Dynamics of commuting systems on two-dimensional manifolds

Annali di Matematica pura ed applicata (IV), Vol. CLXXIII (1997), pp. 213-232

Dynamics of Commuting Systems on Two-Dimensional Manifolds (*).

M. SABATINI

Abstract. - We give a description of the local and global behaviour of orbits of commuting systems in the plane and on two-dimensional compact, connected, oriented manifolds.

Introduction.

Let us consider a couple of differential systems defined on an open, connected subset U of the plane:

(Sv) z = V(z), V=-(Vl ,V2)eC~(U, R2),

(Sw) z = W ( z ) , W = - ( W l , W 2 ) e C ~ ( U , R2).

Let us denote by r z) (F(s, z)) the solution of (Sv) ((Sw)), such that 0(0, z) = z (F(0, z) = z). Since we do not assume that the solutions are defined for every real numbers t or s, we shall consider the local flows defined by the two differential systems [S]. We say that the local flows ~(t, z) and ~f(s, z) commute if, for every couple (t, s) such that both q~(t, ~p(s, z)) and F(s, q~(t, z)) exist, we have:

q~(t, ~(s, z)) = ~(s, ~(t, z)).

I t is easy to verify if a couple of local flows commute. We only have to check if the Lie brackets of V and W vanish identically on U [0, thin. 1.34, A, w 39]. Commutativity is quite strong a condition, and has heavy consequences on the behaviour of the orbits of

(*) Entrata in Redazione il 3 giugno 1995 e, in versione finale, il 6 dicembre 1995. Indirizzo dell'A: Dipartimento di Matematica, Universit~ di Trento, 1-38050 Povo (TN), Italy;

e-mail: [email protected] The author would like to thank M. VILLARINI of the Univ. di Firenze, P. MOSENEDER FRAJRIA

of the Univ. di Trento, and G. MEISTERS of the Univ. of Nebraska, for several interesting conver- sations on the subject of this paper. A special thank to L. MAZZI of the Politecnico di Torino, and J. DEVLIN, of the Univ. of Wales, for a careful reading of a previous version of this paper.

214 M. SABATINI: Dynamics of commut ing systems, etc.

a differential system. Lukashevich [L] and Villarini [V] studied from different view- points a class of commuting systems. Actually, the first author was probably not aware that the systems he considered commute, and proved his results under the only assumption that the components of the vector field are a couple of conjugate harmonic functions. He showed that:

(1) critical points do not have hyperbolic sectors;

(2) if a critical point 0 is nondegenerate, then 0 is a centre if and only if the jacobian of V at O has imaginary eigenvalues;

(3) any centre is isochronous;

(4) the region covered with cycles surrounding a centre is unbounded.

(5) there are no limit cycles. His proofs were based on the the~)ry of analytic transformations. On the other hand, Viltarini showed that under Lukashevich hypotheses, the local flows defined by V and the orthogonal vector field V• commute. Using a topological argument, he re-proved points (3) and (5). Villarini's argument [V, thm 4.5] does not require the orthogonality of the vector fields: points 3, 4 and 5 hold whenever there exists W commuting with V and transversal to V at noncritical points. Such an argument can also be used to give a characterization of isochronous centres [Sab3]: isochronous centres are just centres of commuting systems. In the first section of this paper we re-prove point (1) of Lukashevich paper, and we show that point (2) cannot be extended to general commuting systems. Moreover, we prove that:

(6) every nonempty limit set reduces to a critical point;

(7) the index of a critical point is strictly positive;

(8) a critical point of index 1 can only be a centre or a (negatively) asymptotically stable point;

(9) if O is asymptotically stable, then its region of attraction is unbounded;

(10) if d ivV= 0 in a neighbourhood of a critical point O, then 0 is a centre;

(11) if div V < 0 in a neighbourhood of a critical point O, then 0 is asymptotically stable.

Setting A(z):= vl w2 - v2wl~ we also prove that the functions A(~(t, z)) and A(F(s, z)) are, respectively, solutions to the differential equations

A = AdivV, A = AdivW.

As a consequence of such a property, if a couple of commuting vector fields are transversal at a point z, then they are transversal at every point reachable from z by

M. SABATINI: Dynamics of commuting systems, etc. 215

following orbits of (Sv) and (Sw). Moreover, as explained at the end of section 1, this allows to find invariant curves of commuting systems, as in [Sab2].

In the second section we study some additional properties that can be proved when one of the commuting vector fields is complete. If V defmes a global flow, then

(12) if V has no critical points, then the flow is parallelizable;

(13) if V has a critical point, then it is unique, and it is a global centre or a (negatively) globally asymptotically stable point.

In the last section we study the global properties of commuting flows on two-dimensional, compact, connected, oriented manifolds. Such manifolds can be classified up to homeomorphisms according to their Euler characteristic. Since the Lefschetz index of critical points of commuting systems is positive, using Poincar~-Hopf theorem, we show that only the torus and the sphere admit commuting systems, and give a rough description of the possible global phase portraits.

1. - Definitions and general properties of commuting vector fields.

We are concerned w~th couples of autonomous differential systems in the plane

(Sv) Z ---- V ( z ) , V-(Vl,V2)eC~(U, R2),

(Sw) = W(z), W - (wl, w~) ~ C ~ (U, R2).

with z ---- (x, y) e U, open, connected subset of R 2. Under such assumptions we have local existence and uniqueness of solutions, and continuous dependence on initial data. For sake of simplicity we have assumed the vector fields to be of class C ~, but most of the results in this paper request only the C 2 regularity. We denote by ~b(t, z) (F(s, z)) the solution of (Sv) ((Sw)), such that ~(0, z) = z (F(0, z) = z).We do not assume that the solutions of (Sv) or (Sw) are defined for every real value, so we give next definition in terms of local flows (see [S] for the definition of local flow).

