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Dulac Criteria for Autonomous Systems Having an Invariant Affine Manifold
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
199, 374]390 Ž1996.
0147
Dulac Criteria for Autonomous Systems Having an Invariant Affine Manifold Michael Y. Li* Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, Atlanta, Georgia 30332 Submitted by Hal L. Smith Received April 19, 1994
For a class of higher dimensional autonomous systems that have an invariant affine manifold, conditions are derived to preclude the existence of periodic solutions on the invariant manifold. It is established that these conditions are robust under certain types of local perturbations of the vector field. As a consequence, each bounded semitrajectory on the invariant manifold is shown to converge to a single equilibrium using a C 1 closing lemma for this class. Applications to autonomous systems that are homogeneous of degree 1 are also consid ered. Q 1996 Academic Press, Inc.
1. INTRODUCTION Let X ; R n be an open set and x ¬ f Ž x . g R n a C 1 function defined for x g X. We consider the autonomous system of ordinary differential equations x9 s f Ž x . . Ž 1.1. Let x Ž t, x 0 . denote the solution to Ž1.1. such that x Ž0, x 0 . s x 0 . We are interested in systems Ž1.1. that satisfy the following assumptions: ŽH 1 . There is a constant matrix B such that rank B s r and the Ž n y r .-dimensional affine manifold G s x g X : B Ž x y x . s 04 for some x g X satisfies Bf Ž x . s 0, x g G. ŽH 2 . The Jacobian matrix fr x can be written as f s n Ž x . In= n q A Ž x . , xgG x * Research is supported by a NSERC Postdoctoral Fellowship. 374 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
Ž 1.2.
Ž 1.3.
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DULAC CRITERIA
and BAŽ x . s 0,
x g G,
Ž 1.4.
where x ¬ n Ž x . is a real-valued function. A system Ž1.1. satisfying ŽH 1 . and ŽH 2 . will be called an autonomous system ha¨ ing an in¨ ariant affine manifold, since ŽH 1 . is equivalent to that G is invariant with respect to Ž1.1.. Let a s Bx. We denote the class of such systems by SB, a . When B and a are implied or when their appearance is not essential, we denote this class simply by S . This is a wider class than that considered in w12x, where Ž1.1. was assumed to satisfy: ŽHX1 . The Jacobian matrix fr x can be written as
f x
s yn In= n q A Ž x . ,
xgX
Ž 1.5.
where n is some constant. ŽHX2 . There is a constant matrix B such that rank B s r and BAŽ x . s 0,
x g X,
Ž 1.6.
and the corresponding system Ž1.1. called an autonomous system ha¨ ing an in¨ ariant linear subspace. In this case, the existence of an invariant affine manifold was proved as a consequence of Ž1.5. and Ž1.6.. Systems in S enjoy concrete examples arising from many different areas which are not covered by systems considered in w12x; for instance, the hypercycles which describe self-organization in Mathematical Biology Žsee w8, 11x., and some epidemiological models which allow variable total populations such as models with disease-related death and with vertical transmission Žsee w1, 3x.. A general class of homogeneous systems of degree 1 ŽSection 3. can also be treated in this context. In many cases, our prime interest in systems that belong to S is the behaviour of solutions that stay in the invariant manifold G. The main purpose of the present paper is to derive Dulac type conditions for the nonexistence of periodic solutions in G. Generalizations of the criteria of Bendixson and Dulac to higher dimensions have been obtained in the context of general autonomous systems by many authors w2, 14, 19, 21x. In our situation, one usually tries to restrict Ž1.1. to the invariant manifold G to get a lower dimensional system and then apply these general criteria. However, the effectiveness of this approach depends critically on the choice of coordinates for G. A different technique was developed in w12x which does not require this reduction in dimension and thus is independent of any choice of coordinates for G. As a consequence, Bendixson type
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MICHAEL Y. LI
conditions were derived more coherently and more economically. In the present paper, this technique is further explored in deriving Dulac type conditions for systems more general than those considered in w12x. We also present some applications where the theory in w12x is not applicable. Autonomous convergence theorems were first obtained in w15, 21x, and later in w16x, for general autonomous systems in R n. Certain higher dimensional Bendixson]Dulac criteria were shown to be robust under C 1 perturbations of f and this robustness was used, together with the C 1 closing lemma of Pugh w20x, to show that each bounded semitrajectory converges to an equilibrium under any of these higher dimensional Bendixson]Dulac criteria. In developing this type of result in Section 3 for systems which belong to S , we prove in Lemma 3.3 that a local C 1 closing lemma can be applied to systems in S , namely, we are able to show that the perturbation may be chosen from S . The paper is organized as follows: in Section 2, we derive Dulac criteria for systems in S ; in Section 3, we prove an autonomous convergence theorem on G, after we show the C 1 robustness of our Dulac criteria in a very specific sense and prove a local C 1 closing lemma for systems in S by applying Pugh’s result; in Section 4, we present an application to autonomous homogeneous systems of degree 1.
2. DULAC CRITERIA For a solution x Ž t, x 0 . to Ž1.1. with x 0 g G, the linear variational equation takes the form y9 Ž t . s n Ž x Ž t , x 0 . . y Ž t . q A Ž x Ž t , x 0 . . y Ž t . .
Ž 2.1.
The change of variables uŽ t . s y Ž t .exp yH0t n Ž x Ž s, x 0 .. ds4 leads to u9 Ž t . s A Ž x Ž t , x 0 . . u Ž t .
Ž 2.2.
satisfying the condition BA Ž x Ž t , x 0 . . s 0
for t g R.
Ž 2.3.
Therefore the linear subspace ker B s u g R n : Bu s 04 of R n is invariant with respect to the linear system Ž2.2. in the sense that uŽ0. g ker B implies uŽ t . g ker B for t g R, since Ž BuŽ t ..9 ' 0. Linear differential systems with such invariance properties were investigated in w12x. We first recall a result proved there.
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DULAC CRITERIA
Let w 10 , . . . , wr 0 4 be an orthonormal basis for the orthogonal complement of ker B in R n, and wi Ž t . be the solution to Ž2.2. such that wi Ž0. s wi0 . The following proposition is relation Ž4. in the proof of Theorem 2.5 in w12x. PROPOSITION 2.1.
Let u1Ž t ., u 2 Ž t . be two solutions to Ž2.2.. Then
u1 Ž t . n u 2 Ž t . F w 1 Ž t . n ??? n wr Ž t . n u1 Ž t . n u 2 Ž t . . Here 5 ? 5 is the euclidean norm and n denotes the exterior product in R n. The exterior product z Ž t . s w 1Ž t . n ??? n wr Ž t . n u1Ž t . n u 2 Ž t . satisfies the linear system z9 Ž t . s Aw rq2x Ž x Ž t , x 0 . . z Ž t .
Ž 2.4.
which is the Ž r q 2.nd compound equation of Ž2.2.. Here Aw rq2x is the n Ž r q 2.nd additi¨ e compound matrix of A w17, 19x. It is a = r qn 2 rq2
ž / ž /
matrix, and hence Ž2.4. is a = linear system. For a detailed study on compound matrices and compound equations, we refer the reader to w17, 19x. Let W be the euclidean unit ball in R 2 and let W and W be its closure and boundary, respectively. A function w g LipŽW ª G . will be described as a rectifiable 2-surface in G; a function c g LipŽ W ª G . is a closed rectifiable cur¨ e in G and will be called simple if it is one-to-one. A method for deriving Dulac type conditions for general autonomous systems was developed in w14x by studying the evolution under Ž1.1. of certain functionals which are defined on rectifiable 2-surfaces. If G is simply connected, for a given simple closed curve in LipŽ W ª G ., the set n rq2
n rq2
ž / ž /
S Ž c , G . s w g Lip Ž W ª G . : w Ž W . s c Ž W . 4 is not empty Žsee w14x.. Consider a functional A defined on SŽ c , G . by Aw s
HW
w r1
n
w r2
,
where < ? < is any vector norm in R N , N s r qn 2 . For instance, when the norm is < y < s < y*y < 1r2 , where here and throughout the paper asterisk denotes transposition, then Aw is the usual surface area of w ŽW .. Since G is affine, the following result can be proved as Proposition 2.2 in w14x.
ž /
PROPOSITION 2.2. Suppose c is a simple closed rectifiable cur¨ e on G. Then there exists a d ) 0 such that Aw G d for all w g SŽ c , G ..
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MICHAEL Y. LI
A subset K ; G is absorbing in G for Ž1.1. if each compact subset D ; G satisfies x Ž t, D . ; K for sufficiently large t. We make the following assumption: ŽH 3 . G is simply connected and there is a compact set K in the interior of G which is absorbing for Ž1.1.. Let x ¬ P Ž x . be a nonsingular r qn 2 = r qn 2 matrix-valued function which is C 1 in G. Then < Py1 Ž x .< is uniformly bounded for x g K. Let m be the Lozinskiı˘ measure Žor the logarithmic norm. with respect to < ? < defined by Žcf. w6, p. 41x, or w19x.
ž / ž /
m Ž F . s limq
< I q hF < y 1 h
hª0
for any N = N matrix F, N s y1
E s Pf P
n rq2
ž / . Set
qP
f w rq2x x
Py1 y rn IN=N .
Ž 2.5.
Here Pf is the matrix obtained by replacing each entry pi j of P by Ž pUi j r x . f, its directional derivative in the direction of f, and f w rq2xr x is the Ž r q 2.nd additive compound matrix of the Jacobian matrix fr x. Define the quantity qrq2 s lim sup sup tª`
x 0gK
1
t
H m Ž E Ž x Ž s, x t 0
0
. . . ds.
