Massera's theorem for monotone dynamical systems in three dimensions

J. Math. Anal. Appl. 269 (2002) 607–615

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Massera’s theorem for monotone dynamicalsystems in three dimensions

Juan Campos1

Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain

Received 8 February 2001; received in revised form 24 August 2001; accepted 16 November 2001

Submitted by C.E. Wayne

Abstract

In this paper we give a variant of the classical second Massera’s theorem for three-dimensional periodic systems, that satisfy different types of monotonicity assumptions. 2002 Elsevier Science (USA). All rights reserved.

1. Introduction

Let

x ′ = f (t, x) (1)

be aT -periodic system of differential equations. We assume conditions on thefunction f :R/T Z × R

N → RN in order to have existence, uniqueness and

continuation of the initial value problem. Our main intention is to ensure when theexistence of a solutionx bounded in[0,∞) implies the existence of aT -periodicone.

If N = 1, the existence of a solution bounded in the future implies existenceof a T -periodic solution that is the limit of the bounded one. WhenN = 2, thenthe result is still true but now the periodic solution need not to be the limit ofthe bounded one. WhenN � 3, the result is false unless some more hypotheses

E-mail address:[email protected] (J. Campos).1 Supported by CICYT, PB98-1294, Spain.

0022-247X/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved.PII: S0022-247X(02)00039-2

608 J. Campos / J. Math. Anal. Appl. 269 (2002) 607–615

are imposed. (See [1] for the caseN = 1 andN = 2 and [2] for an examplewith N = 3.)

In this work we assume some additional hypotheses on Eq. (1) to have thisproperty forN = 3. In concrete we prove that, if one solution is bounded to thefuture and the return map is monotone increasing with respect to an abstract order,then there exists a periodic solution.

This result with the usual order inR3 allows to work with cooperative andcompetitive systems coming from the mathematical biology. As an application,we prove that existence of a bounded persistent solution implies the existence ofa periodic one. In those situations the equation is not defined in the whole spacebut in the first octant, however a simple change of variables reduces them to oursetting. In this way I can connect my results with those in [3] (see Remark 3.4 formore details).

Since we use an abstract order inR3, we can connect these results of Massera

type with [4]. In this work, Smith proves an extension of Massera’s theorem byimposing a condition on the derivatives off , that forN = 3 can be interpreted asan abstract order inR3. This was shown in [5] by Ortega and Sánchez, and alsothat the hypotheses can be relaxed. Therefore this gives a generalization for [4] inthree dimensions. Let us observe that the results in [4] hold for any dimension.

To prove these results we do a variant of the Brouwer’s lemma for monotoneoperators. By Brouwer’s lemma we refer to the following result that is a con-sequence of the translation arcs lemma also due to Brouwer.

Lemma 1.1. Let f :R2 → R2 be an orientation preserving homeomorphism

without fixed points. Then every orbit of

xn+1 = f (xn)

tends to infinity.

See [6] for a recent proof.The idea in this work is to reduce the dimension and to apply the two-

dimensional case. This idea of reducing the dimension was used several timesin the bibliography in order to apply some two-dimensional results; see [7,8] forautonomous cases, [3,9] for periodic ones.

The structure of the paper is the following: Section 2 is dedicated to the proofof a variant of the Brouwer’s lemma for abstract monotone homeomorphismsin R

3.Section 3 is dedicated to the concrete application of this Brouwer’s lemma,

first for cooperative systems, second for competitive ones and thirdly for Smith’ssystems.

J. Campos / J. Math. Anal. Appl. 269 (2002) 607–615 609

2. A variant of Brouwer’s lemma

Let C be a closed cone inRN with nonempty interior; this cone induces anorder inR

N that we will denote by�. We will say thatx < y if x � y andx �= y,alsox y if y − x ∈ C.

Let f :RN → RN be an homeomorphism. We will say thatf is increasing if

for everyx � y, f (x) � f (y). This has two consequences:

• For everyx < y, f (x) < f (y).• For everyx y, f (x) f (y). (Sincef is increasing,f (x +C) ⊂ f (x)+

C; now since domain invariancef (x + C) is an open set, then

f (x + C)⊂ f (x)+ C

so the conclusion holds.)