DEFINITION 1.1. - We say that two local flows ~(t, z), ~fl(s, z) commute on the open set U if, for any z ~ U and for any couple of real numbers (t, s), one has :

~(t, ~(s, z)) = ~(s, ~(t, z)),

whenever both r F(s, z)) and ~(s, 0(t, z)) exist.

216 M. SABATINI: Dynamics of commuting systems, etc.

When the two local flows are defined by the solutions of a couple of differential systems of class C ~ in U, we have the following characterization of commutativity [O, thm. 1.34, A, w 39]:

THEOREM 1.1.- The local flows defined by the differential systems (Sv) and (Sw) commute if and only if their Lie brackets [V, W] vanish identically on U.

We recall that the Lie brackets, or commutator, [V, W] - ([V, W]I, [V, W]2) are given by the following formulae:

[ ~Wl ~Vll ( ~wl ~Vl 1 IV, W]I : ( v 1 ~ - w I ~ x ] -[- v2 - ~ y - w 2 9y ] '

aw2 IV, W]2 --- v 1 ~ - t

Villarini [V] has considered the case of a vector field commuting with its orthogonal vector field V• = (v2, -v~). In this case one has:

( v2 vl)( v2 vi) IV, V• ]1 -- Vl ~ x + ~ -~ v2 3y ax '

( v2 Vl)( v2 vi) [ V , V • 9y ax - v 2 ~xx + -~y "

If Vl, v 2 are conjugate harmonic functions, then [V, V~ ] = 0. Lukashevich and Villari- ni's results quoted in the introduction were obtained under such an assumption. Vil- larini's approach works as well in the case of non-orthogonal commuting systems, pro- vided they are transversal at non-critical points. The transversality assumption is necessary, since any vector field commutes with itself.

REMARK 1.1. - An example of a couple of transversal, non-orthogonal polynomial systems commuting with each other is given by

(1.1) 2= - y + x y ,

x + y 2 ,

(1.2) I & = x ( 1 - x ) ,

t?4=Y(1 x).

In this case, we can take U as {(x, y): x < 1}, or as {(x, y): x > 1}. The line x = 1 is invariant for (1.1), and a critical line for (1.2). The origin 0 is an isochronous centre for (1.1). The cycles surrounding 0 are contained in the half-plane {(x, y): x < 1}.

M. SABATINI: D y n a m i c s o f c o m m u t i n g systems, etc. 217

In general, every quadratic system with an isochronous centre commutes with some polynomial system [Sab2].

Examples of hamfltonian systems admitting transversal commuting systems can be obtained from the systems considered in [Sabl]. If :b(x, y) - (fix, y), g(x, y)) is a C ~ map of the plane into itself with detJ:b(z) - 1, then the hamfltonian system having Me = ] ~[2/2 as hamiltonian function:

(He)

8Mr _ f + g 8g 2 -

8114r _ ~ 8g !) = 8x f - g 8-~'

commutes with a transversal system. In fact, when detJr ~ 1, the map :b trans- forms locally (H~) into the linear system:

(Lc)

that commutes with the orthogonal linear system:

(LN) ?=Y.

The map d)-1 takes locally (L N) into a differential system that commutes with (Hv). Such a system is given by the following equations:

(K~)

I ag af Jc = f ~-~y - g ~ y ,

!) = - f + g -~x "

Since we have the explicit form of (K,), we also have the global commutativity of ( H , ) and (Ko). This can be used to give alternative proofs of part of the results obtained in [Sabl].

If we take :b(x, y) = 1 / V ~ ( e ~ / 2 c o s ( y e - X ) - 1, e=/2s in(ye-~)) , then (He) has infinitely many isochronous centres at the points (0, 2nz), n ~ Z. Such example is due to G. MEISTERS [Me].

The geometric approach used in [V] can also be used to prove the above point (1), and t o describe some additional dynamic properties of commuting systems. Point (4) is a consequence of isochronicity, and does not request commutativity to be proved. Property (2) cannot be extended to the class of commuting vector fields, as it is shown

218 M. SABATINI: D y n a m i c s of commut ing systems, etc.

by the commuting systems:

{2 = - y ,

(1.3) ?~ = x;

2 = - y + x (x 2 + y 2 ) ,

(1.4) [ /= x + y (x 2 + y2) .

Both systems have + i as eigenvalues at the origin, but the former one has a centre at O, while the latter one has an unstable focus at O.

Our analysis of the dynamics of commuting systems starts studying its properties at critical points. We follow HARTMAN [H] for the terminology about critical points and their sectors. For sake of brevity, we write ,<0 has a hyperbolic sector, to say that ~,there exists a closed Jordan curve enclosing O, having a hyperbolic sector,. Similar- ly, for elliptic and parabolic sectors. We say that V and W are transversal at a point z, and we write V(z) A W(z) ~ O, if Vl(Z)W2(Z) - v 2 ( z ) w l ( z ) ~ O.

In this paper the following set of assumptions will be referred to as (CT).

i) (Sv) and (Sw) are plane differential systems of class C ~ , having isolated critical points in U;

ii) V(z) = 0 ~ W(z) = 0;

iii) V(z) ~ 0 ~ V(z) A W(z ) ~ 0;

iv) [V(z), W(z)] -- 0 on U.

When studying polynomial systems, point i) is not always satisfied on all of R e. For instance this happens for quadratic systems, as those considered in [Sab2]. In such cases it is sufficient to take U as the complement of the set of nonisolated critical points. In polynomial systems, such a set is an algebraic curve, that is an invariant curve for both V and W (see cor. 1.6 below). At the end of this section we discuss the transversality condition and its relationship with other properties of the vector fields. At the beginning, we draw some topological consequences of the above hypotheses.

