Periodic orbits of autonomous ordinary differential equations: theory and applications

Nonhear Annlysis, Theory, Printed in Great Britain.

Methods

& Apphcorims.

Vol. 5. No. 9, pp. 931-958.

1981.

0362.546X’61/090931-28 @ 1981 Pergamon

$02.0010 Press Ltd.

PERIODIC ORBITS OF AUTONOMOUS ORDINARY DIFFERENTIAL EQUATIONS: THEORY AND APPLICATIONS BINGXI LI Department

of Mathematics,

University of California at Los Angeles. Los Angeles, California 90024, U.S.A.

and Department

of Mathematics, Jinan University, Guangzhou (Canton), The People’s Republic of China (Received 29 September 1980; revised

10 December 1980)

Key words: Dynamical system, existence and nonexistence of periodic orbits, uniqueness of periodic orbits, stability of periodic orbits, torus principle, Poincart criterion for asymptotic stability.

Dedicated to Professor Earl A. Coddington on the occasion of his sixtieth birthday. 1. INTRODUCTION CONSIDER

the autonomous

system i = F(x),

(I)

where x E R”, F: M +- UP,M is a domain of R”, F E C'(M)and such that the solution x(f) = g(t, x0), g(0, x0) = x0, of (1) exists for all t E R. The mappings g’: x --, g(t, x) and (g’)-’ are differentiable, and they form a one-parameter group of diffeomorphisms. This group together with the phase space M is a dynamical system. The orbit I = IQ) through a fixed p E M is the set IQ)

= {X E R”13 (t, p) E R x M with g(t, p) = x}.

Topologically, there are three different types of orbits: (a) a point (b) a Jordan curve, and (c) a homeomorphic image of an open interval; they correspond to a constant, a nonconstant periodic, and a nonperiodic solution of (1) respectively. It is of interest to study existence, nonexistence, uniqueness and stability of nonconstant periodic solutions (periodic orbits) of (1) for n 2 3 both in theory and applications. The aim of this expository paper is to summarize the main results of this topic. The contents of the subsequent sections are: Section 2-Poincare-Bendixson theorem and its generalization, Section 3-existence of periodic orbits, Section 4-nonexistence of periodic orbits, Section S-uniqueness of periodic orbits, and Section c-stability of periodic orbits. For basic concepts and results, see Arnold [ 11, Coddington and Levinson [2], Hale [3], Hartman [4], Hirsch and Smale [5], and Nemytskii and Stepanov [6]. For results concerning the Hopf bifurcation theory, which is not covered in this article, see, for example, Marsden and McCracken [7], Hsu and Kazarinoff [8], and Hassard and Wan [9]. 2. POINCARB-BENDIXSON

THEOREM

AND ITS GENERALIZATION

A basic tool for the detection of periodic orbits in the plane is the famous PoincareBendixson theorem [2-6, 10, 111. 931

BINGXI LI

932

THEOREM 1 (Poincare-Bendixson). is either

(a) a singular

point

If R is a bounded or (b) a periodic orbit.

In 1963, Schwartz [12] was able to extend two-dimensional manifolds.

this theorem

minimal

set of (1) for n = 2, then

to differential

equations

Q

on compact

THEOREM 2 (Schwartz). Let ZJ? be a compact, connected. two-dimensional manifold of class C2. Let #: [w x ZJI + illbe a dynamical system of class C2. Let R C ‘9Jlbe a minimal set. Then Q must be one of the following: a singleton consisting of a singular point; a single, closed orbit homeomorphic to S’; or ;:‘, (c) all of YJ?which is homeomorphic to a torus T’. It is well-known that Theorem 1 can not be extended to differential order without placing some restrictive conditions on the class of equations. 1131 published a counterexample showing that no simple generalization (2 3) dimensional case is possible. His example consists of a third-order

equations of higher In 1961, D’Heedene of this theorem to n system

x = J’, X=.2-x,

i = Z(x,y,

(2) 2).

Aside from three exceptional orbits, each orbit of (2) approaches an entire surface C (homeomorphic to a two-dimensional torus T2) as time t --, m. For each point p E C, C contains the entire orbit I@) passing through p; r 07) itself is dense in C and almost periodic but not periodic. Recently, Smith [14, 151 extended the Poincare-Bendixson theorem to vector differential equations of the formf(D)x + b+(g(D)x) = 0 and Dx = f(x) respectively by placing additional restrictions on them. These will be treated in Section 3. Theorem 1 is frequently interpreted in the following form (annulus principle) in application.

THEOREM 3. If the orbits of (1) for n = 2 that emanate in the boundary points of an annulus region A penetrate in the interior with increasing t, then there exists in A at least one @limit orbit, which is a singular point or a closed orbit. Erle [16] has improved

this result.

Let A C M be a closed annular region which is positively invariant with respect to (1) for n = 2 and free from singular points. Assume dA is the union of two cross-sections (i.e., a curve which intersects each orbit at most once). Then A contains a set 2 which is a closed orbit or a closed annulus of closed orbits such that any neighborhood of 2 in A contains a positively invariant open neighborhood bounded by two cross-sections which are transversal to the vector field of (1).

THEOREM 4 (Erle).

It is natural

to ask whether

this annulus

principle

has any generalization

for dynamical

Periodic orbits of autonomous ordinary differential equations: theory and applications

933

systems in higher dimensions. In this respect, there is Smale’s question ([17, p. 8011, also Fuller [18]): Does every nonvanishing vector field on the solid torus S’ X D* transverse to the boundary have a closed integral curve? In 1974, Schweitzer [19] elegantly gave a negative answer to this question with a vector field of class C’. He also gave a negative answer to Seifert’s question (“Does every nonvanishing vector field on the three-sphere S3 have a closed integral curve?“) with a vector field of class C’. For vector fields of class C(r > 2) these questions remain open. The following is a birds-eye-view of the Schweitzer example answering Smale’s question for vector fields of class C’. Let a vector field Y E C” be defined on the solid torus T3 = S’ x D’ with vectors of Y on LIT3pointing into Int( P), and Y has a unique closed orbit y, which is the central circle in T3. Choose an arbitrary point po E y, modify Y in a neighborhood U(po) of po, such that there is a cylinder W(po) C U’JJ0) centering at PO, and the integral curves in IV@,) are straight lines parallel to Y(po), the vector of the modified vector field at po. Let 7 be the unique closed orbit after modification, the segment of which in W(po) is the axis of that cylinder. The rest of Y remains the same qualitatively. Construct a “block of Schweitzer”, which is embedded in W(po) in order to break the closed orbit 7 with the resulting vector field I’ E C’ (this is a C’ vector field on T3 transverse to a T3 without a closed orbit-the required counterexample). The construction of the “block of Schweitzer” is as follows. Let a C’ vector field of Denjoy on Tz with an exceptional minimal set % be given. Let N = Tz\D*, D* C T*\%. Deform N continuously into a highway overpass, as shown in Fig. 1. Let B = N x D’, and define a C’ vector field on 5, such that: (a) it is free from closed orbits; (b) it has minimal-sets which are copies of % at N ~{a} and N x (-4); (c) every orbit in N X-D’ will either be asymptotic to one of the copies of % or will intersect the boundary of N X D’ at (n, -1) or (n, l), n E N;