Ž 2.6.
We note that, since K ; G, the time average in Ž2.6. is only calculated along solutions on the affine manifold G. Since G is invariant, qrq2 is well defined. A closed rectifiable curve c g LipŽ W ª G . is in¨ ariant with respect to Ž1.1. if the subset c Ž W . of G is invariant. The following result gives a general Dulac criterion in a weak form for autonomous systems in S . THEOREM 2.3.
Assume that ŽH 1 ., ŽH 2 ., and ŽH 3 . are satisfied. If qrq2 - 0,
Ž 2.7.
then no simple closed rectifiable cur¨ e in G can be in¨ ariant with respect to Ž1.1.. Proof. Let e 0 s y 12 qrq2 ) 0. There exists T ) 0 such that t
H0 m Ž E Ž x Ž s, x for t ) T, x 0 g K.
0
. . . ds - ye 0 t
Ž 2.8.
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DULAC CRITERIA
Suppose that, on the contrary, there is a simple closed rectifiable curve c g LipŽ W ª G . that is invariant with respect to Ž1.1.. Let w g SŽ c , G . and w t Ž p . s x Ž t, w Ž p .., where p s Ž p1 , p 2 . g W. Then, from the invariance of c , we know w t g SŽ c , G . for all t, and hence Aw t G d
Ž 2.9.
for some d ) 0 independent of t, by Proposition 2.2. Now, yi Ž t . s w tr pi s Ž x Ž t, w t .r x 0 .Ž wr pi ., i s 1, 2, are solutions to the linear variational equation Ž2.1. with respect to x s w t Ž p ., and hence u i Ž t . s Ž w tr pi .exp yH0t n Ž ws Ž p .. ds4 are solutions to the linear system Ž2.2. with x Ž t . s w t Ž p .. Let w 1Ž t ., . . . , wr Ž t . be the solution to Ž2.2. as in Proposition 2.1. Then u1Ž t . n u 2 Ž t . n w 1Ž t . n ??? n wr Ž t . satisfies the Ž r q 2.nd compound equation Ž2.5.. The following relation holds t
½H
y 1 Ž t . n y 2 Ž t . s u1 Ž t . n u 2 Ž t . exp 2
0
n Ž ws Ž p . . ds
5
Ž 2.10.
and by Proposition 2.1 t
½H
u1 Ž t . n u 2 Ž t . exp 2
0
n Ž ws Ž p . . ds
5 t
½H
F u1 Ž t . n u 2 Ž t . n w 1 Ž t . n ??? n wr Ž t . exp 2
0
5
n Ž ws Ž p . . ds .
Ž 2.11. Direct differentiation yields that t
½H
¨ Ž t . s u1 Ž t . n u 2 Ž t . n w 1 Ž t . n ??? n wr Ž t . exp 2
0
n Ž ws Ž p . . ds
5
satisfies the differential system ¨ 9Ž t . s
ž
f w rq2x x
Ž wt Ž p . . y rn Ž wt Ž p . . IN=N
/
¨ Ž t. .
w rq2x w rq2x Here we use f w rq2xr x s Aw rq2x q In= from Ž1.3. and In=n s Žr q n n 2. IN= N , N s r q 2 . Therefore V Ž t . s P Ž w t Ž p .. ¨ Ž t . satisfies the equation V9Ž t . s EŽ w t Ž p .. V Ž t ., where E is given in Ž2.5.. By a property of the Lozinskiı˘ measure Žsee w6, p. 41x. and by Ž2.8.
ž /
t
H0 m Ž E Ž w Ž p . . . ds F V Ž 0. expŽ ye
V Ž t . F V Ž 0 . exp
s
0t
. , Ž 2.12.
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MICHAEL Y. LI
for p g W and t ) T. Therefore V Ž t . ª 0 as t ª `, uniformly for p g W. Since w t ŽW . ; K for sufficiently large t, and Py1 Ž x . is uniformly bounded for x g K, this implies that ¨ Ž t . ª 0 as t ª `, uniformly for p g W. Thus by Ž2.11. and Ž2.12., < w tr p1 n w tr p 2 < ª 0 exponentially as t ª `, and the exponential rate is uniform for all p g W. As a consequence, Aw t ª 0 as t ª `. This contradicts Ž2.9., completing the proof of Theorem 2.3. Remarks. Ži. Condition Ž2.7. rules out the existence in G of orbits of the following types: Ža. periodic orbits, Žb. homoclinic orbits, Žc. heteroclinic cycles, since each of such orbits gives rise to an invariant simple closed rectifiable curve. Žii. A sufficient condition for Ž2.7. is y1
m Pf P
ž
qP
f w rq2x x
Py1 y rn IN=N - 0
/
on K
Ž 2.13.
which is a general Dulac type condition. Note that conditions Ž2.7. and Ž2.13. provide the flexibility of a choice of N 2 arbitrary functions in addition to the choice of vector norms in deriving suitable conditions. Setting A s I in Ž2.13. leads to the condition
m
ž
f w rq2x x
y rn IN= N - 0
/
Ž 2.14.
which is the Bendixson type condition obtained in w12x. Concrete expressions of Ž2.14. in terms of Lozinsskiı˘ measures with respect to some common vector norms can be found in w12x. Žiii. Conditions Ž2.13. and Ž2.14. take a simple form when r s n y 2. In this case, r q 2 s n, N s r qn 2 s 1, and thus P Ž x . is scalar-valued and f w rq2xr x s divŽ f .. For example, Ž2.14. will read
ž /
div Ž f . - Ž n y 2 . n
on G.
Ž 2.15.
Also note that G is a 2-dimensional affine manifold in this case; any simple closed curve in G encloses a region in G. From the proof of Theorem 2.3 we can see that Ž2.15. need only hold for all x g G except a set of measure zero. Živ. When r s 0, ŽH 1 . and ŽH 2 . pose no restriction on Ž1.1.. In accordance, the condition Ž2.13. agrees with a Dulac type condition for general autonomous systems obtained in w14x. Žv. Assume that Ž1.1. satisfies ŽHX1 . and ŽHX2 . with n s 0. It is proved in w12x that Bx gives r independent linear first integrals for Ž1.1.. In the
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DULAC CRITERIA
special case that n s 3, r s 1, it follows from Theorem 2.3 and Remark Žiii. that no simple closed rectifiable curve on a level surface S defined by Bx s c can be invariant with respect to Ž1.1. if divŽ f . - 0 for almost all x g S. This gives a result of Demidowitsch w7x for the case of a linear first integral. Živ. For many systems which describe population dynamics, the feasin ble region X is usually the nonnegative cone Rq and the affine manifold n G is the portion of a hypersurface bounded in Rq , e.g., the simplex n x g Rq : Ý nis1 x i s 14 ; the assumption ŽH 3 . in such a case is equivalent to the uniform persistence of Ž1.1. in G Žsee w5x..
3. CONVERGENCE OF TRAJECTORIES AND GLOBAL STABILITY In this section, we investigate further implications of condition Ž2.7. for the behaviour of solutions to Ž1.1. which stay on the invariant manifold G. We begin by studying the robustness of condition Ž2.7. under certain smooth perturbations of f. Let < ? < denote a vector norm on R n and the operator norm which it induces for linear mappings from R n to R n. The distance between two functions f, g g C 1 Ž X ª R n . such that f y g has compact support is < f y g < C 1 s sup
½
f Ž x. y gŽ x. q
f x
Ž x. y
g x
5
Ž x. : x g X .
A function g g C 1 Ž X ª R n . is called a C 1 local e-perturbation of f at x 0 g X if there exists an open neighbourhood U of x 0 in X such that the support of Ž f y g . suppŽ f y g . ; U and < f y g < C 1 - e . Given B and x in the assumption ŽH 1 ., set a s Bx. For f g SB, a , let g g SB, a be a C 1 local perturbation of f. We consider the corresponding differential equation x9 s g Ž x . . Ž 3.1. Since g g SB, a , the affine manifold G defined in Ž1.2. is necessarily invariant with respect to Ž3.1.. We say that a Dulac criterion Ž2.7. is robust under C 1 local perturbations of f at x 0 if, for each sufficiently small e ) 0 and neighbourhood U of x 0 , it is also satisfied by C 1 local e-perturbations g such that suppŽ f y g . ; U. Let f G s f < G . Then f G defines a vector field on the affine manifold G from the assumption ŽH 1 .. Denote the differential equation x9 s f G Ž x ., x g G by Ž1.1. G . A point x 0 g G is said to be nonwandering for Ž1.1. G if, for each sufficiently small relative neighbourhood V of x 0 in G, x Ž t, V . l V / B for some t.