We are going to use the orientation property for homeomorphisms. This canbe characterized in terms of topological degree as in [10, Chapter IV, Proposi-tion 5.12].

Let us call

MN = {f :RN → R

N | f is an increasing orientation preservinghomeomorphism

}.

Takef ∈MN and consider the discrete dynamical system

xn+1 = f (xn). (2)

Theorem 2.1.Assumef ∈M3. Then one of the following properties holds:

1. Fix(f ) �= ∅.2. Every orbit diverges(limn→∞ |xn| = +∞).

Before entering into the proof of this theorem we do a simple application ofthis theorem:

Corollary 2.2. Assume thatf = g−1 with g ∈ M3. Then one of the followingproperties holds:

1. Fix(f ) �= ∅.2. Every semiorbit{xn}n�0 is unbounded.

Proof of Corollary 2.2. Let x0 ∈ R3 and suppose that the semiorbitxn is

bounded. Then the omega limit setLω is a nonempty compact invariant set. Hencethere exists an orbit that is bounded but now to the past, then last theorem can beapplied tof−1 = g andf has fixed points. This finishes the proof.✷


To prove Theorem 2.1, we need some preliminary definitions and results.

Definition 2.3. Let K be a proper subset ofRN . Following the terminology of[11] we will say thatK is a CSUC (closed subset upper closed) if it is closed andsatisfies: ifx ∈K, theny ∈K for everyy � x.

The following result gives the structure of this class of sets. For conveniencewe denotex = (x, x) ∈ R

N−1 ×R.

Proposition 2.4. Let K be a CSUC. Then there exists an homeomorphismF :RN →R

N such that

F(K)= {x ∈ R

N | x � 0},

F(∂K)= {x ∈R

N | x = 0},

F(K)= {x ∈ R

N | x > 0}.

Proof. Take e ∈ C fixed in this proof, and consider the orthogonal projectionontoH := e⊥, P :RN → H . First of all, let us prove that the restriction to theboundary ofK is an homeomorphism. It is proved in [12, Proposition 2.6] or[11, Proposition 1.2] thatP|∂K is a Lipschitz-continuous homeomorphism ontoits image. Let us observe that in this case it is onto. To see this, we take somepoint x0 on the boundary and choose anyv ∈ H . Then sincee ∈ C, we havev+ λe � x0 for λ large enough, and thenv+ λe belongs toK. Also for the samereasonv + λe x0 for λ small enough(λ →−∞), and thenv + λe does notbelong toK (otherwisex0 ∈ v + λe+ C ⊂ K).

Then for someλ′, v+λ′e ∈ ∂K. We have seen that for anyv ∈H , we can finda pointv + λ′e belonging to∂K such thatP(v + λ′e)= v. SoP|∂K is onto.

Let us go into the construction ofF now. TakeQ= P−1|∂K and take

F(x)=Q(P(x))+ P(x)− x.

It is easily checked thatF is an involution such that

F(∂K)⊂H, F(H)⊂ ∂K.

This implies thatF is a homeomorphism fromRN onto itself that maps homeo-morphicallyH onto∂K. Thus,RN \ ∂K has two connected components whichare homeomorphic viaF to open halfplanes. If we prove thatK is a connectedcomponent ofRN \∂K, it has to be mapped onto one of the halfplanes ofR

N \H .Then, it only rests to compose with an appropriate rotation.

To prove thatK is a connected component, we only have to prove thatK ispath-connected since every path going out ofK goes through∂K (a connectedcomponent ofRN \H ∼= R

N \ ∂K is path-connected). To prove thatK is path-connected, we will prove that for everyx andy we can find a path-connected


set containing both points. Takex, y ∈ K . Then we can find anε > 0 such thatx − εe, y − εe ∈ K so x, y ∈ (x − εe + C) ∪ (y − εe + C) ∪ K , which is path-connected. (This is the union of two convex sets with nonempty intersection, sinceλe is in both cones forλ large enough.) ✷

Now, we are going to analyze the orientation preserving property. LetK be aCSUC and leth :K → K be an homeomorphism (not necessarily monotone).By the domain invariance theorem,h(K) = K and h(∂K) = ∂K. SinceK ishomeomorphic toRN and∂K to R

N−1, respectively, we have two ways to saywhenh preserves the orientation.