Fig. 1

934

BINGXI LI

Fig. 2

n* E N such that I-@‘) C B, r’(p”) C B; (d) 3p’ = (n*, l),p” = (n*, -l), (e) the vectors on XI are the same as Y(pO). Then embed IEKin IV@,), e; B --, W(&), with p’ ---* ~1, p” + p2 (as shown in Fig. 2), and break the closed orbit p; the resulting vector field Y E C’ is free from closed orbits. In view of the elegant counterexample of Schweitzer, certain additional restrictions should be imposed on the vector field in order to guarantee the existence of closed orbits for a C’ vector field on T3 transverse to dT3. One way to do so is to use the well-known torus principle. 3. EXISTENCE

3.1. The torus principle is a tool frequently n (2 3) dimensional dynamical systems: 5 (torus principle, trajectories of (1) emanating T as t + ~0, T admits an (n trajectories which pass it at reaches S(‘-‘) again for t =

THEOREM

OF PERIODIC

used to detect

ORBITS

the existence

of periodic

orbits

for

Poincare [lo], Pliss [20], Reissig er al. [21]). Suppose that the from the boundary points of a toroidal domain T penetrate into - 1)-dimensional section S (n-1) that forms a local section for the t = 0. Suppose also that every such trajectory g(t, p), p E S(n-” w(p) > 0. g(w, p) E S(n-‘). Then T contains periodic orbits of

(I). The proof of Theorem 5 is based on the Brouwer Fixed Point Theorem [22] (there is an elmentary proof of this theorem by Milnor [22]). For more than three decades the torus principle has been a main tool for studying the existence of periodic orbits of dynamical systems arising in various branches of science and technology, e.g.: (i) Electronics-Friedrichs (1946 [23]), R such (1950 [24]), Li (1977 [25]); (ii) Mechanics and automatic control-Mulholland (1971 [26]), Noldus (1969 [27], 1971 [28]), Kamachkin (1972 [29]), Leonov (1973 [30], 1977 [31]), Williamson (1975 [32], 1976 [33]); (iii) Nuclear physics-Sherman (1965 [34]). Troy (1977 [35]); (iv) Biology and chemistry-Hastings (1974 [36]), Hastings and Murray (1975 [37]), Tyson (1975 [38]), U. an der Heiden (1975 [39]), Hastings, Tyson and Webster (1977, [40]);

Periodic orbits of autonomous ordinary differential equations: theory and applications

935

(v) Others-Vaisbord (1962 [41]), Vinogrado (1965 [42]), Nazarov (1970 [43]), Mamilla and Sedziwy (1971 [44]), Georgeev (1976 [45]). 3.2. One can see that the torus principle is difficult to use in practice because of its complicated geometry. Alternative methods are needed. Smith [ 141 extended the Poincare-Bendixson theorem to vector differential equations of the form f(D)x + b@(g(D)x) = 0, the only restrictions placed on the vector function G(y) are that its Jacobian matrix should be continuous and lies within a suitably chosen elliptic ball. In [15] Smith modified the idea of [14] so as to extend the Poincare-Bendixson theorem to a much larger class of equations. These results provide a general method for proving the existence of periodic solutions of autonomous ordinary differential equations. Consider the automomous differential equation f(D)x + M(g(D)x) in which D=dldt; g(D)=go+glD

= 0,

(3)

f(D)=fo+f,D + . . . +f,,D”, fo,f,,... , fnERmXm with detf,,#O; + . . . +gn-ID”-‘,go,gl, . . . ,gn_lERSXm; x(t) E[W~~‘, t E[W’;

b E[Wrnx ‘; 4: RS-+Iw’, @ E C’(@). Equation (3) is equivalent to DX = AX + B@(GX),

where X = co1 (x, Dx, . . . , Dn-lx), B = co1 (0, . . . , 0, -J;;‘b), and A is the block companion matrix of the polynomial %‘f(D).

(4) G = (go, gl, . . . , g,_l)

6 (Smith [14]). Suppose that (i) BA > 0, a closed elliptic ball 8(C, Li, L2) C 6Yxs and a convex set S C Rs such that

THEOREM

% 1 B(C, Li, Lz) 1 Ra;fe

~#(v)@;

(ii) (4) has a bounded semi-orbit F such that GXES,

VXEI;

(iii) the o-limit set of I, Q(T), contains no singular points of (4); then Q(I) consists of a single periodic orbit of (4). The definitions of %i, !@C, Li, L$ are as follows: ‘%i= {K E @““]det If(z) + bKg(z)] = 0 has two roots z with Re(z) > -A and (mn - 2) roots z with Re(z) < -A}. (When b = -I, g(z) = I, f(z) = lz, det If(z) + bKg(z)] = 0 is the characteristic K). For given C E Qxs, nonsingular Li E Qx’, LZ E Q”““, 8(C,

L1, L2) = {K E @“I

equation of

spectral norm ILl(K - C)L2[ < 1).

(In the special case when L1 = pL2 = I, 23(C, Li, L2) is the closed spherical ball with center C and radius p in the normed space C”“.) Theorem 6 includes the classical Poincare-Bendixson theorem for the plane autonomous

BINGXI LI

936

system (5) in which the real functions ~(6, r,r), q(& q) E C’(R2). This equation is of the form (3) with x = co1 (E, 77), @(x) = co1 (p, q), f(D) = ID, b = -I, g(D) = I, where I denotes unit matrix. A bounded semi-orbit I of (5) lies within some closed circular disc S in the (j, 77) plane. Since the continuous Jacobian matrix &p/lax is bounded in S, it lies within a closed spherical ball 93 in the space C 2x2 having center at the zero matrix and sufficient large radius. Since the eigenvalues of K are bounded for all Kin !8, their real parts all exceed some negative constant -A. then Si >B and therefore (5) satisfies the hypotheses (i), (ii) of Theorem 6. If Q(I) includes no singular points of (5) then Theorem 6 shows that Q(I) consists of a single periodic orbit of (5). This exhibits the classical Poincare-Bendixson theorem as a special case of Theorem 6. In [I51 Smith modified the basic idea of [14] so as to extend the Poincare-Bendixson theorem to the autonomous differential equation Dx =f(x),

(6)

in which f: R” + R”, f E C(R”), D = d/dt. THEOREM

7 (Smith [15]). Suppose that (i) 3 S c ET, S = S, f is locally Lipschitzian in S; (ii) 3k > 0, E > 0 and a quadratic form V(x) such that (a) every pair of solutions xi(t), x2(t) of (6) has DV(xl(t) -x2(t)) + 2AV(xl(t) - x2(t)) s - ~Ix~(f) -x2(t)j2, for all tat which xi(t), x2(t) E S; (b) V(x) = x*Px, where P E R”““, P* = P with 2 negative eigenvalues

and (n - 2) positive eigenvalues; (iii) (6) has a b ounded semi-orbit I C S. If Q(I) contains no singular points of (6), then Q(r) consists of a single periodic orbit of (6). We now show that Theorem 7 includes the Poincart-Bendixson theorem as a special case. Given a bounded semi-orbit I C S, let SObe the closure of I. Since f(x) satisfies (i), there exists a constant p such that If(x) -.NY,_i SPlX -Yl,

kY

(7)