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MICHAEL Y. LI
PROPOSITION 3.1. Assume that f g SB, a . Let qrq2 be defined in Ž2.6. for some ¨ ector norm < ? < and matrix-¨ alued function P. Then the Dulac criterion qrq2 - 0 is robust under C 1 local perturbations in SB, a at any nonequilibrium point x 0 g G that is nonwandering for Ž1.1. G . Proof. This is essentially Proposition 3.3 of w16x. Since qrq2 - 0 implies that Ž1.1. G has no periodic solutions, the minimum return time at a nonequilibrium nonwandering point x 0 , as defined in Ž3.8. of w16x, is q`. Also note that only trajectories on the invariant manifold G are involved in the calculation of qrq2 and that G is affine, the proposition can be proved as Proposition 3.3 in w16x by restricting both Ž1.1. and Ž3.1. to G. Next, we prove an adapted local C 1 closing lemma for systems in SB, a , in which the perturbations g belong to SB, a . The idea is to apply the general local closing lemma of Pugh w20x to the restricted system Ž1.1. G on n G, and then extend the perturbation to Rq using a bump function. Since the extended perturbation has to be in SB, a , we carry out the construction of the bump function in detail. LEMMA 3.2. Let f g SB, a . Suppose that x 0 g G is nonwandering for Ž1.1. G and that f Ž x 0 . / 0. Then, for each e ) 0 and each neighbourhood U of x 0 in X, there exists C 1 e-perturbations g of f such that Ž1. g g SB, a and suppŽ f y g . ; U, Ž2. the system Ž3.1. has a nonconstant periodic solution whose trajectory lies in G and passes through x 0 . Proof. Since G is affine, without loss of generality, we may assume that G lies on the coordinate plane Ž x 1 , . . . , x nyr . and that x 0 is at the origin. More precisely, write X s X 1 [ X 2 , such that X 1 ( R ny r , X 2 ( R r. Then we assume G ; X 1. If x s Ž x 1 , . . . , x n ., we write x Ž1. s Ž x 1 , . . . , x nyr . and x Ž2. s Ž x ny rq1 , . . . , x n . so that x s Ž x Ž1., x Ž2. .. In this setting, the matrix B may be chosen as the form B s Ž0, Ir=r .. Now f G is a mapping from G to X 1 , and thus system Ž1.1. G is a n y r dimensional autonomous system d dt
x Ž1. s f G Ž x Ž1. . ,
x Ž1. g G ; X 1 .
Ž 3.2.
Let U1Ž0, d . ; R ny r be a euclidean ball centered at the origin and of radius d . We choose d small enough so that U1Ž0, d . ; U and V s U1 Ž 0, d . = x Ž2. g X 2 : < r x Ž2. < - 1 4 ; U
for some r ) 1. Ž 3.3.
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DULAC CRITERIA
Apply to Ž3.2. the local C 1 Closing Lemma of Pugh as formulated in w10x Žalso see w20x.. There exists a h G g C 1 Ž G ª X 1 . such that Ža. < h G < C 1 - er8 r and suppŽ h G . ; U1Ž0, d ., Žb. Ž drdt . x Ž1. s f G Ž x Ž1. . q h G Ž x Ž1. . has a nonconstant periodic solution whose trajectory passes through 0. Since this periodic trajectory necessarily stays in G, the lemma is proved if we can extend h G to a function h defined on X so that g s f q h satisfies Lemma 3.2Ž1.. For r ) 1 in Ž3.3., define a function x ¬ a Ž x . from R n to R by
a Ž x. s
½
Ž 1 y < r x Ž2. < 2 .
2
< r x Ž2. < F 1
,
0,
otherwise,
where x s Ž x Ž1., x Ž2. ., and a function x ¬ hŽ x . from R n to R n by hŽ x . s
ž
h G Ž x Ž1. . . 0
/
Then < a < - 1 and < h < C 1 F < h G < C 1 - er8 r . Moreover, the gradient =a Ž x . s
½
y4 r 2 Ž 1 y < r x Ž2. < 2 . Ž 0, x Ž2. . *, 0,
< r x Ž2. < F 1, otherwise,
and BhŽ x . s 0 for all x g X. Set g s f q a h. Then g g C 1 Ž X ª R n ., g < G s f G q h G , and suppŽ f y g . ; V ; U, where V is given in Ž3.3.. In particular, Bg Ž G . s 0. Moreover g y f s ah
g x
y
f x
sa
h x
q h =a *.
Thus < g y f < C1 F < a h< q a F
e 4r
h x
q < h=a * <
Ž1 q 2 r . - e ,
and B Ž gr x y fr x . s 0 for all x g X. Therefore g g SB, a and satisfies Ž1. and Ž2..
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MICHAEL Y. LI
PROPOSITION 3.3. Under the assumptions ŽH 1 ., ŽH 2 ., and ŽH 3 ., if qrq2 0, then the dimension of the stable manifold of any equilibrium of Ž1.1. G is at least n y r y 1; if an equilibrium of Ž1.1. G is not isolated, then its stable manifold has dimension n y r y 1 and it has a center manifold of dimension 1 which contains all nearby equilibria of Ž1.1. G . Proof. Without loss of generality, we assume the same set up as in the proof of Lemma 3.2 such that B s Ž0, Ir=r ., G ; X 1 ( R ny r , and the Jacobian matrix fr x may be written in the block form
f x
s
n Ž x . Ir=r A1
0 . A2
It is easy to see that Ž fr x .Ž x 0 . has an eigenvalue n Ž x 0 . with r independent eigenvectors in X 1. At an equilibrium x 1 of Ž1.1. G , qrq2 - 0 implies
m P
ž
f w rq2x x
y rn IN= N Py1 - 0
/
Ž 3.4.
since f Ž x 1 . s 0 implies Pf Ž x 1 .Ž x 1 . s 0. Let l1 , . . . , l nyr be the eigenvalues of Ž fr x .Ž x 1 . with respect to the invariant subspace X 1 , such that Re l1 G ??? G Re l nyr . Then
l1 q l2 q rn is an eigenvalue of Ž f w rq2xr x .Ž x 1 . Žsee w19x.. Therefore the matrix f w rq2xr x y rn IN= N has an eigenvalue l1 q l2 . Thus Ž3.4. implies Žsee w6, p. 41x. Re l1 q Re l2 - 0. It then follows that 0 ) Re l2 G ??? G Re l nyr ; only l1Ž x 1 . can possibly have nonnegative real part. The stable manifold of x 1 in G has dimension at least n y r y 1. If x 1 is not isolated, Ž fr x .Ž x 1 .< X 1 has a singular matrix representation, l1Ž x 1 . s 0 so that its stable manifold in G has dimension n y r y 1 and there is a 1-dimensional center manifold in G which contains all nearby equilibria in G, by the Center Manifold Theorem Žcf. w9, p. 48x.. THEOREM 3.4. Under the assumptions ŽH 1 ., ŽH 2 ., and ŽH 3 ., if qrq2 - 0, then Ž1. E¨ ery nonwandering point of Ž1.1. G is an equilibrium. Ž2. E¨ ery nonempty alpha or omega limit set of Ž1.1. G is a single equilibrium.
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385
Proof. Under the condition qrq2 - 0, Ž1.1. G has no nonconstant periodic point by Theorem 2.3. Let x 0 g G be a nonwandering point for Ž1.1. G and f Ž x 0 . / 0. Then by Lemma 3.2, there exist arbitrarily small local C 1 perturbations g g SB, a of f at x 0 such that Ž3.1. has a nonconstant periodic solution. However, any such g satisfies qrq2 - 0, by Proposition 3.1, and thus cannot have any nonconstant periodic solution. This contradiction establishes Ž1.. To prove the assertion that each nonempty alpha or omega limit set is a single equilibrium, first observe that since each limit point is nonwandering, it is an equilibrium. Let x 1 g v Ž x 0 ., the omega limit set of x 0 for Ž1.1. G . If x 1 is an isolated equilibrium for Ž1.1. G , then x 14 s v Ž x 0 ., since v Ž x 0 . is connected. If x 1 is not isolated, then Proposition 3.3 implies that there is a 1-dimensional center manifold associated with x 1 in G containing all nearby equilibrium for Ž1.1. G . Since v Ž x 0 . contains a continuum of equilibria, we may choose x 1 and a sufficiently small neighbourhood U of x 1 so that in U any of such local center manifolds consists entirely of equilibria and every trajectory of Ž1.1. G which intersects U is asymptotic to a trajectory in the center manifold. Thus lim t ª` x Ž t, x 0 . s x 1 so that v Ž x 0 . s x 14 in this case also. The proof that a nonempty alpha limit set is a single equilibrium is the same. COROLLARY 3.5. Assume that ŽH 1 ., ŽH 2 ., and ŽH 3 . are satisfied. Suppose Ž1.1. has a unique equilibrium x 0 in the interior of G. If qrq2 - 0 for some ¨ ector norm < ? < and matrix-¨ alued function P Ž x ., then x 0 is globally asymptotically stable in the interior of G. Proof. Clearly x 0 4 is globally attracting in the interior of G since Theorem 3.4 implies that it is the omega limit set of every trajectory starting in the interior of G. Moreover, x 0 4 is stable since otherwise it would be both the alpha limit set and the omega limit set of some homoclinic trajectory g ; G, which gives rise to a simple closed invariant curve that is also rectifiable Žsee w15, the proof of Corollary 2.6x.. This has been ruled out by qrq2 - 0. 4. HOMOGENEOUS SYSTEMS OF DEGREE 1 In this section, we consider autonomous system Ž1.1. under the assumption that f Ž x . is homogeneous of degree 1, namely, f Ž lx. s lf Ž x.
for l ) 0, x g R n .
Ž 4.1.
Such systems have been used in modeling of population dynamics and the spread of infectious diseases Že.g., see w3, 18x., where components x i of x
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MICHAEL Y. LI
are restricted to be nonnegative. In the same spirit, we assume that each n component f i of f satisfies f i < x is0 G 0 so that the nonnegative cone Rq is positively invariant with respect to Ž1.1.. n Let x Ž t, x 0 . be a solution to Ž1.1. with x 0 g Rq R 0. It is customary in the study of homogeneous differential systems to consider the projected variable yŽ t. s
xŽ t. NŽ t.
n
where N Ž t . s
,
Ý xi Ž t . .
Ž 4.2.
is1
Then Ý nis1 yi Ž t . s 1 for all t and N9Ž t . s Ý nis1 f i Ž x Ž t ... We thus have the relation n
N9 Ž t .
s
NŽ t.
Ý fi Ž y Ž t . . .