Proposition 2.5.The following statements are equivalent:

1. h|K : K → K preserves the orientation.2. h|∂K : ∂K → ∂K preserves the orientation.

Proof. By the last proposition we can assume that

K = {x ∈R

N | x � 0},

and leth be defined by

h(x,0)= (h(x),0

).

Now since the orientation property can be defined in terms of topological degree(see [10, IV.5.12]), we only have to prove that for any(x, x) with x > 0,

deg(x,x) h|K = degx h.

Let us defineh andh as

h(x)={h(x), x � 0,(h(x), x), x � 0,

h(x)= (h(x), x

).

Then

deg(x,x) h|K = deg(x,x) h= deg(x,−x) h= deg(x,−x) h= degx h.

First and third equalities follow from the local character of the topological degree.The second equality follows from the independence of the orientation propertyof the base point (see [10, IV.5.12] again). The last equality follows from theproduct formula sinceh= h⊕1R, where 1R is the identity map (see [10, IV.5.13,Exercise 3]). ✷

Now, we return with to dynamical system (2) withf ∈ MN . The followingdefinition was given in [13].

Definition 2.6. A nonempty subsetS of RN is called balanced if any two points

in S are not strongly comparable.(x, y ∈ S ⇒ x � y andy � x.)


Lemma 2.7. Let S be an invariant balanced set inRN . Then there exists aninvariant CSUCD such thatS ⊂ ∂D.

Proof. Let us take

U = {x ∈R

N | ∃(n, s) ∈ N× S with f n(x)� s}

and takeD = U . It is not difficult to prove, using the invariance ofS, that alsoU is invariant. HenceD is an invariant CSUC, and we only need to show thatS ⊂ ∂D.

First let observe thatS ⊂D. In factx ∈ S; then forλ > 0 small,x+ λe ∈ U sox ∈D. To show thatS ⊂ ∂D it is sufficient to notice thatS ∩ D. Then we argueby the way of contradiction; ifx ∈ D we can take a small positiveλ such thatx − λe ∈ D. Now, sinceD = U , we can takeyk → x − λe, with yk ∈ U . Thenfor k large enough,yk x, so sincef is increasing,x ∈U . This is contradictorywith the definition ofU , becausef n(x) ∈ S for all n andS is balanced. ✷Lemma 2.8. Let {xn}n∈N be a nonbalanced orbit. Then one of the followingproperties hold:

1. |xn| −→n→+∞∞.

2. xn − yn −→n→+∞0, where{yn}n∈N is a periodic orbit.

Proof. Sincexn is not balanced there existp,q such thatxp xq . Assume thatp < q ; the other case is similar. Then sincef is increasing,{xn(q−p)+p}n�0 isstrongly increasing; so, if it is bounded, it converges to aq − p periodic point(q − p need not to be the minimal period). From here the situation 2 will followeasily. Let us observe that necessarily the sequences

{xn(q−p)+p+k}n∈N (3)

with k = 1,2, . . . , q − p − 1 will also converge. Also, if one of this sequencesconverges, then all converge. The only possibility not to be in the second situationis that all sequences in (3) were unbounded. Since they are all increasing, allconverge to infinity, and we are in the first situation.✷Proof of Theorem 2.1. Let us take finallyf ∈M3. Let us assume by the wayof contradiction that there is no fixed points. We will show that all orbits areunbounded. Let us take an orbit{xn}n∈N and distinguish two cases on whether theorbit is balanced or not.

If {xn}n∈Z is balanced, using Lemma 2.7 we can takeD an invariant closedset, upper closed withxn ∈ ∂D. By Proposition 2.4,∂D is homeomorphic to theplane, and by Proposition 2.5,f|∂D preserves the orientation, so we can applyTheorem 0.3 of [3] obtaining that the orbit has no accumulation points on∂D.