We need the following consequence of the definition of commutativity.

LEMMA 1.1. - L e t ( CT) hold. A s s u m e that the nontr iv ial arcs of orbit ~ 1, ~ inter-

sect the nontr iv ia l arcs of orbit ~1 , ~ 2 . Let us set a : = q~l A F1- I fS l , se , tl , te are the

least positive values such that:

b:= ~1N yJ~ = q~(tl, a) ,

c:= 4'2 F1 ~1 = ~(sl, a),

d := q~2 A ~2 = q~(t2, c) -- ~(s2, b).

Then sl = s2, t l "-~ t 2 .


PROOF. - By contradiction. Without loss of generality we can assume that s~ < s~, t~ ~< t~ (possibly exchanging V and W). Then ~fl(sl, b) = ~f(s~, ~(tl, a)) and q~(tl, C) = = q~(tl, ~f(s~, a)) exist and coincide, by the commutativity of V and W. This contradicts the minimality of se. �9

In lemma 1.1, q~, ~e (~p~, We) could be arcs of the same orbit. This can occur in pres- ence of asymptotically stable critical points. To assume that si, t~ > 0, i = 1, 2, is not a restriction, since we can replace V with - V, or W with - W, and we still have a couple of commuting vector fields.

If we do not assume the minimality of the involved parameters, the above statement is false: any commuting system with a centre provides a counterexampte.

In next theorem we denote by w(r (a(q~)) the positive (negative) limit set of the orbit q~.

THEOREM 1.2. - Let (CT) hold. Assume that 0 is a critical point of (Sv). Then 0 cannot have hyperbolic sectors.

PROOF. - By contradiction. Let B[O, ~/] be a closed disk not containing other critical points than O. Let us assume that 0 has a hyperbolic sector, bounded by two separatrices 01 and q)2, satisfying w ( r 1 6 2 Let us choose Zlar ~]], z2er ~/] such that ~{ := {q~(t, zl): t t> 0}cB[O, ~]], q)~ := := {q~(t, z2): t ~< 0} cB[O, 7/]. By hypothesis, there exist positive (possibly changing W into - W ) sl , t12, ~2 such that r (~P(el, Zl))) = ~fl(ee, z2), and every q~-orbit starting at a point of the curve ~((0, ~1),zl) intersects ~((0, ee),z~). For z:= ~P(sl, Zl), let us set

2:1 :---- { tE [0, t12]: 3S* ~< 0: 'f(S*, q~(t, Z)) e ~ { , 'p((S*, 0), ~(t, Z)) N ( ~ U ~ ) = 9}.

•1 is nonempty, since 0 e2:1. Moreover, sup 2:1 < t12, by the continuity of ~f. Let us set a : = sup 2:1. Then the negative semi-orbit ~fj starting at r z) does not intersect q~ {, otherwise any other orbit in a neighbourhood of r z) would do the same, contradicting ~ = sup2:1. Analogously, ~fj does not intersect q ~ . Hence ~ j , contained in the part of plane bounded by q~{, O, q~ , ~f((0, e2), z2), q~([0, t12], z), ~(el , Zl), has compact closure. This implies that the corresponding ~f-solution exists for all s ~< 0. By the eommutativity of V and W, we have:

~( -- El, r Z)) : ~(O', ~/)(--t~l, Z)) = r Z l ) .

This is a contradiction, since ~p( - sl , r z)) lies on ~p~, while q~(a, ~p(-Sl, z)) lies on q ~ , that does not intersect ~u~. �9


REMARK 1.2. - A commuting vector field can have a critical point with elliptic sectors, as showed by the following example (see [SC, pag. 80]):

[ ~ = - 2 x y , (Se) ! ?~ = X 2 __ y2 .

(Se) has two conjugate harmonic functions as components, hence it commutes with its orthogonal system. The origin is a critical point with two elliptic sectors, having the half-lines {(x, y): x = 0, y < 0} and {(x, y): x = 0, y > 0} as separatrices. Every orbit nonintersecting the y-axis is homoclinic. Its orthogonal system has the same phase-portrait, rotated by z /2 .

COROLLARY 1.1. - Let (CT) hold. I f a semi-orbit q~ § c U has non-empty positive limit set w(q~ + ), then there exists a critical point P e U such that w(q~ +) = {P}.

PROOF. - By Poincar~-Bendixson theorem, since (Sv) has no limit cycles, (o(~ § can only be a critical point or a generalized cycle F. In the latter case, F should contain a critical point with a hyperbolic sector, contradicting Theorem 1.2. []

In next corollary, j(O) denotes the index of 0 with respect to V(see [H] for the definition of index of a critical point).

COROLLARY 1.2. - Let (CT) hold. Let 0 be a critical point of V. Then j(O) > O.

PROOF. - Let C be a circumference centred at O, not containing other critical points than O. By [H, ch. VII, thin. 9.1], the index of 0 is:

= 1 ( 2 + ne - nh), j(o)

where ne is the number of elliptic sectors on C, nh is the number of hyperbolic sectors on C. Then the thesis comes from Theorem 1.2. []

DEFINITION 1 . 2 . - If 0 is a centre, we denote by No the largest open connected set covered with cycles surrounding O.

If 0 is asymptotically stable, we denote by Ao its region of attraction. Assume that there exists a curve y, transversal to V, such that every ~-orbit

meeting y is a cycle. We denote by B the union of the q~-orbits meeting ~. We say that B is a central band of (Sv) if there exists no centre 0 such that B c No.

COROLLARY 1.3. - Let (CT) hold. Then (Sv) has no central bands.