Ž 4.3.
is1
As a consequence, y Ž t . satisfies the following system of differential equations n
y9 Ž t . s y
ž
Ý fi Ž y . is1
/
y q f Ž y. .
Ž 4.4.
n System Ž4.4. is well-defined in Rq , though the variable y defined in Ž4.2. stays in the simplex
½
n D s y g Rq :
n
Ý yi s 1 is1
5
.
Ž 4.5.
In fact, by adding the equations of Ž4.4., one may check that D is invariant with respect to Ž4.4.. When restricted to the invariant simplex D, system Ž4.4. describes the projected flow of the homogeneous system Ž1.1. onto D. In particular, for each solution y Ž t . of Ž4.4. with y Ž0. g D, there exists a solution x Ž t . to Ž1.1. such that x Ž t . and y Ž t . satisfy the relation Ž4.2.. In fact, by the uniqueness of solution, such a x Ž t . may be obtained by setting the initial conditions x i Ž0. s yi Ž0.. From these considerations, we want to study the behaviour of solutions of Ž4.4. on D. n Observe that D is an affine manifold in Rq defined by
Ž 1, . . . , 1 . y s 1. Thus the assumption ŽH 1 . in Section 1 is satisfied with X being the n interior of Rq , G being D, B s Ž1, . . . , 1., r s 1, and x s Ž1rn, . . . , 1rn.*.
387
DULAC CRITERIA
The Jacobian matrix J Ž y . of Ž4.4. can be written as J Ž y . s n 1 Ž y . In=n q F Ž y . , where n
n 1Ž y . s y Ý fi Ž y .
Ž 4.6.
is1
and FŽ y. s
f y
n
y Ž y 1 , . . . , yn . *
ž
Ý is1
fi y1
fi
n
,...,
Ý is1
yn
/
.
Ž 4.7.
Since n
Ž 1, . . . , 1 . F Ž y . s 1 y
ž
Ý yi is1
n
/ž
Ý is1
fi yi
n
,...,
Ý is1
fi yn
/
one may see that ŽH 2 . is also satisfied. Theorem 2.3 can now be applied to system Ž4.4. to yield conditions which rule out the existence of solutions x Ž t . to Ž1.1. for which y Ž t . is periodic in D. Assume that Ž4.4. is uniformly persistent in D and let K 1 be the compact absorbing set in D. We can define q3 as in Ž2.6. for r s 1, n s n 1 , and for some N = N matrix-valued function P Ž x . and some vector norm < ? < in R N , N s nr3. The next result follows from Theorem 2.3 and the fact that y Ž t . is periodic if x Ž t . associated with y Ž t . by Ž4.2. is periodic. THEOREM 4.1. If q3 - 0, then no simple closed rectifiable cur¨ e in D can be in¨ ariant with respect to Ž4.3.. In particular, Ž1.1. has no periodic solutions n in Rq . Remark. The condition q3 - 0 also rules out the types of orbits of Ž1.1. listed in Remark Ži. following the proof of Theorem 2.3. For an application of Theorem 4.1 to a SEIRS epidemiological model with a homogeneous incidence function and a varying total population, we refer the reader to w13x. In the following, we consider a special case. Assume n s 3 and f takes the form f Ž x . s Ax q Ž diag x . f Ž x . ,
Ž 4.8.
where A s Ž a i j . 3=3 is a nonnegative matrix, namely, ai j G 0
i/j
Ž 4.9.
and f Ž x . is homogeneous of degree 0, that is, f Ž l x . s f Ž x ., for l ) 0 and x g R 3. System Ž1.1. with f given in Ž4.8. has been used in population models Žsee w3, 4x.. In this case, we have the following result.
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MICHAEL Y. LI
THEOREM 4.2. Assume that n s 3 and that f satisfies Ž4.8. and Ž4.9.. Then the conclusions of Theorem 4.1 hold if
f i
3
xi
Ý
xi
is1
3
-0
Ý x i s 1.
when
Ž 4.10.
is1
Proof. Set P Ž y . s Ž y 1 y 2 y 3 .y1 , n s 3, r s 1, and n s n 1 in Ž2.5.. Then the matrix E can be calculated with respect to system Ž4.4. as E s Pg Py1 q div Ž g . y n 1 , where g denotes the vector field of Ž4.4.. Relation Ž4.3. leads to Pg Py1 s Pf Py1 q
P
3
ž
Ý
yi
is1
yi n 1 Py1
/
s Pf Py1 y 3n 1 and 3
div Ž g . s div Ž f . y
fi
3
Ý Ý js1 is1
yj
y j q 3n 1 .
Differentiating Ž4.1. with respect to l at l s 1 leads to 3
Ý js1
fi yj
yj s f i ,
i s 1, 2, 3.
Therefore E s Pf Py1 q div Ž f . s Py1 div Ž Pf . . For f given in Ž4.8. div Ž Pf . s y Ý a i j i/j
yj yi
3
q
Ý is1
f i yi
yi ,
and thus for y g K ; S
m Ž E . s Py1 y Ý a i j
ž
i/j
yj yi
3
q
Ý is1
f i yi
/
yi F yd 1 - 0
for some d 1 ) 0. Therefore Theorem 4.2 follows from Theorem 2.3.
DULAC CRITERIA
389
Remarks. Ž1. When condition Ž4.10. is satisfied, a system Ž1.1. with f Ž x . s Ždiag x . f Ž x . is said to be self-regulating. For 3-dimensional self-regulating systems, a result of Butler, Schmid, and Waltman w4x implies that 3-dimensional volumes decay exponentially along solutions of Ž1.1., whereas Theorem 4.2 implies that if Ž1.1. is also homogeneous of degree 1, then 2-dimensional surface areas as well as 3-dimensional volumes, when projected onto D, decay exponentially. Moreover, the projected dynamics of Ž1.1. onto D is trivial in the sense that every nonwandering point is an equilibrium and bounded solutions either converge to equilibria or approach the boundary. Ž2. Homogeneous systems Ž1.1. with vector fields given in Ž4.8. was also considered in Busenberg and van den Driessche w2x, where conditions which preclude the existence of periodic solutions and closed phase polygons D are derived without the uniform persistence assumption. One of the conditions given in w2x is
f i xi
q
f j xj
y
ž
f i xj
q
f j xi
/
F 0,
i / j, i , j s 1, 2, 3, on D . Ž 4.11.
While Ž4.10. may be easier to verify than Ž4.11., it is not apparent that one implies the other.
ACKNOWLEDGMENTS The final draft of this paper was completed when the author was visiting the Center for Dynamical Systems and Nonlinear Studies ŽCDSNS. at Georgia Institute of Technology as a NSERC postdoctoral fellow. He is grateful for Professors J. K. Hale and S. N. Chow for their hospitality and support and for the support of the NSERC Postdoctoral Fellowship. He also thanks an anonymous referee whose criticism and suggestions helped to improve the paper.
REFERENCES 1. S. Busenberg and K. L. Cooke, The population dynamics of two vertically transmitted infections, Theoret. Population Biol. 33 Ž1988., 181]198. 2. S. Busenberg and P. van den Driessche, A method for proving the nonexistence of limit cycles, J. Math. Anal. Appl. 172 Ž1993., 463]479. 3. S. Busenberg and P. van den Driessche, Analysis of a disease transmission model in a population with varying size, J. Math. Biol. 28 Ž1990., 257]271. 4. G. J. Butler, R. Schimid, and P. Waltman, Limiting the complexity of limit sets in self-regulating systems, J. Math. Anal. Appl. 147 Ž1990., 63]68. 5. G. J. Butler and P. Waltman, Persistence in dynamical systems, Proc. Amer. Math. Soc. 96 Ž1986., 425]430.
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6. W. A. Coppel, ‘‘Stability and Asymptotic Behavior of Differential Equations,’’ Heath, Boston, 1965. 7. W. B. Demidowitsch, Eine Verallgemeinerung des Kriteriums von Bendixson, Z. Angew. Math. Mech. 46 Ž1966., 145]146. 8. M. Eigen and P. Schuster, ‘‘The Hypercycles}A Principle of Natural Selforganization,’’ Springer-Verlag, Berlin, 1979. 9. J. Guckenheimer and P. Holmes, ‘‘Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,’’ Appl. Math. Sci., Vol. 42, Springer-Verlag, New York, 1983. 10. M. W. Hirsch, System of differential equations that are competitive and cooperative. VI. A local C r closing lemma, Ergodic Theory Dynamical Systems 11 Ž1991., 443]454. 11. J. Hofbauer, A general cooperation theorem for hypercycles, Mh. Math 91 Ž1981., 233]240. 12. M. Y. Li, Bendixson’s criterion for autonomous systems with an invariant linear subspace, Rocky Mountain J. Math. 25 Ž1995., 351]363. 13. M. Y. Li, Nonexistence of periodic solutions in a SEIRS epidemiological model with a varying total population, preprint No. 2171, Centre des Rechereches Mathematiques, ´ 1994. 14. Y. Li and J. S. Muldowney, On Bendixson’s criterion, J. Differential Equations 106 Ž1994., 27]39. 15. M. Y. Li and J. S. Muldowney, On R. A. Smith’s autonomous convergence theorem, Rocky Mountain J. Math. 25 Ž1995., 365]379. 16. M. Y. Li and J. G. Muldowney, A geometric approach to global stability problems, preprint, No. 2160, Centre des Recherches Mathematiques, 1994. ´ 17. D. London, On derivations arising in differential equations, Linear and Multilinear Algebra 4 Ž1976., 179]189. 18. J. Mena-Lorca and H. W. Hethcote, Dynamic models of infectious diseases as regulators of population size, J. Math. Biol. 30 Ž1992., 693]716. 19. J. S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mountain J. Math. 20 Ž1990., 857]872. 20. C. C. Pugh, The closing lemma, Amer. J. Math. 89 Ž1967., 856]1009. 21. R. A. Smith, Some applications of Hausdorff dimension inequality for ordinary differential equations, Proc. Roy. Soc. Edinburgh Sect. A 104 Ž1986., 235]259.
