dynamical Systems” (Z. Nitecki and C. Robinson, Eds.), pp. 393-409. Lecture Notes in Mathematics No. 819. Springer-Verlag. New York, 1980. 7. S. SCHECTER AND M. F. SINGER, Separatrices at singular points of planar vector fields. Acra Math. 145 (1980), 47-78. 8. R. SVERDLOVE AND M. SAIN, Closed-loop systems with no real poles. to appear.