PROOF. - i ) Let B be a central band. By definition, its interior boundary a~B does not reduce to a critical point. Let z E 9iB be a non critical point. The solution starting at z exists for all t, since it is contained in a compact set. There is a sequence z~ e B


converging to z. Working as in the proof of Villarini's theorem, one can show that the cycles in B have the same period T. Hence

0 ( T , z ) = l im q~(T, z~) = l im z~ = z ,

that is, z belongs to a cycle 5. Since 5 cannot be interior to a band of cycles, then either it is a limit cycle, or it has a neighbourhood containing infinitely many limit cycles, contradicting Theorem 4.5 in [V]. �9

If a critical point O has index 1, it cannot have elliptic sectors, by the formula in the proof of Corollary 1.2. Since hyperbolic sectors are not allowed, if 0 is not a centre, it can only have one parabolic sector. In order to prove that it is asymptotically stable (negatively asymptotically stable), we need the following lemma.

LEMMA 1.2. - Let (CT) hold, and 0 be a critical point of (Sv). I f there exists a neighbourhood Uo such that for any z ~ Uo, co(z)= {O} and 0 ~ a(z), then 0 is asymptotically stable.

PROOF. - By contradiction. Assume 0 is not asymptotically stable. Since by hypothesis it is attractive, it cannot be stable. Then there exist s > 0, z~-~ --) 0, t.~ e R, t~ >I O, such that Ir z n ) 1 --~ S and B[ O , ~ ] r Uo . Without loss of generality we can assume that

(o) t n : = i n f { t ~ O : IO(t~,z~)l = s } .

The sequence d~ -= r z~) has a subsequence (that we call again d~) converging to a point d*. The negative semiorbit 5~. is contained in the closed disk B[O, s]. Other- wise, there exists t~ > 0 such that 10 ( - t~, d*)l > ~, and, by the continuity of q~,

O(-t~, d* ) = l im r d~) = l im O(t~ - t~, Zn).

In this case for large values of n we have Iq~(t~ - t~, z~)[ > e, contradicting (o). Hence a(d*) c B[O, ~]. a(d*) cannot be a cycle, since in this case 0 should be en-

closed by it, and 6~. should cross it to go to O. Hence a(d*) = {O}, contradicting the hypothesis. �9

REMARK 1.3. - In general, it is not sufficient to request that any orbit in a neighbourhood of O is attracted to O to ensure its asymptotic stability (see [BS], pag. 59).

THEOREM 1.3. - Let (CT) hold. I f O e U is a critical point of index 1, then either 0 is a centre or it is (negatively) asymptotically stable.

PROOF. - Let us assume that O is not a centre. Since (Sv) has no limit cycles, by [H, ch. VII, lemma 8.1], there exists an orbit q~l having {O} as its positive (negative) limit set. If there exists an orbit r having {0} as negative (positive) limit set, then O has an elliptic or a hyperbolic sector, contradicting Theorem 1.2 or Corollary 1.2. Hence q~ 1


belongs to a parabolic sector. 0 cannot have more than one parabolic sector, since two parabolic sectors are separated by one or more elliptic or hyperbolic sectors. Hence the hypotheses of lemma 1.2 are satisfied. "

In next theorem we show how commutativity can be also used to prove the existence of centres. In order to do this, we need the following definition.

DEFINm0N 1.3. - We say that {V(0) e 6~ ~ (U, R2), 0 e R } is a complete family of commuting vector fields on U, if V(O) is a complete family of rotated vector fields (see [D] for the definition), and V01, 02eR, one has [V(0x), V(02)]-= 0 on U.

We recall that the definition of a complete family of vector fields requests the transversatity of any couple of non-opposite, distinct members of the family.

If V and W satisfy (CT), the family defined as follows is a complete family of commuting vector fields.

(Fvw) {V(0):= cos (0)V+ sin(0)W: 0 ~ R}

By [D, thm. 1], the index of a critical point with respect to V(O) is indipendent of 0. Let us denote by S(0) the differential system associated to the vector field V(O).

THEOREM 1.4. - Let V(O) be a complete farnily of commuting vector fields on U. It" 0 E U is a critical point of index I of V(O),for any 0 ~ R, then there exists 0* e [0, z) such that

(1) 0 is a centre for V(O*) and for V(O* + z);

(2) 0 is asymptotically stable (negatively asymptotically stable) V 0 e (0", 0" + z);

(3) 0 is negatively asymptotically stable (asymptotically stable) V 0 ~ [0", 0" + Jr].

PROOF. - First, we prove that there exists a parameter 0 E [0, 2z) such that 0 is a centre for V(O). By Thin. 4 in [D], 0 cannot be a centre for all 0 ~ [0, 2z). Let 0' E [0, z) be a parameter for which 0 is, say, asymptotically stable. Since, from the definition of complete family of vector fields, V(O'+ ~)= -V(O'), 0 is negatively asymptotically stable for V(O' + ~). Let us set

0 : = sup {0 E [0 ' , 0 ' + Jr]: 0 is asymptotically stable for V(0)},

_0:= inf{0 E [0 ' , 0 ' + Jr]: 0 is neg. asymptotically stable for V(0)}.

If 0 ~ _0, then, by Theorem 1.3, for any 0 e (0,_0), 0 is a centre, contradicting thin. 4 in [D]. Let us set 0* = 0 = 0. If 0 is asymptotically stable for v(O* ), the hypotheses of Theorem 2.2 in [MNSS] hold, and a family of bifurcating limit cycles exists for small positive values of 0 - 0" , contradicting Theorem 4.5 in [V]. Similarly, we have a contradiction, if 0 is negatively asymptotically stable for V(O * ). Hence, 0 is a centre for V(O* ).