ES@

It is sufficient to find a quadratic form V(x) which satisfies (ii) when n = 2. For this purpose we take V(x) = -?(A - p)-‘[XI’, where p is the constant in (7). If the constant il exceeds p then P = -&(A - p))’ I satisfies (ii) (b) because n = 2. If xi(f), x2(t) are solutions of (6) then (7) gives (A - P) (D + 2A)V(x, -

x2) =@I

- x2)* tf(x2) -.f(xd

s (P -

- 4x,-

~91

A) 1x1- x212,

for all t at which xi(t), x2(t) E So. All the hypotheses of Theorem 7 therefore hold with S, V(x) theorem then replaced by So, -&(A - p)-l]x12. The conclusion of the Poincare-Bendixson follows at once from Theorem 7. Theorem 6 is used to obtain sufficient conditions for the existence of a non-constant periodic

Periodic orbits of autonomous ordinary differential equations: theory and applications

937

solution x(t) of (3) [14, Section 41, Theorem 7 is used to obtain similar results [15, Section 51. These results are better than that obtained by Burkin and Leonov [31]. For example, when I = s = 1, Theorem 2 of [14] is essentially the same as the result in [31], except that [31] replaced condition (i) of that theorem by a more restrictive hypothesis and restated condition (iii) as a frequency domain condition, but the restriction r = 1 seems important for Leonov’s argument. Remark. Noldus [27] first used the torus principle to obtain frequency domain conditions for the existence of periodic solutions of the autonomous feedback control equations. His work was simplified and extended by Burkin and Leonov [31]. For example, the following system can be studied by the method of [31] but not by that of [27]. i = Ax - btp(c*x),

(8)

where x = co1 (xi, x2, x3), A=(_!

_%

_;).

b = co1 (0, 0, l), c* = (-y, 1, O), (Y> 0, /3 > 0, 6 > 0, cup > 6, a’ > 4/Y?,y > 0, Q?(O)= 0, (p’(O) > (@l - @/(a! + y), q’(o) 3 0. 3.3 While the Brouwer Fixed Point Theorem is the basis of the torus principle, an attempt has been made to apply the Banach Fixed Point theorem to attack the existence and uniqueness of periodic orbit of system (1) by Korolev (461. Consider the autonomous system ii =

Fi(XlyX2,.

. . 7x3,

i =

1.. . . , ?I

(9)

where Fi E C(R”), n 2 3. Korolev imposed the following conditions on (9): (I) 3x;, . . . , xi ER such that the vector field defined by the system

il = Fhx2,x;,

. . . ,.d,

(10)

i2 = F2(x,, x2, x;, . . . , xi). points toward Int (K) on dK, where K is a star-shaped region containing the origin in the x1x2-plane (i.e., every ray from 0 intersects aK in exactly one point). Let V be the domain consisting of the topological product Int (K) X Z3 X . . . X I,, where z, = [xl, =J), s = 3, . . . , n; the boundary of V consists of the following hypersurfaces:

S = 6K x Z3 x . . . x I, S3 = Int (K) x xi X Z4. . . X I,, S, = Int (K) X Z3 X . . . X I,-, X xi X Z,_i. . . X I,, S,=Int(K)xZ3x... x1F2

(11)

-

x2F1

+

x:

*cu>o, x:

xZ,_ixx~. V(xl,...,xn)EV;

r = 4,. . . , n - 1,

BINGXI LI

938 (III)

Fk>O,~(~l,...,x,)ESk,k =3,...,n;

(lV)~F1-~F,sO.

V(xl,.

. . ,x,)

ES,

k=3,...,n

k

Korolev asserted that: (I), (II), (III), (IV) + the vector field defined by (9) points toward Int (V) on dV and there exists a homeomorphism &: n ---, n, defined by the trajectories of (9), = 0)n V.Then one more condition (very complicated) is where n = {(x1, ... ,x,)E jW"~x~ imposed on (9) in order to make ti a contracting mapping and to pave the way for applying Banach Fixed Point Theorem. Unfortunately the above assertion is wrong. There is a counterexample (Li [25]): X1 =

F,(x,,x2.xli) = -x1-x2 +xg3,

i2= Fz(xl,x2,xg) =x1 -x2 +xxp3,

(11)

i3= Fs(x1,x2,xg) = 1 +x3 Choosingx; =O, K: xi + xi 5 a2 (s > 0), we can see that (I), (II), (III), (IV) are satisfied, the vector field defined by (11) is not always pointing toward Int (V) on dV,because > 0,

but

ifx3 > 1,

(F1,F2,F$*(x1,x?,0) =u'.(xJ-1)= =O, ifx3=1, i CO,

ifx3 < 1,

V(xl, x2, x3) E S. Hence the assertion of Korolev is not true. Moreover, this system possesses no nontrivial periodic solutions. A revised version of this theorem appeared in [21]. but the conditions are so complicated that the theorem is very difficult to use. 3.4. The process of finding a positively invariant solid torus can be very difficult. For equations that model ecological systems, in particular, it is much more feasible to find a positively invariant disk. In 1977, Grasman [47] applied the theory of the Brouwer degree of a mapping to solve this problem along this direction and obtained a set of sufficient conditions for the existence of a nontrivial periodic orbit of an n-dimensional autonomous system (n 2 3). THEOREM

8 (Grasman).

Consider

the autonomous

system

-6 = fi(x,, . . . xn), 3

(12)

where fi E C’(W), fi(0) = 0, 1 s i s n. Suppose that M C R” which is star-shaped form 0 (i.e., 0 E Int (i) there exists a compact neighborhood (M) and every ray from 0 intersects 8M in exactly one point) and is positively invariant under (12) ; and exactly (ii) 0 is the only singular point of (12) in M, this singular point is hyperbolic, two eigenvalues of fX(0) have positive real parts; (iii) the stable manifold of 0 contains {(x1, x2,

. .. 7x,)E Klx, = x2 = 0)n M;

Periodic orbits of autonomous ordinary differential equations: theory and applications

(iv) (12) can be transformed

939

by Xl

=

x2 =

Yz COSYI, y2

Xi = yj,

sin

(13)

yl,

3 < i S n,

to ji

=

gi(ylv.

. . ,y,),

1 =Si Gn,

where gi E C’(M), gl # 0 on MC, and MC = b E [w”]c(y) E M, c:y + x is the map defined by (13)); then (12) has a nontrivial periodic orbit in M. This theorem was applied to an ecological model of three competing species

NI = N*(l - Nr - LyN*- /3N,) + E, N; = Nz(1 - ~NI - N2 - oNj) + E,

(14)

N; = Ns(1 - (YNI- /3N2- N3) + E. (See also May and Leonard [48].) 3.5. Let p be a point in the phase of space of (1). Suppose S is an open subset of an (n 1)-dimensional subspace transverse to the vector field at p. Following trajectories from one point of S to another, defines a C’ map h: SO* S, where So is open in S and contains p. We call h the Poincare map. The orbit through a fixed point p* of h is a closed orbit. Several authors have used the Poincare map to study the existence of closed orbits of (1). This method is widely used to investigate piecewise linear systems. For example, Gaushus [49] used this method to establish the existence of periodic orbits for relay systems of 3 and 4 dimensions, e.g.: dz dx dy -= -k(z), &= -k(z) + 52 sgnx + y, (15) dt ” dt= where 1, if 2 E (a, m), k(z) = 0, if 2 E [-a, a], -1, if 2 E (-m, -a), and a 5 0. [l + (h - l)F(u + a)] 2 + E = F(-u

- a),

(1 + (h - l)F(a - u)] 2 + r/ = F(u - a),

(16)

du -=z+sSE-sq, dt where a 3 0, h 2 0, S 5 0, and 1 F(x)={O:

ifx E (0, m) ifxE(-m,;)).