199, 374]390 Ž1996.
0147
Dulac Criteria for Autonomous Systems Having an Invariant Affine Manifold Michael Y. Li* Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, Atlanta, Georgia 30332 Submitted by Hal L. Smith Received April 19, 1994
For a class of higher dimensional autonomous systems that have an invariant affine manifold, conditions are derived to preclude the existence of periodic solutions on the invariant manifold. It is established that these conditions are robust under certain types of local perturbations of the vector field. As a consequence, each bounded semitrajectory on the invariant manifold is shown to converge to a single equilibrium using a C 1 closing lemma for this class. Applications to autonomous systems that are homogeneous of degree 1 are also consid ered. Q 1996 Academic Press, Inc.
1. INTRODUCTION Let X ; R n be an open set and x ¬ f Ž x . g R n a C 1 function defined for x g X. We consider the autonomous system of ordinary differential equations x9 s f Ž x . . Ž 1.1. Let x Ž t, x 0 . denote the solution to Ž1.1. such that x Ž0, x 0 . s x 0 . We are interested in systems Ž1.1. that satisfy the following assumptions: ŽH 1 . There is a constant matrix B such that rank B s r and the Ž n y r .-dimensional affine manifold G s x g X : B Ž x y x . s 04 for some x g X satisfies Bf Ž x . s 0, x g G. ŽH 2 . The Jacobian matrix fr x can be written as f s n Ž x . In= n q A Ž x . , xgG x * Research is supported by a NSERC Postdoctoral Fellowship. 374 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
Ž 1.2.
Ž 1.3.
375
DULAC CRITERIA
and BAŽ x . s 0,
x g G,
Ž 1.4.
where x ¬ n Ž x . is a real-valued function. A system Ž1.1. satisfying ŽH 1 . and ŽH 2 . will be called an autonomous system ha¨ ing an in¨ ariant affine manifold, since ŽH 1 . is equivalent to that G is invariant with respect to Ž1.1.. Let a s Bx. We denote the class of such systems by SB, a . When B and a are implied or when their appearance is not essential, we denote this class simply by S . This is a wider class than that considered in w12x, where Ž1.1. was assumed to satisfy: ŽHX1 . The Jacobian matrix fr x can be written as
f x
s yn In= n q A Ž x . ,
xgX
Ž 1.5.
where n is some constant. ŽHX2 . There is a constant matrix B such that rank B s r and BAŽ x . s 0,
x g X,
Ž 1.6.
and the corresponding system Ž1.1. called an autonomous system ha¨ ing an in¨ ariant linear subspace. In this case, the existence of an invariant affine manifold was proved as a consequence of Ž1.5. and Ž1.6.. Systems in S enjoy concrete examples arising from many different areas which are not covered by systems considered in w12x; for instance, the hypercycles which describe self-organization in Mathematical Biology Žsee w8, 11x., and some epidemiological models which allow variable total populations such as models with disease-related death and with vertical transmission Žsee w1, 3x.. A general class of homogeneous systems of degree 1 ŽSection 3. can also be treated in this context. In many cases, our prime interest in systems that belong to S is the behaviour of solutions that stay in the invariant manifold G. The main purpose of the present paper is to derive Dulac type conditions for the nonexistence of periodic solutions in G. Generalizations of the criteria of Bendixson and Dulac to higher dimensions have been obtained in the context of general autonomous systems by many authors w2, 14, 19, 21x. In our situation, one usually tries to restrict Ž1.1. to the invariant manifold G to get a lower dimensional system and then apply these general criteria. However, the effectiveness of this approach depends critically on the choice of coordinates for G. A different technique was developed in w12x which does not require this reduction in dimension and thus is independent of any choice of coordinates for G. As a consequence, Bendixson type
376
MICHAEL Y. LI
conditions were derived more coherently and more economically. In the present paper, this technique is further explored in deriving Dulac type conditions for systems more general than those considered in w12x. We also present some applications where the theory in w12x is not applicable. Autonomous convergence theorems were first obtained in w15, 21x, and later in w16x, for general autonomous systems in R n. Certain higher dimensional Bendixson]Dulac criteria were shown to be robust under C 1 perturbations of f and this robustness was used, together with the C 1 closing lemma of Pugh w20x, to show that each bounded semitrajectory converges to an equilibrium under any of these higher dimensional Bendixson]Dulac criteria. In developing this type of result in Section 3 for systems which belong to S , we prove in Lemma 3.3 that a local C 1 closing lemma can be applied to systems in S , namely, we are able to show that the perturbation may be chosen from S . The paper is organized as follows: in Section 2, we derive Dulac criteria for systems in S ; in Section 3, we prove an autonomous convergence theorem on G, after we show the C 1 robustness of our Dulac criteria in a very specific sense and prove a local C 1 closing lemma for systems in S by applying Pugh’s result; in Section 4, we present an application to autonomous homogeneous systems of degree 1.
2. DULAC CRITERIA For a solution x Ž t, x 0 . to Ž1.1. with x 0 g G, the linear variational equation takes the form y9 Ž t . s n Ž x Ž t , x 0 . . y Ž t . q A Ž x Ž t , x 0 . . y Ž t . .
Ž 2.1.
The change of variables uŽ t . s y Ž t .exp yH0t n Ž x Ž s, x 0 .. ds4 leads to u9 Ž t . s A Ž x Ž t , x 0 . . u Ž t .
Ž 2.2.
satisfying the condition BA Ž x Ž t , x 0 . . s 0
for t g R.
Ž 2.3.
Therefore the linear subspace ker B s u g R n : Bu s 04 of R n is invariant with respect to the linear system Ž2.2. in the sense that uŽ0. g ker B implies uŽ t . g ker B for t g R, since Ž BuŽ t ..9 ' 0. Linear differential systems with such invariance properties were investigated in w12x. We first recall a result proved there.
377
DULAC CRITERIA
Let w 10 , . . . , wr 0 4 be an orthonormal basis for the orthogonal complement of ker B in R n, and wi Ž t . be the solution to Ž2.2. such that wi Ž0. s wi0 . The following proposition is relation Ž4. in the proof of Theorem 2.5 in w12x. PROPOSITION 2.1.
Let u1Ž t ., u 2 Ž t . be two solutions to Ž2.2.. Then
u1 Ž t . n u 2 Ž t . F w 1 Ž t . n ??? n wr Ž t . n u1 Ž t . n u 2 Ž t . . Here 5 ? 5 is the euclidean norm and n denotes the exterior product in R n. The exterior product z Ž t . s w 1Ž t . n ??? n wr Ž t . n u1Ž t . n u 2 Ž t . satisfies the linear system z9 Ž t . s Aw rq2x Ž x Ž t , x 0 . . z Ž t .
Ž 2.4.
which is the Ž r q 2.nd compound equation of Ž2.2.. Here Aw rq2x is the n Ž r q 2.nd additi¨ e compound matrix of A w17, 19x. It is a = r qn 2 rq2
ž / ž /
matrix, and hence Ž2.4. is a = linear system. For a detailed study on compound matrices and compound equations, we refer the reader to w17, 19x. Let W be the euclidean unit ball in R 2 and let W and W be its closure and boundary, respectively. A function w g LipŽW ª G . will be described as a rectifiable 2-surface in G; a function c g LipŽ W ª G . is a closed rectifiable cur¨ e in G and will be called simple if it is one-to-one. A method for deriving Dulac type conditions for general autonomous systems was developed in w14x by studying the evolution under Ž1.1. of certain functionals which are defined on rectifiable 2-surfaces. If G is simply connected, for a given simple closed curve in LipŽ W ª G ., the set n rq2
n rq2
ž / ž /
S Ž c , G . s w g Lip Ž W ª G . : w Ž W . s c Ž W . 4 is not empty Žsee w14x.. Consider a functional A defined on SŽ c , G . by Aw s
HW
w r1
n
w r2
,
where < ? < is any vector norm in R N , N s r qn 2 . For instance, when the norm is < y < s < y*y < 1r2 , where here and throughout the paper asterisk denotes transposition, then Aw is the usual surface area of w ŽW .. Since G is affine, the following result can be proved as Proposition 2.2 in w14x.
ž /
PROPOSITION 2.2. Suppose c is a simple closed rectifiable cur¨ e on G. Then there exists a d ) 0 such that Aw G d for all w g SŽ c , G ..
378
MICHAEL Y. LI
A subset K ; G is absorbing in G for Ž1.1. if each compact subset D ; G satisfies x Ž t, D . ; K for sufficiently large t. We make the following assumption: ŽH 3 . G is simply connected and there is a compact set K in the interior of G which is absorbing for Ž1.1.. Let x ¬ P Ž x . be a nonsingular r qn 2 = r qn 2 matrix-valued function which is C 1 in G. Then < Py1 Ž x .< is uniformly bounded for x g K. Let m be the Lozinskiı˘ measure Žor the logarithmic norm. with respect to < ? < defined by Žcf. w6, p. 41x, or w19x.
ž / ž /
m Ž F . s limq
< I q hF < y 1 h
hª0
for any N = N matrix F, N s y1
E s Pf P
n rq2
ž / . Set
qP
f w rq2x x
Py1 y rn IN=N .