By Theorem 1.2 in [MS], (S(O*)) admits a C ~ first integral I(O*) in a neighbourhood of O, that we can use as a Liapunov function for 0 ~ 0" , 0* + z. Indeed, ff 0 ~ 0" , 0* + z, the derivative ie of I along the solutions of (S(O)) is:

io= 91v~(0)+ 9I v2(o) = (-v2(O*)vl(O) + vl(o*)v2(o))

where fl is a function that does not vanish on No(O*)\{O}. Hence for any 0 ~ 0" , 0* + z, O is either asymptotically stable or negatively asymptotically stable. Since, as 0 varies, we pass from an asymptotically stable system to a negatively asymptotically stable system only crossing a centre, the theorem is proved. []

REMARK. - 1.4. - Theorem 1.4 can be used to prove the existence of centres. If we find a couple of commuting, transversal vector fields having the same critical point of index 1, then the complete family (Fyw) has exactly two (opposite) centres.

COROLLARY 1.4. - Let (CT) hold and 0 be a critical point of V. Then

i) i f 0 is a centre, then No is unbounded;

ii) i f 0 is asymptotically stable, then Ao is unbounded.

PROOF. - i) Let O be a centre. If No is bounded, then also aNo is bounded. The argument used in the proof of Corollary 1.3 applies to prove that aNo is a cycle 6. By the absence of limit cycles, it is interior to a band of cycles, contradicting the assumption that 6 r 9No.

ii) Let O be asymptotically stable. Let us consider the family Fvw defined above. By Theorem 1.4 there exists 0* such that 0 is a centre for (So.). Its central region No (0*) is unbounded. Without loss of generality, we may assume that the C-orbits enter No (0"). Any C-orbit entering No (0*) remains inside it, by the transversality condition. Working as in the second part of the proof of Theorem 1.4, we can show that No (0*)r which gives the unboundedness of Ao. []

Going back to the transversality of V and W, the vanishing of Lie brackets allows us to check ff VA W ~ 0, by working only on suitable open subsets of the plane, as we show below. This is useful when dealing with centres, since in a neighbourhood of the critical point the transversality can be checked by comparing the linear parts. Let us denote by A(z) the function vlw2 - V2Wl. In Theorem 1.5, Corollary 1.5 and Corollary 1.6 we do not assume that (CT) holds. In particular, we do not assume that V A W ~ O .


THEOREM 1.5. - Let V, W e C~ (U, R2). If [V, W] =-0, then

) A(O(t, z)) = A(z)exp div V(g)(r, z)) dr , 0

( f ) A(~s , z)) -- A(z)exp div W(~(r, z)) dr .

PROOF. - We prove the second statement. The first one can be proved similarly. It is sufficient to prove that A(y)(s, z)) is a solution to the differential equation

; i = A div W .

Let us compute the derivative A of A = VlW2 - v2wl along the solutions of ~ = W(z). We have:

A

IgVl ~W2(~V2 ~Wl t I ~vl ~W2 "~ ~X w2 4- V 1 ~X ~X wl -- v2 ~ wl -~- ~ y 'w2 "~- vl ~

av2 3wl ) ay wl -- v2 ~ w2 ~-

2w2 3ve ) = w~ vl -~x - w l -~x + w l w2

~V 1 ~V 2 ~ ~W2 ~W 1 9X ~y ) + vl w2 -~y -- V2 Wl ~---~ --

~W 1 ~V 1 ) -w2 v2 -~y - w2 -~y �9

Since [V, WI - O, the first term above is -Wl v2 ~ - w2 ~ , while the last term ( 3w, 9v~)

is w2 v l - ~ x - w l ~ �9 Hence:

2~ ---- - - W 1 V 2 9y - ~ y w2] + wl w2 Sx ay -]- vl w2 - ~ y -- v2 w l ~ +

( awl avl ) /awl 9w2) -~-W2 Vl ~X~ ~X wl =(vlw2-v2wl)[--~-x + -~y =AdivW.

There is an evident similarity between the formula in the above theorem and the formula appearing in Liouville's theorem on the derivative of the phase area along the local flow.

[]

DEFINITION 1.4. -- Let us define the reachable set R(z) of z E U as the set of points of U that can be connected to z by finitely many arcs of orbits of q) and ~.

We have the following corollaries:


COROLLARY 1 .5 . - Let V, W e C ~ ( U , R2), and [V, W] = 0. We have:

(1) i f V(z) A W(z) ~ 0, then V(z') A W(z ' ) ~ 0 for any z' �9 R(z);

(2) i f V(z) A W(z) = 0, then V(z' ) A W(z' ) = 0 for any z' �9 R(z).

PROOF. - IfA(z) ;~ 0, then by Theorem 1.5 A cannot vanish both along C-orbits and along tp-orbits connecting z to z ' . Similarly, if A(z )= 0 it cannot assume non-zero values both along q~-orbits and along V-orbits connecting z to z' �9

COROLLARY 1.6. - Let V, W e C ~ (U, R2). I f [V, W] =- 0, then both {z �9 U: V(z) A W(z) ~ 0} and {z �9 U: V(z) A W(z) = 0} are both V-invariant and W-invariant.

PROOF. - Immediate. �9

REMARK 1.5. - In Theorem 1.5 and Corollaries 1.5 and 1.6 we do not ask for all of (CT) conditions to hold. If V and W are polynomial vector fields, and V A W is not identically zero, {z e U: Y(z) A W(z) = 0} is an algebraic curve. Hence Corollary 1.6 can be used to find invariant curves of plane systems with nonisolated critical points. This is the case of some quadratic systems, as those studied in [Sab2]. Critical curves of W are invariant curves of V, and in some cases they are part of the boundary of a central region of V.