BINGXILI

940

Systems (15), (16) arise in the study of orientation control of airplanes and space vehicles. Glass and Pasternack [50] have considered the following piecewise linear systems which have been proposed as models for biological control systems -

dt

= Ai(X1, ,X72,. . . , x;hi)- X;,

i = 1,

7

N,

(17)

where Ai is nowhere zero and for each i the sign of &(x1, X2, . . , iN) is independent of xi, the Boolean variables f; are defined by f; = 1 if xi > 0, 2, = 0 if xi < 0. A curve in N-dimensional phase space (xl(t), x*(t), . . . , x~(t)) is a solution to (17) provided the xi(t) are continuous, piecewise differentiable and satisfy (17) whenever all x, # 0. A non-singular solution is one in which for each i, xi(t) = 0 only at isolated values of t. Only non-singular solutions are considered. They proved the existence of a closed orbit for (17) by first explicitly computing the form of the Poincare map of the flow, h: S + S, and showed it is of the form f(z) =

Az

1 + b#&2)’

where A is an (N - 1) x (N - 1) positive matrix, $J is a non-negative (N - 1) vector, z E S, and ( ., a) represents the inner product of two vectors. Then they provedf(z) has a non-zero fixed point iff p > 1, where p is the dominant eigenvalue of A. BruSlinskaya [51] studies the following system i = A(E)x + Q(x, E),

(18)

where x E R”, Q is sufficiently smooth, and

44 A(&)

=

--b(4 ; 0 44 : b(E) _____._____________~_________ : C(E) 1 0

with a(0) = 0, a’(O) > 0, b(0) > 0, the eigenvalues of the (n - 2) order matrix C(0) have negative real parts. The main result is THEOREM 9 (Bruslinskaya). There exists a neighborhood II of the origin, independent of E, such that for all E E (0, ~g), there is a unique periodic orbit in U. This closed orbit is a stable limit cycle. There exists a homeomorphically flat invariant manifold MS which contains the origin and the limit cycle. The part of Mz bounded by the cycle is filled out by orbits tending toward the limit cycle as t + CCand toward the origin as t + - 00. The orbits tending to the origin as t --, cQ fill out an (n - 2)-dimensional manifold M’fW2.Any orbit which originates on Lr\Mzm2tends to the limit cycle as T + co.

The proof of this theorem is based on a study of the Poincare map, induced by (18), of a certain (n - l)-dimensional region in the phase space. Robbins [52] considers the Lorenz equations dw dr

-=R-zy-w,

Periodic orbits of autonomous ordinary differential equations: theory and applications

941

(19)

which describe convection in a circular fluid loop and the behaviour of a modified homopolar dynamo (Robbins [53]). The existence of a periodic solution for finite R is established by means of proving the existence of a fixed point of a Poincare map. The Poincare map is also used to study the closed orbits of the three-body problem, e.g., the Moser closed orbit theorem (Theorem 36.7 in Abraham and Marsden [54]), etc. 3.6. For piecewise linear systems and systems with switching some authors made use of directed graph to investigate the existence of periodic orbits. Glass and Pasternack [50] classified the flows in phase space determined by (17) by a directed graph, called a state transition diagram, on an N-cube. Each vertex of the N-cube corresponds to an orthant in phase space and each edge corresponds to an open boundary between neighboring orthants. If the state transition diagram contains a certain configuration called a cyclic attractor, then all trajectories of (17) corresponding to the cyclic attractor either (i) approach the origin, as t + 03, or (ii) approach a unique stable limit cycle attractor. An algebraic criterion is given to distinguish the two cases by studying the Poincare map. Prikhod’ko [55] employed directed graphs and the Brouwer Fixed Point Theorem (and Browder Fixed Point Theorem) to study the existence of periodic orbits of systems with switching. 3.7. Consider the two-dimensional

system

i* =

-a1

+

fi(Xl,

x2),

where fi(xi, x2) are higher order terms at (xi, x2) = (0, 0). This system need not necessarily have periodic solutions (e.g., [2, p. 3811). A classical result due to Poincare [56] states that if fi and A satisfy the symmetry conditions fl@l,

-32)

=

-fl(Xl,

f2h1,

--x2)

=f2(x1.

x2),

x2),

then (20) has periodic solutions with period near 21tllal in a neighborhood Podolak and Westreich (571 consider the analogous n-dimensional situation i = Ax + f(x),

(21)

of the origin.

(22)

where If(x)] = 0( Ix]) near x = 0 and show that if the nonlinear terms have similar symmetry properties, the system possesses periodic orbits. They obtain this result by translating the problem to a bifurcation problem and showing that bifurcation occurs. The approach is a direct, finite-dimensional analysis of the problem.

BINGXI LI

942

Assume that the matrix A in (22) has the form

AZ0

Al [

A2

0

1

where Al, A2 E Rkxk with AZ invertible and Al not necessarily invertible. Assume that A has pure imaginary eigenvalues +i and possibly integer multiples of those eigenvalues. (22) can be written as XI = Al~z

+ f&1,

x2),

i2 = A2x1

+ fib17

x2),

BINGXI LI

944

case, the gi are assumed to begin with forms of degree not lower than three. Pasynkova [72] generalizes these results. Berger [73] has also obtained several extensions of Liapunov’s theorem by reformulating the periodic solutions of 2 e

dtz

+

grad U = 0

(where x E IF?, U: R” + R, U is a C’ even function, grad U(0) = 0), as critical points of an isoperimetric problem in the calculus of variations. One of the results is independent of any Hamiltonian assumptions. However in that theorem only a sequence of periodic solutions is obtained instead of a family. In 1973, Weinstein [74], [75] p resented two extremely deep and brilliant theorems on the existence of periodic solutions of Hamiltonian system of equations of the form i = JH,,

946

then there is an asymptotically ic the nl-limit wt nf u(t)

948

2 = col(r~.

. . ,-72n), J =

BINGXI LI

stable periodic solution y(t) of (1) such that the orbit I of y(t)

BINGX LI

Additional theorems discuss a similar behavior for the derivatives of x and u with respect to the initial conditions. In case the system (36) is autonomous, the same result relating the existence of periodic solution for (36) to (37) holds. Wasow [91] considers the nth order differential equation 8x(“) = F(x, x’, . . . ) x@); t; &),

(38)

Periodic orbits of autonomous ordinary differential equations: theory and applications

943

where the matrix A has purely imaginary eigenvalues fai, ‘c&2, . . . , 2 cxnu,, and f is analytic and begins with quadratic terms. A famous theorem of Liapunov [61, 621 states the following: 11. (Liapunov). Let &al, . . . , + an be the purely imaginary eigenvalues of A, and assume that akuk/alis not an integer, k = 2, . . . n. Moreover, assume that the system (24) possesses an integral with nondegenerate Hessian. Then, there exists a one-parameter family of periodic solutions with period near 2niloi. If &k/al f integer, k # 1, then n such families exist.