Ž 2.5.
Here Pf is the matrix obtained by replacing each entry pi j of P by Ž pUi j r x . f, its directional derivative in the direction of f, and f w rq2xr x is the Ž r q 2.nd additive compound matrix of the Jacobian matrix fr x. Define the quantity qrq2 s lim sup sup tª`
x 0gK
1
t
H m Ž E Ž x Ž s, x t 0
0
. . . ds.
Ž 2.6.
We note that, since K ; G, the time average in Ž2.6. is only calculated along solutions on the affine manifold G. Since G is invariant, qrq2 is well defined. A closed rectifiable curve c g LipŽ W ª G . is in¨ ariant with respect to Ž1.1. if the subset c Ž W . of G is invariant. The following result gives a general Dulac criterion in a weak form for autonomous systems in S . THEOREM 2.3.
Assume that ŽH 1 ., ŽH 2 ., and ŽH 3 . are satisfied. If qrq2 - 0,
Ž 2.7.
then no simple closed rectifiable cur¨ e in G can be in¨ ariant with respect to Ž1.1.. Proof. Let e 0 s y 12 qrq2 ) 0. There exists T ) 0 such that t
H0 m Ž E Ž x Ž s, x for t ) T, x 0 g K.
0
. . . ds - ye 0 t
Ž 2.8.
379
DULAC CRITERIA
Suppose that, on the contrary, there is a simple closed rectifiable curve c g LipŽ W ª G . that is invariant with respect to Ž1.1.. Let w g SŽ c , G . and w t Ž p . s x Ž t, w Ž p .., where p s Ž p1 , p 2 . g W. Then, from the invariance of c , we know w t g SŽ c , G . for all t, and hence Aw t G d
Ž 2.9.
for some d ) 0 independent of t, by Proposition 2.2. Now, yi Ž t . s w tr pi s Ž x Ž t, w t .r x 0 .Ž wr pi ., i s 1, 2, are solutions to the linear variational equation Ž2.1. with respect to x s w t Ž p ., and hence u i Ž t . s Ž w tr pi .exp yH0t n Ž ws Ž p .. ds4 are solutions to the linear system Ž2.2. with x Ž t . s w t Ž p .. Let w 1Ž t ., . . . , wr Ž t . be the solution to Ž2.2. as in Proposition 2.1. Then u1Ž t . n u 2 Ž t . n w 1Ž t . n ??? n wr Ž t . satisfies the Ž r q 2.nd compound equation Ž2.5.. The following relation holds t
½H
y 1 Ž t . n y 2 Ž t . s u1 Ž t . n u 2 Ž t . exp 2
0
n Ž ws Ž p . . ds
5
Ž 2.10.
and by Proposition 2.1 t
½H
u1 Ž t . n u 2 Ž t . exp 2
0
n Ž ws Ž p . . ds
5 t
½H
F u1 Ž t . n u 2 Ž t . n w 1 Ž t . n ??? n wr Ž t . exp 2
0
5
n Ž ws Ž p . . ds .
Ž 2.11. Direct differentiation yields that t
½H
¨ Ž t . s u1 Ž t . n u 2 Ž t . n w 1 Ž t . n ??? n wr Ž t . exp 2
0
n Ž ws Ž p . . ds
5
satisfies the differential system ¨ 9Ž t . s
ž
f w rq2x x
Ž wt Ž p . . y rn Ž wt Ž p . . IN=N
/
¨ Ž t. .
w rq2x w rq2x Here we use f w rq2xr x s Aw rq2x q In= from Ž1.3. and In=n s Žr q n n 2. IN= N , N s r q 2 . Therefore V Ž t . s P Ž w t Ž p .. ¨ Ž t . satisfies the equation V9Ž t . s EŽ w t Ž p .. V Ž t ., where E is given in Ž2.5.. By a property of the Lozinskiı˘ measure Žsee w6, p. 41x. and by Ž2.8.
ž /
t
H0 m Ž E Ž w Ž p . . . ds F V Ž 0. expŽ ye
V Ž t . F V Ž 0 . exp
s
0t
. , Ž 2.12.
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MICHAEL Y. LI
for p g W and t ) T. Therefore V Ž t . ª 0 as t ª `, uniformly for p g W. Since w t ŽW . ; K for sufficiently large t, and Py1 Ž x . is uniformly bounded for x g K, this implies that ¨ Ž t . ª 0 as t ª `, uniformly for p g W. Thus by Ž2.11. and Ž2.12., < w tr p1 n w tr p 2 < ª 0 exponentially as t ª `, and the exponential rate is uniform for all p g W. As a consequence, Aw t ª 0 as t ª `. This contradicts Ž2.9., completing the proof of Theorem 2.3. Remarks. Ži. Condition Ž2.7. rules out the existence in G of orbits of the following types: Ža. periodic orbits, Žb. homoclinic orbits, Žc. heteroclinic cycles, since each of such orbits gives rise to an invariant simple closed rectifiable curve. Žii. A sufficient condition for Ž2.7. is y1
m Pf P
ž
qP
f w rq2x x
Py1 y rn IN=N - 0
/
on K
Ž 2.13.
which is a general Dulac type condition. Note that conditions Ž2.7. and Ž2.13. provide the flexibility of a choice of N 2 arbitrary functions in addition to the choice of vector norms in deriving suitable conditions. Setting A s I in Ž2.13. leads to the condition
m
ž
f w rq2x x
y rn IN= N - 0
/
Ž 2.14.
which is the Bendixson type condition obtained in w12x. Concrete expressions of Ž2.14. in terms of Lozinsskiı˘ measures with respect to some common vector norms can be found in w12x. Žiii. Conditions Ž2.13. and Ž2.14. take a simple form when r s n y 2. In this case, r q 2 s n, N s r qn 2 s 1, and thus P Ž x . is scalar-valued and f w rq2xr x s divŽ f .. For example, Ž2.14. will read
ž /
div Ž f . - Ž n y 2 . n
on G.
Ž 2.15.
Also note that G is a 2-dimensional affine manifold in this case; any simple closed curve in G encloses a region in G. From the proof of Theorem 2.3 we can see that Ž2.15. need only hold for all x g G except a set of measure zero. Živ. When r s 0, ŽH 1 . and ŽH 2 . pose no restriction on Ž1.1.. In accordance, the condition Ž2.13. agrees with a Dulac type condition for general autonomous systems obtained in w14x. Žv. Assume that Ž1.1. satisfies ŽHX1 . and ŽHX2 . with n s 0. It is proved in w12x that Bx gives r independent linear first integrals for Ž1.1.. In the
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DULAC CRITERIA
special case that n s 3, r s 1, it follows from Theorem 2.3 and Remark Žiii. that no simple closed rectifiable curve on a level surface S defined by Bx s c can be invariant with respect to Ž1.1. if divŽ f . - 0 for almost all x g S. This gives a result of Demidowitsch w7x for the case of a linear first integral. Živ. For many systems which describe population dynamics, the feasin ble region X is usually the nonnegative cone Rq and the affine manifold n G is the portion of a hypersurface bounded in Rq , e.g., the simplex n x g Rq : Ý nis1 x i s 14 ; the assumption ŽH 3 . in such a case is equivalent to the uniform persistence of Ž1.1. in G Žsee w5x..
3. CONVERGENCE OF TRAJECTORIES AND GLOBAL STABILITY In this section, we investigate further implications of condition Ž2.7. for the behaviour of solutions to Ž1.1. which stay on the invariant manifold G. We begin by studying the robustness of condition Ž2.7. under certain smooth perturbations of f. Let < ? < denote a vector norm on R n and the operator norm which it induces for linear mappings from R n to R n. The distance between two functions f, g g C 1 Ž X ª R n . such that f y g has compact support is < f y g < C 1 s sup
½
f Ž x. y gŽ x. q
f x
Ž x. y
g x
5
Ž x. : x g X .
A function g g C 1 Ž X ª R n . is called a C 1 local e-perturbation of f at x 0 g X if there exists an open neighbourhood U of x 0 in X such that the support of Ž f y g . suppŽ f y g . ; U and < f y g < C 1 - e . Given B and x in the assumption ŽH 1 ., set a s Bx. For f g SB, a , let g g SB, a be a C 1 local perturbation of f. We consider the corresponding differential equation x9 s g Ž x . . Ž 3.1. Since g g SB, a , the affine manifold G defined in Ž1.2. is necessarily invariant with respect to Ž3.1.. We say that a Dulac criterion Ž2.7. is robust under C 1 local perturbations of f at x 0 if, for each sufficiently small e ) 0 and neighbourhood U of x 0 , it is also satisfied by C 1 local e-perturbations g such that suppŽ f y g . ; U. Let f G s f < G . Then f G defines a vector field on the affine manifold G from the assumption ŽH 1 .. Denote the differential equation x9 s f G Ž x ., x g G by Ž1.1. G . A point x 0 g G is said to be nonwandering for Ž1.1. G if, for each sufficiently small relative neighbourhood V of x 0 in G, x Ž t, V . l V / B for some t.