COROLLARY 1.7. - Let V, W e C ~ (U, R2). Let (CT) hold~ and 0 be a critical point of V. Then:

(1) i f

(2) i f

(3) /f

divV < 0 in U, then 0 is asymptotically stable;

d i v V - O , then 0 is a centre;

div V > 0 in U, then 0 is negatively asymptotically stable.

PROOF. - ( 1 ) Let us use A(z) as a Liapunov function. By (CT), it is positive definite with respect to O. Since, by Theorem 1.5, the derivative of A along the nontrivial solutions of (Sv) is strictly negative, 0 is asymptotically stable.

(2) Let us use A(z) as a first integral. By Theorem 1.5, A is constant along the solutions of (Sv). Since, by (CT), A is positive definite with respect to O, O is a centre.

Point (3) can be proved as point (1). �9

Examples of systems verifying the hypotheses of the above corollary can be obtained from systems (H~) and (Kr (Hv) is a hamiltonian system, has divergence identically zero, and A - f e + g2, which is a first integral. The divergence of (Kr is 2, and (K~) has a negatively asymptotically stable point at O.


2. - C o m p l e t e c o m m u t i n g vec tor f ie lds in the plane.

A vector field is said to be complete if it defines a global flow, that is if the solutions of the associated differential system are defined for every t e R. The completeness of a vector field V in R e is ensured if there exist constants a, b ~ R such that I V(z)l <~ a + b!z I . Every vector field defined on a compact set invariant for its local flow is complete. Moreover, if V and W commute, and V has a first integral I , t h e n / W commutes with V [Sab3]. For some vector fields W it is possible to find first integrals I of (Sv) such that I W is complete. An example is given by system (1.4) , for which we can choose I(x, y ) = a(x2+y2) , where a is any continuous function such that

rlim ra(r) = 1.

In this section we study some additional properties of commuting vector fields that can be obtained as a consequence of their completeness. For sake of simplicity, we assume that U --- R2.

THEOREM 2.1. - Let (CT) hold, and V be complete. I f V has no critical points, then the flow defined by (Sv) is parallelizable.

PROOF. - By Thm. 5 in [M], it is sufficient to prove that (Sv) has no separatrices. By the definition of separatrix [M, pag. 50] it is sufficient to show that (Sv) has no inseparable orbits. By contradiction, let us assume that there exist two inseparable orbits 0 , , q~ 2- Then, chosen zl e q) 1, z2 e 0 2, there exist ~ 1, E2 > 0 ( < 0), such that any q~- orbit intersecting F((0, el), Z l ) also meets F((0, e2), z2). Le t us set z:= ~(~1, zl ), for some ~]1 e (0, el). There exist t12 e R , ~]2 E (0, ~2) such that

~)(t12, z) = q~(t12, ~f(~]l, Zl)) = ~(~12, z2) .

~f T]I > r]2 , then ~(-~]2 , z) exists, and, by the commutativity, we have:

Z2 = ~fl( -- r]2, q~(tl2, Z)) = q~(t12, ~f l ( - - r ]2 , Z)) ,

~)(- - t12 , Z2) : ~/)(--7]2 , Z) : ~ ( / ] 1 -- ~]2, Z l ) ) .

Hence (~1 and q)e intersect the same transversal orbit, which contradicts their insepa- rability. Similarly, we cannot have ~/1 < ~/e, so that ~71 = ~/2 =: r/. Hence:

z2 = ~ ( - ~, q~(tle, z)) = ~(t~2, ~ ( - ~, z)) = q~(t~2, z l ) ,

that is, z~ and z2 belong to the same orbit, a contradiction. �9

THEOREM 2.2. - Let (CT) hold, and V be complete. I f V has a critical point O, then:

i) either 0 is a centre or it is (negatively) asymptotically stable;

ii) i f O is a centre, then No =- R2; i f O is asymptotically stable, then it is globally asymptotically stable.


PROOF. - i) Let C be a circumference centred at 0 and not containing other critical points. By contradiction, let us assume that 0 has an elliptic sector and define B[O, r]], q~l, r Zl, z2 as in the :proof of Theorem 1.2. By hypothesis, there exist positive

e l , t12, e2 such that

~(t12, ~)(t?2, Z2)) = ~)(~1, Z l )

(notice that here a 0-orbit first meets ~((0, e2), z2 ) and then y)((0, e i ), zl ), different- ly from the proof of Theorem 1.2). By absurd, assume that el > e2. If the couple {V, W} is positively oriented (similarly, if {V, W} is negatively oriented), the positive ~-semiorbits starting at z~, z2 enter any region bounded by an homoclinic orbit of the elliptic sector and O, hence the corresponding solutions are defined for all s I> 0. By the completeness of V, the point ~(t12, Z2) exists. By the commutativity of y) and q~, we have:

~fl(e2, (b(t12, Z2)) = q~(tl2, ~fl(e2, Z2)) = ~D(~I, Z1)

q~(t12, Z2) = ~)(~1 - ~2, Zl ) .

But the point q~(t12, ze) lies on q~2, while ~(e~ - e2, Zl) is inside the elliptic sector, a contradiction. In a similar way one proves that el r e2. Then e~ = e2 =: e. As a consequence,

q~(t12, z2) = ~(e~ - e2, Zl) : ~P(0, Z1) : Z 1 �9

a n d ~1 = ~ 2 . That is, the boundary of the elliptic sector is a homoclinic orbit. Let us consider the negative yz-semiorbits starting at zl, z2. By the continuous de-

pendence on initial data, there exists /~ < 0 such that any ~-orbit intersecting ~((tt, 0), zl) enters C. If, for any n > 0 there exists tt~ e (0, it~n) such that the q~-orbit starting at Y)(/~n, Z~ ) exits from C after having entered it, then 0 has a hyperbolic sector, contradicting Theorem 1.2. Hence there exists tt~ such that any orbit crossing ~((tt~', 0), zl ) has {O} as its positive limit set. The same argument can be applied to z2 to show that there exists tt~' < 0 such that any orbit crossing F((tt~', 0), z2 ) has {0} as its negative limit set. Let us set it* := max {#~', Z~ }. Then F(s, z~), i = 1, 2, exist for any s e (~*, 0). We have:

0(t12 :, ~fl(S, Z 2)) = ~fl(S, ~b(t12 , Z 2)) = ~fl(S, Z l ) .