THEOREM

One question is what happens if CQ/C~I is an integer. Liapunov’s proof, which involves a series construction, appears to fall through in this case. About ten years ago Roels [63], [64] showed that these series could be salvaged if the ratio CQ/CU, was at least three. Shortly thereafter, Schmidt and Sweet [65], using a bifurcation theory developed by Hale [66], presented a new proof of these results, and also obtained some results for the ratios 1 and 2. (See also Ryabov [67].) Another question is what happens if the matrix A has two zero eigenvalues with a nonsimple elementary divisor. Pliss [68] demonstrates with certain additional assumptions that if the system has an analytic integral of a special kind, then there also exists a continuous family of periodic solutions. This family is, in general, not analytic. Nustrov [69] generalizes the result of Liapunov to the following real system:

j = -AZ +

Y(Xl,. ..,x/;y,z;u1,.

i=Ay+z(xl)...)

. . ,um),

x~;y,z;ul,...)ud,

(25)

di=bjlUl+...+bi,U,+V,(Xl,...,X/;y,Z;U1,...,U~, l;i=l,...,

j=l,...,

m,

where Xi, Y, Z, Vi are analytic nonlinear functions and the eigenvalues associated with the matrix ]]bjk]]are not integer multiples of +ti. The above system is assume to possess a first integral of the form N=yz+Zz+Li(xi

)...

,x,) +Ldu1,...

,um)

+qxt...

,x/;y,z;ut....

,u&

where the Li are quadratic forms and S contains terms of order not less than three. An application is made to the motion of a balanced gyroscope in a Cardan suspension. There exists extensions of Liapunov’s theorem in which the Hamiltonian condition is not needed. First, there is an extension due to Krasnosel’skii [70] on the following system: d2xi ~+OZgi(Xl,...,X,)=O,

i=l,...,

n.

(26)

Pliss [71] presents conditions for the existence of families of symmetric periodic solutions of 2 t-E! + #Xi = gi(Xl, a . .

dt2

,x3,

i=l,...,

n,

(27)

where & are positive numbers, gi are analytic and begin with quadratic terms. In the resonance

BINGXILI

944

case, the gi are assumed to begin with forms of degree not lower than three. Pasynkova [72] generalizes these results. Berger 1731 has also obtained several extensions of Liapunov’s theorem by reformulating the periodic solutions of e t grad U = 0 dt? (where x E 1w”,U: IR”-+ iw, U is a C’ even function, grad u(O) = 0), as critical points of an isoperimetric problem in the calculus of variations. One of the results is independent of any ~amiltonian assumptions. However in that theorem only a sequence of periodic solutions is obtained instead of a family. In 1973, Weinstein [74], 1753 presented two extremely deep and brilliant theorems on the existence of periodic solutions of Hamiltonian system of equations of the form i = JH,,

2 =

co1(21, * . . , z&J,

J =

0 i -I

I

1

0’

(29)

He studied C’ perturbations of a Hamiltonian vector field on a manifold M which possessed a submanifold n consisting entirely of periodic orbits. Under small perturbations, only a finite number of periodic orbits can be expected to survive. in general. Weinstein reduced the determination of the minimal number of su~ivi~g periodic orbits to studying the intersection of two close Lagrange manifolds, and this problem in turn, is reduced to estimating the number of critical points on some orbit manifold. Then the foIlowing theorem was obtained. 12 (Weinstein). If H E C’ near z = 0, and the Hessian matrix is positive definite, then for sufficiently small E, any energy surface

THEOREM

W(z) = H(0) -t i! contains at least n periodic orbits of (29) with periods close to those of the linearized system. Later in 1976, Moser [76] presented an alternate proof of Weinstein’s results. In 1978, Rabinowitz [77] carried out the search for periodic solutions of the Hamiltonian system of ordinary differential equations:

where H E C’(Iw*“, [iB),and pI 4 E iw”,.Z = @, 4). First he looked for solutions of (30) having prescribed energy and obtained: THEOREM~~ (Rabinowitz). Suppose r-i-‘(b) is radially homeomorphi~ to S2n-’ for some ii f 0 (i.e., H-‘(b) is appropriately star-shaped with respect to the origin), and (c, H,(c))nz” f 0 for 5 E N-‘(b) (here (t, .)nzOdenotes the inner product in rW&).Then (30) possesses a periodic solution on H-‘(b).

Secondly he looked for periodic solutions with fixed period and proved: THEOREM

14 (Rabinowitz).

Suppose (i) H(z) = 0(]zj2) at L = 0, (ii) H 3 0 and 0
Periodic orbits of autonomous ordinary differential equations: theory and applications

etz,K)R~ forI4 3

f, 0 E (0,1). Then, for any T > 0, there exists a non-constant

945

T-periodic

solution of (30). For proofs and brief comments on related works by Seifert, Weinstein, Moser, Chow and Maller-Paret, see [77].

Berger,

Gordon,

Clark,

3.9. Several authors have studied the existence of periodic obrits near a certain invariant set of system (1). Neimark [78], Gavrilov and Shil’nikov [79], and Ivanov [BO] considered the three dimensional system dw - = W(w), (31) dt where W E C’. Assume that (31) has a periodic solution w(t, 0) whose smallest period is t, hence Ii = {w = w(t, O)lt E [0, t]} is a closed orbit, and the two-dimensional manifolds W = {WI/dist [w(t, w,), rl] -0,

ast--,

a},

W = {wzl dist [w(t, wz), rl] + as t + -m},

are the stable and unstable manifolds of Ii respectively. Suppose that there is a trajectory I homoclinic to Ii, i.e., a trajectory whose (Y-and *limit sets coincide with I,. The result of [78] shows that in the case of transversal layering of W and W with respect to I, there are closed orbits in the neighborhood of Ii U r. Results in [79] imply that when W and W are tangent to the first order with respect to I, this is not always true. In [BO], Ivanov examined the case of arbitrary tangency and presented a necessary and sufficient condition for the existence of closed orbits in a small neighborhood of Ii U r. Chernshev [Bl] investigates the system (1) with a sufficient smooth function F(x) and assumes that the system has a smooth invariant torus T, and the rotation number on the torus is irrational. Chernyshev has proved that if the rotation number is well approximated by rationals and F(X) satisfies certain conditions, then each neighborhood of T contains an infinite number of closed orbits. As to the existence of invariant tori, the elegant and important works of Kolmogorov [82], Arnold [83], and Moser [84] should be mentioned. 3.10. Cronin [85] puts forward the following definitions. A solution x(t) = g(t, x0) of (1) is uniformly stable if there exists a number K such that E > 0 implies there is a positive b(e) so that if u(t) is a solution of (1) and if there exists numbers tl, t2 such that t2 2 K and such that lu(tl)

- x(t2)l < b(&),

then for all t Z- 0,

lu(t + tJ - x(t + tj ( < E

A solution x(t) is asymptotically stable if x(t) is uniformly stable and moreover, of the definition of uniform stability, there exists a number t3 such that

in the notation

lim 124(t)- x(t + t3)I = 0.