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PROPOSITION 3.1. Assume that f g SB, a . Let qrq2 be defined in Ž2.6. for some ¨ ector norm < ? < and matrix-¨ alued function P. Then the Dulac criterion qrq2 - 0 is robust under C 1 local perturbations in SB, a at any nonequilibrium point x 0 g G that is nonwandering for Ž1.1. G . Proof. This is essentially Proposition 3.3 of w16x. Since qrq2 - 0 implies that Ž1.1. G has no periodic solutions, the minimum return time at a nonequilibrium nonwandering point x 0 , as defined in Ž3.8. of w16x, is q`. Also note that only trajectories on the invariant manifold G are involved in the calculation of qrq2 and that G is affine, the proposition can be proved as Proposition 3.3 in w16x by restricting both Ž1.1. and Ž3.1. to G. Next, we prove an adapted local C 1 closing lemma for systems in SB, a , in which the perturbations g belong to SB, a . The idea is to apply the general local closing lemma of Pugh w20x to the restricted system Ž1.1. G on n G, and then extend the perturbation to Rq using a bump function. Since the extended perturbation has to be in SB, a , we carry out the construction of the bump function in detail. LEMMA 3.2. Let f g SB, a . Suppose that x 0 g G is nonwandering for Ž1.1. G and that f Ž x 0 . / 0. Then, for each e ) 0 and each neighbourhood U of x 0 in X, there exists C 1 e-perturbations g of f such that Ž1. g g SB, a and suppŽ f y g . ; U, Ž2. the system Ž3.1. has a nonconstant periodic solution whose trajectory lies in G and passes through x 0 . Proof. Since G is affine, without loss of generality, we may assume that G lies on the coordinate plane Ž x 1 , . . . , x nyr . and that x 0 is at the origin. More precisely, write X s X 1 [ X 2 , such that X 1 ( R ny r , X 2 ( R r. Then we assume G ; X 1. If x s Ž x 1 , . . . , x n ., we write x Ž1. s Ž x 1 , . . . , x nyr . and x Ž2. s Ž x ny rq1 , . . . , x n . so that x s Ž x Ž1., x Ž2. .. In this setting, the matrix B may be chosen as the form B s Ž0, Ir=r .. Now f G is a mapping from G to X 1 , and thus system Ž1.1. G is a n y r dimensional autonomous system d dt
x Ž1. s f G Ž x Ž1. . ,
x Ž1. g G ; X 1 .
Ž 3.2.
Let U1Ž0, d . ; R ny r be a euclidean ball centered at the origin and of radius d . We choose d small enough so that U1Ž0, d . ; U and V s U1 Ž 0, d . = x Ž2. g X 2 : < r x Ž2. < - 1 4 ; U
for some r ) 1. Ž 3.3.
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DULAC CRITERIA
Apply to Ž3.2. the local C 1 Closing Lemma of Pugh as formulated in w10x Žalso see w20x.. There exists a h G g C 1 Ž G ª X 1 . such that Ža. < h G < C 1 - er8 r and suppŽ h G . ; U1Ž0, d ., Žb. Ž drdt . x Ž1. s f G Ž x Ž1. . q h G Ž x Ž1. . has a nonconstant periodic solution whose trajectory passes through 0. Since this periodic trajectory necessarily stays in G, the lemma is proved if we can extend h G to a function h defined on X so that g s f q h satisfies Lemma 3.2Ž1.. For r ) 1 in Ž3.3., define a function x ¬ a Ž x . from R n to R by
a Ž x. s
½
Ž 1 y < r x Ž2. < 2 .
2
< r x Ž2. < F 1
,
0,
otherwise,
where x s Ž x Ž1., x Ž2. ., and a function x ¬ hŽ x . from R n to R n by hŽ x . s
ž
h G Ž x Ž1. . . 0
/
Then < a < - 1 and < h < C 1 F < h G < C 1 - er8 r . Moreover, the gradient =a Ž x . s
½
y4 r 2 Ž 1 y < r x Ž2. < 2 . Ž 0, x Ž2. . *, 0,
< r x Ž2. < F 1, otherwise,
and BhŽ x . s 0 for all x g X. Set g s f q a h. Then g g C 1 Ž X ª R n ., g < G s f G q h G , and suppŽ f y g . ; V ; U, where V is given in Ž3.3.. In particular, Bg Ž G . s 0. Moreover g y f s ah
g x
y
f x
sa
h x
q h =a *.
Thus < g y f < C1 F < a h< q a F
e 4r
h x
q < h=a * <
Ž1 q 2 r . - e ,
and B Ž gr x y fr x . s 0 for all x g X. Therefore g g SB, a and satisfies Ž1. and Ž2..
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MICHAEL Y. LI
PROPOSITION 3.3. Under the assumptions ŽH 1 ., ŽH 2 ., and ŽH 3 ., if qrq2 0, then the dimension of the stable manifold of any equilibrium of Ž1.1. G is at least n y r y 1; if an equilibrium of Ž1.1. G is not isolated, then its stable manifold has dimension n y r y 1 and it has a center manifold of dimension 1 which contains all nearby equilibria of Ž1.1. G . Proof. Without loss of generality, we assume the same set up as in the proof of Lemma 3.2 such that B s Ž0, Ir=r ., G ; X 1 ( R ny r , and the Jacobian matrix fr x may be written in the block form
f x
s
n Ž x . Ir=r A1
0 . A2
It is easy to see that Ž fr x .Ž x 0 . has an eigenvalue n Ž x 0 . with r independent eigenvectors in X 1. At an equilibrium x 1 of Ž1.1. G , qrq2 - 0 implies
m P
ž
f w rq2x x
y rn IN= N Py1 - 0
/
Ž 3.4.
since f Ž x 1 . s 0 implies Pf Ž x 1 .Ž x 1 . s 0. Let l1 , . . . , l nyr be the eigenvalues of Ž fr x .Ž x 1 . with respect to the invariant subspace X 1 , such that Re l1 G ??? G Re l nyr . Then
l1 q l2 q rn is an eigenvalue of Ž f w rq2xr x .Ž x 1 . Žsee w19x.. Therefore the matrix f w rq2xr x y rn IN= N has an eigenvalue l1 q l2 . Thus Ž3.4. implies Žsee w6, p. 41x. Re l1 q Re l2 - 0. It then follows that 0 ) Re l2 G ??? G Re l nyr ; only l1Ž x 1 . can possibly have nonnegative real part. The stable manifold of x 1 in G has dimension at least n y r y 1. If x 1 is not isolated, Ž fr x .Ž x 1 .< X 1 has a singular matrix representation, l1Ž x 1 . s 0 so that its stable manifold in G has dimension n y r y 1 and there is a 1-dimensional center manifold in G which contains all nearby equilibria in G, by the Center Manifold Theorem Žcf. w9, p. 48x.. THEOREM 3.4. Under the assumptions ŽH 1 ., ŽH 2 ., and ŽH 3 ., if qrq2 - 0, then Ž1. E¨ ery nonwandering point of Ž1.1. G is an equilibrium. Ž2. E¨ ery nonempty alpha or omega limit set of Ž1.1. G is a single equilibrium.
DULAC CRITERIA
385
Proof. Under the condition qrq2 - 0, Ž1.1. G has no nonconstant periodic point by Theorem 2.3. Let x 0 g G be a nonwandering point for Ž1.1. G and f Ž x 0 . / 0. Then by Lemma 3.2, there exist arbitrarily small local C 1 perturbations g g SB, a of f at x 0 such that Ž3.1. has a nonconstant periodic solution. However, any such g satisfies qrq2 - 0, by Proposition 3.1, and thus cannot have any nonconstant periodic solution. This contradiction establishes Ž1.. To prove the assertion that each nonempty alpha or omega limit set is a single equilibrium, first observe that since each limit point is nonwandering, it is an equilibrium. Let x 1 g v Ž x 0 ., the omega limit set of x 0 for Ž1.1. G . If x 1 is an isolated equilibrium for Ž1.1. G , then x 14 s v Ž x 0 ., since v Ž x 0 . is connected. If x 1 is not isolated, then Proposition 3.3 implies that there is a 1-dimensional center manifold associated with x 1 in G containing all nearby equilibrium for Ž1.1. G . Since v Ž x 0 . contains a continuum of equilibria, we may choose x 1 and a sufficiently small neighbourhood U of x 1 so that in U any of such local center manifolds consists entirely of equilibria and every trajectory of Ž1.1. G which intersects U is asymptotic to a trajectory in the center manifold. Thus lim t ª` x Ž t, x 0 . s x 1 so that v Ž x 0 . s x 14 in this case also. The proof that a nonempty alpha limit set is a single equilibrium is the same. COROLLARY 3.5. Assume that ŽH 1 ., ŽH 2 ., and ŽH 3 . are satisfied. Suppose Ž1.1. has a unique equilibrium x 0 in the interior of G. If qrq2 - 0 for some ¨ ector norm < ? < and matrix-¨ alued function P Ž x ., then x 0 is globally asymptotically stable in the interior of G. Proof. Clearly x 0 4 is globally attracting in the interior of G since Theorem 3.4 implies that it is the omega limit set of every trajectory starting in the interior of G. Moreover, x 0 4 is stable since otherwise it would be both the alpha limit set and the omega limit set of some homoclinic trajectory g ; G, which gives rise to a simple closed invariant curve that is also rectifiable Žsee w15, the proof of Corollary 2.6x.. This has been ruled out by qrq2 - 0. 4. HOMOGENEOUS SYSTEMS OF DEGREE 1 In this section, we consider autonomous system Ž1.1. under the assumption that f Ž x . is homogeneous of degree 1, namely, f Ž lx. s lf Ž x.
for l ) 0, x g R n .
Ž 4.1.
Such systems have been used in modeling of population dynamics and the spread of infectious diseases Že.g., see w3, 18x., where components x i of x
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MICHAEL Y. LI
are restricted to be nonnegative. In the same spirit, we assume that each n component f i of f satisfies f i < x is0 G 0 so that the nonnegative cone Rq is positively invariant with respect to Ž1.1.. n Let x Ž t, x 0 . be a solution to Ž1.1. with x 0 g Rq R 0. It is customary in the study of homogeneous differential systems to consider the projected variable yŽ t. s
xŽ t. NŽ t.
n
where N Ž t . s
,
Ý xi Ž t . .