Hence any q~-orbit meeting ~((/~*, 0), Zl ) is a homoclinic orbit, contradicting the fact that r t is part of the boundary of the elliptic sector.

ii) Assume that 0 is a centre and aNo ~ 0. Since the solutions of (Sv) exist for all t e R, we can apply the argument of Corollary 1.3 to prove that 9No is a cycle, contradicting Corollary 1.4.

By contradiction, assume that O is asymptotically stable, but it is not globally asymptotically stable. Then 9-4o # 0. Let us call W* the vector field of the family Fvw defined in the previous section, having a centre at O. Since 0 is asymptotically stable with respect to (Sv), the b-orbits enter its central region N3, so that N3 cAo. More-


over, 8Ao M zNo = 0, since otherwise an orbit r contained in 8.4o would be attracted to O. By the transversality of V and W*, every ~-orbit in a neighbourhood of r would be attracted to O, contradicting r r 8Ao. Let z ~ 8Ao, and s e R be such that Fw* (s, z) exists and lies in Ao. Then r YJw* (s, z)) e Ao for any t e R. By the asymptotic stability of O with respect to V, there exists t e R large enough to have z* = = q~(t, ~fw*(S, z)) e N 3 . We have

z = ~w*( - s, q)(-t , z*)) = r - t, F w . ( - s , z*)),

where Y)w*(-s, z*) exists, since the z* is on a cycle, and r - t, ~ w * ( - s , z*)) exists by the completeness of V. Hence we have

q)w. ( - s , z*) = r z) E 3Ao,

a contradiction, since 8Ao cannot cross any cycle of the centre. �9

REMARK 2.1. - The hypothesis of completeness cannot be deleted, in order to prove point i). The separatrices of system (Se) of the previous section are the support of solutions not defined for all t e R. The hypothesis of completeness cannot be deleted as well in point ii), as the following example shows (see also [C J, p. 278]).

(S~) ( ~ = 2xy ,

~) 3 - x 2 + y2 .

(S~) is symmetric with respect to the y-axis. It has two isochronous centres, and the y- axis is an invariant line, which is the support of a solution not defined for all t e R .

REMARK 2.2. - Systems (1.3) and (1.4) of the previous section define a family of commuting vector fields whose unique complete vector field is (1.3), that has a global centre at the origin.

In next corollaries we show an unusual relationship between the index of a closed curve and the existence of solutions not defined for all t e R.

COROLLARY 2.1. - Let (CT) hold. I f there exists a closed Jordan curve F having index greater than one with respect to V, then (Sv) has a solution that goes to infinity in a finite time.

PROOF. - I f / " has index greater than one, then either the bounded region having F as boundary contains a critical point of index greater than one, or it contains more than one critical point. Both possibilities contradict Theorem 2.2, hence the vector field cannot be complete. �9

COROLLARY 2.2. - Let (CT) hold, and Fyw be the complete family of commuting vector fields defined in the previous section. I f there exists a closed Jordan curve F


having index greater than one with respect to V, then every field of the family has a solution that goes to infinity in a finite time.

PROOF. - Since the index of a given curve with respect to every vector field of a complete familty is independent of the parameter 0, we can apply the previous corollary to every vector field of the family. �9

3. - C o m m u t i n g vec tor f ie lds on c o m p a c t c o n n e c t e d or i en ted t w o - d i m e n s i o n a l man i fo lds .

Let S be a compact, connected, oriented, two-dimensional manifold (from now on, just surface). Every surface is homeomorphic to a sphere with handles, and two such spheres are homeomorphic if and only ff they have the same number of handles [GP, p. 124, W, Thin. 7.1]. The Euler characteristic of a sphere with p handles is 2 - 2p, so that the sphere has characteristic 2 and the torus has characteristic 0. Every other surface has negative characteristic. In this section we determine which surfaces admit commuting vector fields. For such surfaces, we give some information about the possible global behaviour of orbits.

THEOREM 3.1. - I f a surface has negative Euler characteristic, then it cannot have couples of commuting vector fields satisfying (CT).

PROOF. - Given a surface S and any vector field Vwith isolated critical points on S, by Poincar~-Hopf theorem [GP, p. 134], the Euler characteristic of S is the sum of the indices of the critical points of V. Since every point of S belongs to an open set diffeo- morphic to an open subset of the plane, Theorem 1.2 and Corollary 1.2 hold. If, by absurd, a couple of commuting vector fields existed, then we should have at least a critical point with negative index, a contradiction. �9

Both the torus and the sphere admit couples of commuting vector fields, as we show in the following.

For arbitrary commuting flows on the torus, we have the following theorem.

THEOREM 3.2. - Let (CT) hold on the tomes. Then the flows on the tomes are obtained as quotient flows of a couple of parallelizable flows in the plane.

PROOF. - Let V and W be the vector fields on T. The corresponding plane vector fields Vp, Wp are x-periodic and y-periodic, hence they are bounded. This implies that they are complete, so by Theorem 2.1, they are parallelizable. �9

It is easy to construct examples of commuting systems on the torus. For sake of simplicity we refer to the square Sq with vertices (0, 0) (0, 1) (1, 1) (1, 0) in the plane. The torus is obtained by Sq by identifying opposite sides. For 0v, Ow E [0, z),


let us consider the differential systems:

(Tv) a} = cos 0 V

sin 0 v,

(Tw) { ~ = cos Ow

= sin Ow.