I-m THEOREM

15 (Sell [86], Cronin). If x(t) is a bounded asymptotically

stable solution of (l),

BINGXILI

946

then there is an asymptotically is the w-limit set of x(t).

stable periodic solution y(t) of (1) such that the orbit I of y(t)

This result is applied to generalized been obtained by Cronin [87].

Volterra equations [85]. Extension of Theorem

15 has

3.11. A singular perturbation problem is. roughly speaking, the problem of solving an ndimensional system in which one or more of the terms _&is multiplied by a parameter e which is close to zero. An example of such a problem would be: x’=f(x,y)7

Ei, =&,y),

(32)

where y E R, x E R”, m 3 1. For definiteness, let m = 2. Suppose that y = h(x) is a function such that g[x, h(x)] = 0. The reduced system of (32) is .t =

f[x,W)l

(33)

Now suppose that x(t) is a periodic solution of (33). Then y(t) = h[x(t)] is a periodic function with the same period and x(t), y(t) satisfy (32) if E = 0. It is natural to ask: does there exist a periodic solution (x(t, E), y(t, E)) of (32) which converges to (x(t), y(f)) as E + O? Friedrichs and Wasow [88] have considered the system Xi=Fi(Xi,. c% = F&l,.

. . ,X,),

i=l,.

. . ,?Z -1

(34)

. . 3X”),

where FiE C’, j = 1, . . . , n, along with the reduced system Xi = Fi(Xl,.. ,Xn)y

(35)

0 = F,(x,, ...,x,). It is assumed that (35) has a periodic solution (xi, . . . , x,) 2= (ui(t), . . . , u,,(t)) with period T,the “base solution”. The main theorem of [88] is: 16 (Friedrichs and Wasow). If the system (35) possesses a non-degenerate solution (xi, . . . , x,) = (us, . . . , u,,(t)) satisfying the condition THEOREM

aF,(ui,.

. . , u,)

au,

periodic

+ 0,

then, for every sufficiently small value of E, say, (E] < ~0, the system (34) admits exactly one closed orbit (Ui(t; E), . . . , U,(t; E)) with the following properties: (a) (Ui(t; c), . . . , u,(t;

d) -+(4(t),

. . . , u,,(t)) , as E -+ 0. The convergence

(b) tU,(t; E), . . . , U,,(t; E)) possesses a period T + Z(E), t(c) + 0,as E ---*0; (c) the Uj(t; E) are continuous functions of E for (E( < Q.

is uniform in

Periodic

orbits

of autonomous

ordinary

differential

equations:

theory

and applications

941

Levinson [89,90] has obtained important results in case the “base solution” is discontinuous. Let x E R”, u E R. The system

i==li+ql,

(36)

&ii+ gri + h = 0, is considered in [89, 901 for E > 0. The functions f, Q,are C’ vector functions depending on X, U, t, E and g, h are C’ scalar functions of the same variables. For E = 0, (36) reduces to

Y =fi.J+ cp,

(37)

gi+h=O,

where X, u are replaced by y, v in order to distinguish between the two systems. Since g may vanish at certain points, the solutions of (37) may have discontinuities. Functions y(t), v(t), m G t < p are called a solution of (37) if (i) with the exception of a finite number of values q of t, ff < Zl -=c. . . < tN < /I?,y and v are continuous and the limits of y and v exist at q k 0; (ii) in each of the subintervals, y and v satisfy the reduced system (37); (iii) at t = (Y, p the function g is greater than 0 and in each subinterval g is greater than 0 and tends to zero at t + q - 0; (iv) the jumps at each t are obtained by solving dyldv = f(y, v, t, 0) from v(t - 0), y(t - 0) for increasing or decreasing v, depending on the sign of h, until the first value of v for which the integral g(y(v), v, r, 0) dv, I extended from v(t - 0) to v, vanishes. This value of v is v(r + 0) and the corresponding value of y is y(t + 0). It is assumed that C fi(aglayi) + (ag/av) f 0 at t - 0 and that h does not vanish there. 17 (Levinson). Let the reduced system (37) have a solution SOin the sense defined above, for cr G t s /3. Let E > 0, S 1 and S, are small enough there is a solution x(t), u(t) of (36) over ((u, p) for any set of initial values satisfying

THEOREM

lx(a) - Y(4

+ 144

- 4cu>l =G61,

Moreover as E, 6i and &J tends to zero, the curve representing u, t)-space tends to So. In particular for any fixed S > 0 Ix(t) - r(t)1 + 144 - v(r) I -

the solution x(t), u(t) in (x,

0

uniformly over the intervals (Yc t c tl - 6, tl + 6 C t C q - 6, . . . , tN + 6 d t d @, as E, 6, and 62 + 0. Also

uniformly over the above-mentioned ((d2uldt2) - (d2v/dt2)).

intervals as E, S1 and 62 +

0. The same is true for

BINGXJ LI

948

Additional theorems discuss a similar behavior for the derivatives of x and u with respect to the initial conditions. In case the system (36) is autonomous, the same result relating the existence of periodic solution for (36) to (37) holds. Wasow [91] considers the nth order differential equation ?x(“) = F(x, x’, . . . ( xy

t; E),

(38)

where k > 0, n > M 2 0 are integers, and the right side is analytic. He shows the work of Volk [Akad. Nuuk SSSR, Prikl. Mat. Mech. 10, 559-574 (1946); 11, 433-444 (1947); 12, 29-38 (1948)] contains a serious error. He uses the formal scheme of Volk in combination with ideas of his earlier paper [J. Math. Phys. 23, 173-183 (1944)] and thereby obtains a rigorous theory. Periodic solutions of singular perturbation problems are also studied by several Russian mathematicians, e.g., Pontryagin [92]. See Cesari [93] and Cronin [94] for further references. 4. NONEXISTENCE

OF PERIODIC

ORBITS

Bendixson’s criterion [ll], [95] can be used to determine whether second-order autonomous systems have no closed orbits or closed contours consisting of orbits. As to the case of higher dimensions, few works have been published. Leonov [96] derived a similar criterion for the following third-order equation: I = cp(P,i,x),

or the equivalent

(39)

system il

=

&1,X2,x3),

(40)

x2 =

Xl,

x3

x2,

=

where 47 E C’(Iw3). Leonov used a result of Yakubovich

[97] to prove the following

THEOREM 18 (Leonov).