Ž 4.2.
is1
Then Ý nis1 yi Ž t . s 1 for all t and N9Ž t . s Ý nis1 f i Ž x Ž t ... We thus have the relation n
N9 Ž t .
s
NŽ t.
Ý fi Ž y Ž t . . .
Ž 4.3.
is1
As a consequence, y Ž t . satisfies the following system of differential equations n
y9 Ž t . s y
ž
Ý fi Ž y . is1
/
y q f Ž y. .
Ž 4.4.
n System Ž4.4. is well-defined in Rq , though the variable y defined in Ž4.2. stays in the simplex
½
n D s y g Rq :
n
Ý yi s 1 is1
5
.
Ž 4.5.
In fact, by adding the equations of Ž4.4., one may check that D is invariant with respect to Ž4.4.. When restricted to the invariant simplex D, system Ž4.4. describes the projected flow of the homogeneous system Ž1.1. onto D. In particular, for each solution y Ž t . of Ž4.4. with y Ž0. g D, there exists a solution x Ž t . to Ž1.1. such that x Ž t . and y Ž t . satisfy the relation Ž4.2.. In fact, by the uniqueness of solution, such a x Ž t . may be obtained by setting the initial conditions x i Ž0. s yi Ž0.. From these considerations, we want to study the behaviour of solutions of Ž4.4. on D. n Observe that D is an affine manifold in Rq defined by
Ž 1, . . . , 1 . y s 1. Thus the assumption ŽH 1 . in Section 1 is satisfied with X being the n interior of Rq , G being D, B s Ž1, . . . , 1., r s 1, and x s Ž1rn, . . . , 1rn.*.
387
DULAC CRITERIA
The Jacobian matrix J Ž y . of Ž4.4. can be written as J Ž y . s n 1 Ž y . In=n q F Ž y . , where n
n 1Ž y . s y Ý fi Ž y .
Ž 4.6.
is1
and FŽ y. s
f y
n
y Ž y 1 , . . . , yn . *
ž
Ý is1
fi y1
fi
n
,...,
Ý is1
yn
/
.
Ž 4.7.
Since n
Ž 1, . . . , 1 . F Ž y . s 1 y
ž
Ý yi is1
n
/ž
Ý is1
fi yi
n
,...,
Ý is1
fi yn
/
one may see that ŽH 2 . is also satisfied. Theorem 2.3 can now be applied to system Ž4.4. to yield conditions which rule out the existence of solutions x Ž t . to Ž1.1. for which y Ž t . is periodic in D. Assume that Ž4.4. is uniformly persistent in D and let K 1 be the compact absorbing set in D. We can define q3 as in Ž2.6. for r s 1, n s n 1 , and for some N = N matrix-valued function P Ž x . and some vector norm < ? < in R N , N s nr3. The next result follows from Theorem 2.3 and the fact that y Ž t . is periodic if x Ž t . associated with y Ž t . by Ž4.2. is periodic. THEOREM 4.1. If q3 - 0, then no simple closed rectifiable cur¨ e in D can be in¨ ariant with respect to Ž4.3.. In particular, Ž1.1. has no periodic solutions n in Rq . Remark. The condition q3 - 0 also rules out the types of orbits of Ž1.1. listed in Remark Ži. following the proof of Theorem 2.3. For an application of Theorem 4.1 to a SEIRS epidemiological model with a homogeneous incidence function and a varying total population, we refer the reader to w13x. In the following, we consider a special case. Assume n s 3 and f takes the form f Ž x . s Ax q Ž diag x . f Ž x . ,
Ž 4.8.
where A s Ž a i j . 3=3 is a nonnegative matrix, namely, ai j G 0
i/j
Ž 4.9.
and f Ž x . is homogeneous of degree 0, that is, f Ž l x . s f Ž x ., for l ) 0 and x g R 3. System Ž1.1. with f given in Ž4.8. has been used in population models Žsee w3, 4x.. In this case, we have the following result.
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MICHAEL Y. LI
THEOREM 4.2. Assume that n s 3 and that f satisfies Ž4.8. and Ž4.9.. Then the conclusions of Theorem 4.1 hold if
f i
3
xi
Ý
xi
is1
3
-0
Ý x i s 1.
when
Ž 4.10.
is1
Proof. Set P Ž y . s Ž y 1 y 2 y 3 .y1 , n s 3, r s 1, and n s n 1 in Ž2.5.. Then the matrix E can be calculated with respect to system Ž4.4. as E s Pg Py1 q div Ž g . y n 1 , where g denotes the vector field of Ž4.4.. Relation Ž4.3. leads to Pg Py1 s Pf Py1 q
P
3
ž
Ý
yi
is1
yi n 1 Py1
/
s Pf Py1 y 3n 1 and 3
div Ž g . s div Ž f . y
fi
3
Ý Ý js1 is1
yj
y j q 3n 1 .
Differentiating Ž4.1. with respect to l at l s 1 leads to 3
Ý js1
fi yj
yj s f i ,
i s 1, 2, 3.
Therefore E s Pf Py1 q div Ž f . s Py1 div Ž Pf . . For f given in Ž4.8. div Ž Pf . s y Ý a i j i/j
yj yi
3
q
Ý is1
f i yi
yi ,
and thus for y g K ; S
m Ž E . s Py1 y Ý a i j
ž
i/j
yj yi
3
q
Ý is1
f i yi
/
yi F yd 1 - 0
for some d 1 ) 0. Therefore Theorem 4.2 follows from Theorem 2.3.
DULAC CRITERIA
389
Remarks. Ž1. When condition Ž4.10. is satisfied, a system Ž1.1. with f Ž x . s Ždiag x . f Ž x . is said to be self-regulating. For 3-dimensional self-regulating systems, a result of Butler, Schmid, and Waltman w4x implies that 3-dimensional volumes decay exponentially along solutions of Ž1.1., whereas Theorem 4.2 implies that if Ž1.1. is also homogeneous of degree 1, then 2-dimensional surface areas as well as 3-dimensional volumes, when projected onto D, decay exponentially. Moreover, the projected dynamics of Ž1.1. onto D is trivial in the sense that every nonwandering point is an equilibrium and bounded solutions either converge to equilibria or approach the boundary. Ž2. Homogeneous systems Ž1.1. with vector fields given in Ž4.8. was also considered in Busenberg and van den Driessche w2x, where conditions which preclude the existence of periodic solutions and closed phase polygons D are derived without the uniform persistence assumption. One of the conditions given in w2x is
f i xi
q
f j xj
y
ž
f i xj
q
f j xi
/
F 0,
i / j, i , j s 1, 2, 3, on D . Ž 4.11.
While Ž4.10. may be easier to verify than Ž4.11., it is not apparent that one implies the other.
ACKNOWLEDGMENTS The final draft of this paper was completed when the author was visiting the Center for Dynamical Systems and Nonlinear Studies ŽCDSNS. at Georgia Institute of Technology as a NSERC postdoctoral fellow. He is grateful for Professors J. K. Hale and S. N. Chow for their hospitality and support and for the support of the NSERC Postdoctoral Fellowship. He also thanks an anonymous referee whose criticism and suggestions helped to improve the paper.
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6. W. A. Coppel, ‘‘Stability and Asymptotic Behavior of Differential Equations,’’ Heath, Boston, 1965. 7. W. B. Demidowitsch, Eine Verallgemeinerung des Kriteriums von Bendixson, Z. Angew. Math. Mech. 46 Ž1966., 145]146. 8. M. Eigen and P. Schuster, ‘‘The Hypercycles}A Principle of Natural Selforganization,’’ Springer-Verlag, Berlin, 1979. 9. J. Guckenheimer and P. Holmes, ‘‘Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,’’ Appl. Math. Sci., Vol. 42, Springer-Verlag, New York, 1983. 10. M. W. Hirsch, System of differential equations that are competitive and cooperative. VI. A local C r closing lemma, Ergodic Theory Dynamical Systems 11 Ž1991., 443]454. 11. J. Hofbauer, A general cooperation theorem for hypercycles, Mh. Math 91 Ž1981., 233]240. 12. M. Y. Li, Bendixson’s criterion for autonomous systems with an invariant linear subspace, Rocky Mountain J. Math. 25 Ž1995., 351]363. 13. M. Y. Li, Nonexistence of periodic solutions in a SEIRS epidemiological model with a varying total population, preprint No. 2171, Centre des Rechereches Mathematiques, ´ 1994. 14. Y. Li and J. S. Muldowney, On Bendixson’s criterion, J. Differential Equations 106 Ž1994., 27]39. 15. M. Y. Li and J. S. Muldowney, On R. A. Smith’s autonomous convergence theorem, Rocky Mountain J. Math. 25 Ž1995., 365]379. 16. M. Y. Li and J. G. Muldowney, A geometric approach to global stability problems, preprint, No. 2160, Centre des Recherches Mathematiques, 1994. ´ 17. D. London, On derivations arising in differential equations, Linear and Multilinear Algebra 4 Ž1976., 179]189. 18. J. Mena-Lorca and H. W. Hethcote, Dynamic models of infectious diseases as regulators of population size, J. Math. Biol. 30 Ž1992., 693]716. 19. J. S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mountain J. Math. 20 Ž1990., 857]872. 20. C. C. Pugh, The closing lemma, Amer. J. Math. 89 Ž1967., 856]1009. 21. R. A. Smith, Some applications of Hausdorff dimension inequality for ordinary differential equations, Proc. Roy. Soc. Edinburgh Sect. A 104 Ž1986., 235]259.