By Proposition 2.4, the orbit has no accumulation points inR3, and we finish this

case.If xn is not balanced, we will apply Lemma 2.8 and observe that the situation 2

there is not possible. The argument is that a periodic orbit is always balanced(a nonbalanced invariant set contains always a strict monotone sequence) and weare in the other case.✷Remark 2.9.Theorem 2.1 and Corollary 2.2 are also valid if we considerf :K →K, with K being a CSUC ofR3. However, in the caseK = C the results are trivialbecause the monotonicity implies that the origin is a fixed point.

3. Applications

In this section we show how these results can be applied to some specificsituations. During this section, let us consider system (1) wheref is continuousandT -periodic in thet variable. We will assume uniqueness and continuation ofthe initial value problem.

Let us callf = (f1, f2, f3) andx = (x1, x2, x3). We will say that the system(1) is cooperative (respectively, competitive) if for everyi �= j , the functionfi isincreasing (decreasing) with respect to thexj variable.

Theorem 3.1.Assume thatf :R × R3 → R

3 and that the system is cooperative.Then one of the following properties holds:

1. There exists aT -periodic solution of(1).2. Every solution of(1) tends to infinity:

limt→+∞|x(t)| = +∞.

Proof. It is shown in [7] that the return map

x0 → x(T , x0)

is in this case an increasing homeomorphism. Also it preserves the orientationsince the return map is always isotopic to the identity. Then, we use Theorem 2.1and obtain that, if there are noT -periodic solutions,

limn→∞|x(nT , x0)| =∞.

Now, the thesis follows from standard arguments.✷Corollary 3.2. Assume thatf :R ×R

3 → R3 and that the system is competitive.

Then, one of the following properties holds:


1. There exists aT -periodic solution of(1).2. Every solution of(1) is unbounded.

Proof. We are in a similar situation. The inverse of the return map is an increasinghomeomorphism, so we have to apply Corollary 2.2 instead of Theorem 2.1.✷

Now we are going to apply these last results to biological systems. Takef :R × (0,∞)3 → R

3. We will say that a solutionx is persistent if there existε > 0 such thatxi(t) � ε for everyi = 1,2,3 andt � 0.

Theorem 3.3.Assume that the system is cooperative or competitive and that thereexists a persistent bounded solution. Then there exists a periodic one(this iscalled a coexistence state).

Proof. The change of variablesyi = lnxi , i = 1,2,3, reduces the problem to asystem of one of the last two results.✷Remark 3.4. Theorem 3.3 is connected with the work [3] on the competitivecase. Here we do not assume dissipativity, also the monotonicity assumptionof fi is only for i �= j and need not to be strict. However, there, they do notassume continuation of the solutions and their conclusion is stronger. Assumingdissipativity, it is easy to prove using Theorem 3.3 that either we have a periodicsolution or for every solution

limt→∞ infmin

{x1(t), x2(t), x3(t)

} = 0,

instead of the condition obtained in [3]

limt→∞min

{x1(t), x2(t), x3(t)

}= 0.

Following the ideas of [4,5] we fix a nonsingular symmetric matrixP with twonegative eigenvalues and a positive one.

Theorem 3.5.Assume that there exist a functionλ :R×R3 →R such that

P∂f

∂x(t, ξ)+ ∂f

∂x(t, ξ)∗P + λ(t, ξ)P (4)

is positive semidefinite(respectively, negative semidefinite). Then the conclusionof Theorem3.1 (respectively, Corollary3.2) holds.

Proof. Following the argument in [5] now the return map (respectively, the in-verse of the return map) is increasing with respect to an abstract order defined bythe cone

K = {ξ ∈ R

3 | (P ξ, ξ) � 0, (ξ, e+) � 0}.


Heree+ is an eigenvector corresponding to the positive eigenvalue ofP (see [5]for the detail). Now the conclusion goes as in the previous results.✷Remark 3.6. It was shown in [5] that the condition imposed in [4] is more re-strictive. In fact, in [4] it is imposed (4) forλ constant and negative. Then thesetheorems are generalizations of the results there. However, the results in [4] workin any dimension.

Acknowledgments

I thank E. Norman Dancer, Rafael Ortega and Luis A. Sánchez for many suggestions about proofsand references. I thank also Rafael Ortega for reading the first version of this paper.

References

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Massera's theorem for monotone dynamical systems in three dimensions

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