@$x,7

x2,

x3)

. adO.

x2,

x3)

<

o

ax3

ax1

’

then the system (40) does not possess any nontrivial closed orbits. Demidowitsch [98] presents a sufficient condition for dynamical systems in (w3to have no closed orbits. He considers the system clx z = P(X,Y,X),

g=

Q(x,r~,,~=R(x,y,z),

(41)

where P, Q, R E C’(G), G C [w3is a simply connected region, and shows that if the system (41) admits a first integral H(x, y, z) = h and the divergence of the vector field is of constant sign in G (except for a set of measure zero), then (41) does not have a closed orbit in that

Periodic orbits of autonomous ordinary differential equations: theory and applications

949

region. The existence of the first integral reduces the problem to two dimensions and so the theorem is essentially the same as the well-known theorem of Bendixson. Medvedev [99] investigates sufficient conditions for dynamical systems on n-dimensional manifolds M” to have no k-dimensional integral cycles. A singular chain Lk on a manifold M” is called a singular integral chain of the dynamical system if the vector field determining the dynamical system on M” induces a tangent vector field on Lk. Some criteria for the absence of integral singular cycles homologous to zero on M” are derived. They are generalizations of the criteria of Poincare-Bendixson. For example, Theorem 3 of [99] reads as follows: Let a dynamical system (A) of class @ or the analytic class be given on M” (i.e., in each region H of the covering Z there is a dynamical system duldt = X(x), X is a contravariant vector on M”. At each point of M” a vector X(x) is defined, lying in the tangent space to M”), and let there be a smooth or analytic function 4x) such that the divergence of the vector fl g VX has constant sign on M”:

where g = Det Igijl, and gij i, j = 1, . . . , n) are the components of a contravariant tensor of M”. Then (A) has no (n - 1)-dimensional integral singular cycles homologous to zero on M”. Cronin [lOO] presents some theorems showing how changes of parameters or the introduction of a periodic term may result in the suppression of a periodic solution and the appearance of a stable equilibrium point. One theorem shows that the addition of a constant term can suppress oscillations, another shows that the addition of linear terms can suppress oscillations. 5. UNIQUENESS

OF PERIODIC

ORBITS

5.1. Goran Borg [loll observes that the periodic orbit of (1) in a bounded region D of the phase space M can be shown to be unique and stable if the distance between solution paths

decreases with increasing time. 19 (Borg): Assume that: (i) it) f a;;‘; (x - y)) d -Allx - yll*, Vx, y E D, where D C M is a bounded region, - y is orthogonal to the vector field of (1) at y, i.e., (F(y), x - y) = 0; (ii) E is free from singular points; (iii) P+(p) C D, Vp E 0; then D contains a unique periodic orbit y, and Q(p) = y, VP E D. THEOREM

Sherman [34] successfully applies Theorem 19 to prove the uniqueness and stability of a periodic orbit of a 3-dimensional autonomous system arising from a nuclear spin generator: f = /3x + y, j = -x - /3y(l - kz)

(42)

i = /3[o(l - z) - ky2], where /3 5 0 and /3a 3 0 are linear damping terms, and /3k is proportional

to the amplifier

BINGXI LI

950

gain in the voltage feedback. To establish the contracting property, Sherman uses a suitably defined metric which depends on a theorem by Liapunov [102. pp. 187-1901. 5.2. Robbins [52] proves the uniqueness of a periodic orbit of the Lorenz equation (19) by demonstrating the existence of a unique fixed point for the corresponding Poincare map. This is another method of proving the uniqueness of a closed orbit for (l), although it is not always easy to apply. 5.3. Whether or not a closed orbit of (1) is unique can be a delicate question. The theorem by Borg and the method of the Poincare map are quite difficult to use. Sometimes it is necessary to gear the tricks to the specific situation. The following papers serve as examples. Chin [103] studies the third order system of Rauch [24]: k,

2 +(k2 + kg(x)) $ +

+ g(x) g + x = 0.

(43)

Introducing the notations D = didt, D” = d”ldr”, one can rewrite the left side of (43) in the following form

=[k3~+1]

If k2 -(kllk3)

~~?+~(X)-k~}~+~]x+(k~-~+kj)D;. i 3

+ k: = 0, (43) can be decomposed

[

3

into

2 D’ + (g(x) - k3) D + 1 x = u, 1

and [kjD + l] U = 0.

The latter can be integrated: U(t) = C exp( -t/k3).

Hence, Chin reduces the investigation of (43) to studying the second order nonlinear equation $D’+(g(x)-kj)D+l

[ 3

I

x=Cexp(-t/k3),

where C is an arbitrary constant. Among the results obtained by Chin is the following: THEOREM

20 (Chin). Suppose that:

(i) kz - 2 + k: = 0; (ii) the second order nonlinear equation i has a unique nontrivial periodic solution.

2 0’ + (g(x) - k3) D + 1 periodic

solution.

1x=0

Then (43) possesses a unique nontrivial

Periodic orbits of autonomous ordinary differential equations: theory and applications Kamachkin

[29] considers

a relay system

with hysteresis

i = Ax + Bf(a), where’x

E R”, the constant

matrix

951

(44)

of A have negative real A E Rnx”, all the eigenvalues vectors. The nonlinear function f(u) is given

parts, u = T*x, B, 0 # I E KY are constant by

where

ml
>12,

-I’*A-‘Bm2

< 11. A distance

function

p

between x1 and x2 is introduced: &1,x2)

=(a4

-

x2@))”

qm-4

-

X2(0),

and the existence of the positive definite matrix V (such that dpldt < 0) is guaranteed by a theorem of Liapunov on stability matrices. It is proved that if A*T = cC(O # CYE R) then (44) possesses a unique closed orbit The n-dimensional system 1, = l/(1 +x3

- CYlXl,

is considered by Li [104]. This mathematical model was proposed, for negative feedback control processes arising in the study of a genetic regulatory mechanism in the synthesis of proteins, by several authors [105], [106], [403. System (45) possesses at least one periodic orbit in a certain closed and bounded region T (Hastings et al. (401). Li proves that if mj # aj, i # j, the periodic orbit of (45) is unique and asymptotically stable using the theory of nonlinear Volterra integral equation (Miller [ 1071). See also Hastings [ 1081, where the uniqueness and asymptotic stability of the periodic orbit of (45) for n = 3 is proved under some restrictive conditions on the parameters. Li [109] applies a theorem due to Miller [107] to solve the uniqueness and stability problem for a closed orbit of the system (46) considered by U. an der Heiden [39]: i = a/[1 + exp{-f)‘=x-PY,

b(q - p)z}] - ax, (46)

i=y-qz,

where x(r), 0 < x(t) < 1, is the normalized axonal impulse frequency at time t, f = f(t) is the input of the nerve cell. The constants a > 0. 0 < p K q and b are characteristic for the individual neuron or special type of neurons. The system (46) is an improved neural model for studying a neural network, see Stein, Leung, Mangeron and Ogustoreli (1101. Dai [ill] successfully uses the techniques developed in [ 1081 to treat the uniqueness and global asymptotic stability of the periodic solution of the modified Michaelis Menten mechanism.