We consider only the case Oy ~ Ow, so that VA W ~ 0. They are constant vector fields, hence they commute. The corresponding vector fields on the torus also commute. Let us consider some cases.

1) Ov= 0, Ow = ~/2. In this case we have only V-cycles and W-cycles (on the torus). Every V-cycle

meets every W-cycle exactly at a single point.

2) 0 < 0v < ~/2, Ow = zc/2. It is well-known that if tan Ov is rational, the q~-orbits are all cycles. Villarini's argument works also in this case, so that every cycle has the same period. Since we have chosen vector fields of modulus one, the period measures the length of the orbit, and it is connected to the number of times the orbit winds around the torus. The number of intersections of a V-cycle with W-cycles depends on Oy.

If tan Ov is not rational, the 0-orbits are never cycles, and each one of them is dense on the torus. Each one of them meets every W-cycle infinitely many times.

In general, for Ov ~ Ow, both arbitrary, we can have any combination of the situa- tions described at points 1 and 2.

We conclude considering commuting flows on the sphere. We recall that we can multiply V or W by -1, in order to change stability into negative stability, or a-limits into w-limits.

THEOREM 3.3. - Let S be the sphere, and (CT) hold. Only three phase portraits are possible:

(1) there is just one singular point and every orbit is homoclinic (for both vector fields);

(2) there are two critical points 01, 02; every non-trivial orbit of V is a cycle, all having the same period; every non-trivial orbit of W has 01 as a-limit, 02 as w-limit;

(3) there are two critical points 01, 02 ; every non-trivial orbit of V has 01 as a- limit, 02 as w-limit; the same holds for W, possibly exchanging the words ~,a-limit, and ~o~-limit..


PROOF. - The sphere's characteristic is 2. Since the indices of critical points of commuting fields are positive, there are only two possibilities: either a single point of index 2, or a couple of points of index 1.

In the former case, such a critical point has exactly two elliptic sectors, and every orbit has the unique critical point as positive and negative limit set. This holds for both V and W, since their critical points coincide.

In the latter case, every critical point has index 1, and can only be a centre or a (negatively) asymptotically stable point. We cannot have two centres at the same critical point, by the transversality, but we can have a centre and an asymptotically stable point, or a couple of (negatively) asymptotically stable points. �9

W e show here some examples of commuting systems on the sphere.

1) Two centres and an attractor-repellor couple.

Consider the unit sphere ~::= {(x, y, z) e R3: x 2 + y2 + z 2 = 1} in R ~ . The following vector fields V and W are transversal at noncritical points and commute. V(xl, x2, x3): = ( - x 2 , xl, 0) has two centres at S : = (0, 0, - 1 ) and N : = (0, 0, 1). W(xl, x2, x~):= (0, 0, x~ - 1) has an asymptotically stable point at S := (0, 0, - 1 ) and a negatively asymptotically stable point at N : = (0, 0, 1).

2) Two attractor-repellor couples. Take V and W of the previous point and consider the corresponding complete family Fvw. There are infinitely many parameters for which we have attractor-repellor couples.

[A]

[ B S ]

[CJ]

[D]

[GP]

[H] [L]

[m] [Me] [MS]

REFERENCES

V. I. ARNOLD, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York (1989). N. P. BHATIA - G. P. SZEGO, Stability Theory of Dynamical Systems, Springer-Verlag, New York (1970). C. CmCONE - M. JACOBS, Bifurcation of critical periods for plane vector fields, Trans. A.M.S., 312, 2 (1989), pp. 433-486. G. F. D. DUFF, Limit cycles and rotated vector fields, Ann. Math., 57 (1953), pp. 15-31. V. GUILLEMIN - A. POLLACK, Differential Topology, Prentice-Hall, Englewood Cliffs, New Jersey (1974)o P. HARTMAN, Ordinary Differential Equations, Birkhauser, Boston (1982). N. A. LUKASHEWCH, Isochronicity of center for certain systems of differential equations, Diff. Equations, 1 (1965), pp. 220-226. L. MARKUS, Parallel dynamical systems, Topology, 8 (1969), pp. 47-57. G. H. MEISTERS, Personal communication. L. MAZZl - M. SABATINI, A characterization of centres via first integrals, Jour. Diff. Eq., 76 2 (1988), pp. 222-237.


[MNSS]

[o]

IS]

[Sabl]

[Sab2] [Sab3]

[sc]

IV]

[w]

F. MARCHETTI ~ P. NEGRINI - L. SALVADORI - M. SCALIA, Liapunov Direct Method in Approaching Bifurcation Problems, Ann. Mat. Pura Appl., (IV), CVIII (1976), pp. 211-226. P. J. OLVER, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York (1986). G. R. S E L L , Topological Dynamics and Ordina~?t Differential Equations, Van Nos- trand Reinhold, New York (1971). M. SABATINI, A connection between isochronous Hamiltonian centres and the Jacobian conjecture, preprint, Trento (1994). M. SABATINI, Quadratic isochronous centres commute, preprint, Trento (1995). M. SABATINI, Characterizing isochronous centres by Lie brackets, preprint, Trento (1995). G. SANSONE - R. CONTI, Equazioni differenziali nonlinear~ Cremonese, Roma (1956). M. VILLARINI, Regularity properties of the period function near a center of a planar vector field, Nonlinear Analysis, T.M.A., 19, 8 (1992), pp. 787-803. A. WALLACE, Differential Topology, W. A. Benjamin, New York (1968).

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Dynamics of commuting systems on two-dimensional manifolds