BINGXILI

952

6. STABILITY OF PERIODIC ORBITS 6.1. Let y be a closed orbit of period T > 0 of the dynamical system (1). Let p E y. Suppose that it - 1 of the eigenvalues of the linear map Dgr(p) are less than 1 in absolute value. Then y is orbitally asymptotically stable, see Hirsch and Smale [5, p. 2771. Or if n - 1 of the characteristic exponents have negative real parts, then y is orbitally asymptotically stable. Floquet theory may be used to find characteristic exponents if the periodic orbit is known explicitly, see Hartman [4]. These propositions are not convenient to use since they require information about the solution of the system, not merely of the vector field, though they are of great importance. Franke and Selgrade [112] demonstrate how techniques of global analysis and numerical analysis enable a computer to prove certain qualitative properties of orbits to autonomous ordinary differential equations. They provide criteria for locating a connected, attracting invariant set and for proving this set is a periodic orbit with one zero characteristic exponent, and the remaining characteristic exponents having negative real parts. Such a periodic orbit is orbitally asymptotically stable. In [112], (1) is considered with M = R”. The system (1) determines a system of ordinary differential equations on R” x R” given by

i = F(x), d = DF(x)u,

(47)

where (x, u) E R” x Iw”, and DF(x) is the derivative matrix of F at x. The solution curve to (47) with initial condition (x, v) E [w” X [w”is q’(x, u) = (g’(x), Dg’(x)u). Here g’(x) is the solution to (1) and Dg’(x) is the derivative matrix of g’ at x. q\Ir’is called the tangent flow. Let A C IX”be an invariant set of the flow g’, for each x E A, the vector field at x determines two subspaces of 53” x R” called the tangent and normal subspaces: E, = {(x, u) E R” x Iw”1u = cuF(x) for some (YE R},

N, = {(x, u) E R” x R”](u, F(x)) = 0). Here (. , .) denotes the usual inner product on PI”. Let 0,: {x} X R”,+ N, be the orthogonal projection of {x} x R” onto N,. Explicitly, 0,(x, v) =(x, G(V)) , where

%(‘)

=

:‘-

((u, F(x)) F(x)l(F(x),

F(x))),

if F(x) = 0, if F(x) # 0.

THEOREM 21 (Franke and Selgrade). Suppose that (i) A is a compact, connected, invariant set of d; (ii) for each x E A the set {IG,(D~‘(x)z))(t ~0) is unbounded V(x, u) E A x [w”- E, where E =X&E,. Then A is a periodic orbit or a rest point whose nontrivial characteristic exponents

have negative real parts.

A sufficient condition for (ii) is: (ii’) There is a T > 0 so that for each x E A and (x, w) E N, with )WI = 1, l@(gT(x)w)l < 1.

Periodic orbits of autonomous ordinary differentialequations: theory and applications

953

Franke and Selgrade use the computer to verify the condition (ii’) for the modified Van der Pol equation i=y,

j=a(l-x2-y*)y-X.

(Here (his taken to be 0.5.) Hence, Theorem 21 shows that the invariant set within the annulus is a periodic orbit with nontrivial characteristic exponents having negative real parts. 6.2. An alternative method of guaranteeing for n = 2 is the well-known result:

the asymptotical

stability of closed orbit of (1)

Poincarb Criterion. Let F: lR2+ lR* be a C’-vector field, and let y: R -+ R* be a periodic integral curve of F. If Jr div(F) < 0, then y is asymptotically orbitally stable, with asymptotic phase [2-4].

Churchill and Selgrade [113] discuss an n-dimensional generalization of the above criterion. They observe that by interpreting the integral in a slightly different way, the result admits a useful generalization. Consider system (1) with M = 58”. Let T(R”) = R” X R” be the tangent bundle of R”, and let Dg’: T(W) + T(W) be the tangent flow, i.e., the flow induced by the linearized equations (47). Let T’(W) denote the unit sphere bundle of R”, and let EL denote the orthogonal complement of any subbundle E C T(W). 22 (Churchill and Selgrade). Assume I is a closed orbit of (1). Let E C Tt-(R”) be the line bundle generated by F, and for p E r, (p, w) E EL fl T@!“), t E Iw, define

THEOREM

v(r) = V(p, w; t) =

(F(gyP)),F(gyP)))@g’(p)w,Q’(p)w) - u’(gf.d), @d&d

Now assume that: (*) V(q, p; t) increases as t--, --co for each (q, y) E Ef = {(p, w) E Tr(Iw”)I Iw/ = 1, w E El}, q E I?. Then the nontrivial characteristic exponents of I have negative real parts. That means, I is asymptotically orbitally stable with asymptotic phase. Remarks

(i) If ri < 0, then (*) obviously holds; this is the first thing to check in applications. (ii) For n = 3, V(t) can be expressed as IF@‘(p)) X Dg’(p)l*, where x denotes the usual cross product. (iii) For n = 2 one computes that v = 2 div(F)V; hence in this case V(t) = V(0) exp(2 1 div(F)).

Suppose I is a periodic orbit of primitive period t > 0, and that J6 div(F) = p < 0. Let MO =inf{exp( - 2 po div (F)) 10 0, and for t < 0 let k = k(t) be the greatest integer

BINGXILI

954

in {-r/r}. Then from the above expression of V(t) we have V(t) = V(0) exp(-

j”div(flJ

= V(0) exp( -2/-*.

div(F) - 2 rkrdiv(F))

I

2 V(0) M. exp( -2kp). Since k -+a as t+ - m. we conclude the hypothesis (*) must hold. Theorem 22 thus includes the Poincare criterion. Churchill and Selgrade also apply Theorem 22 to determine stability of a rest point of the system: ir = X4(1 + X3) - LrXr, X? =x1 - pxz,

(48)

x3 =x2 - yx3,

CY> 0, p > 0, y > 0, which was proposed by Griffith [114] to model a cellular process for control of gene expression by positive feedback. Chicone and Swanson [ 1151 presents a generalized Poincart stability criterion for an abstract dynamical system, which is different from that of Churchill and Selgrade. They associate a linear semigroup with a given dynamical system. If a smooth vector bundle flow (@‘, $‘) acts on a smooth Riemannian vector bundle (E, M, n), a one-parameter group @I”of bounded operators on the complex Banach space B(E) of continuous sections q of E is defined:

The most important algebraic property of this semigroup is hyperbolicity. The pair (@I, Q’) is hyperbolic if E splits as a continuous Whitney sum of invariant subbundles E = E’ @ Esuch that +’ contracts E’ and expands E-. Equivalently [116, p. 81, (+‘, @‘) is hyperbolic if & for any f # 0 is a hyperbolic operator, i.e., has spectrum a(@:‘), disjoint from the unit circle. The generator of q5? is a closable operator L, densely defined in B(E) by setting

An operator is called infinitesimally hyperbolic if its spectrum is disjoint from the imaginary axis. The main result of (115] is: THEOREM

23 (Chicone and Swanson). (+‘, r$‘) is hyperbolic if and only if L is infintesimally

hyperbolic. This theorem offers a generalization of a flow on the plane. 6.3. For special class of nonlinear

of Poincare’a hyperbolicity

systems, an equivalent

criterion for a periodic orbit

necessary and sufficient condition

Periodic orbits of autonomous ordinary differential equations: theory and applications

955

for the asymptotic orbital stability of a closed orbit was given by Cartianu and Murgan [117]. They consider an automomous nonlinear systems of the form (associated with a certain nonlinear feedback system)

where P(S) =s” +pn-&-l + . . . +pts + pa, Q(s) = q,_rs”-’ + . . . +qls + qo, f: R+ R’ is a nonlinear map. They treat the problem by the method of first variation and the method of harmonic linearization. Acknowledgement-I

am indebted to the referee for helpful suggestions.

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