The aim of this section is to apply Theorem 4.8 and Theorem 4.11 to the autonomous delay differential equation (4.2) to conclude that the generic solution converges to equi-librium. To Φ, defined by (4.3), we associate C, S and E, denoting respectively the sets of convergent, stable and equilibrium points. The main result of this section is the following.
Theorem 4.12 Let f ∈ C1(D), (4.2) be cooperative on the p-convex open subset D of C or Cr and satisfy:
(a) the hypotheses of Theorem 4.8 or of Theorem 4.11 hold.
(b) every positive semiorbit of Φ has compact closure in D and D = AC ∪ BC.
Then
(i) C ∩ S contains a dense open subset of D, consisting of points whose trajectories converge to equilibria.
(ii) If E is compact there is a stable equilibrium, and an asymptotically stable equilib-rium when E is finite.
Proof: For definiteness, suppose that (4.2) is cooperative on the p-convex open subset D of C and that the hypotheses of Theorem 4.8 hold. The other case is proved similarly. As-sumption (a) ensures that Φ is eventually strongly monotone. Moreover, the derivative of Φt(φ) with respect to φ exists and Φ0τ(φ)χ = yτ(t0, χ), where y(t, t0, χ) is the solution of the variational equation (4.8). As our hypotheses ensure that L(t) = ∂f∂φ(xt(φ)) sat-isfies (STD), we conclude from Theorem 4.6 that Φ0τ(φ) is strongly positive for τ ≥ 3r.
Compactness of Φ0τ(φ) : C → C for τ ≥ r follows from the fact that a bound for yτ(t0, χ), uniform for χ belonging to a bounded set B ⊂ C, can be readily obtained so, using (4.8), we may also find a uniform bound for y0(t, t0, χ), τ − r ≤ t ≤ τ . See e.g. Hale [58], Theorem 4.1.1 for more detail.
The hypotheses of Theorem 2.26, with X = D, are fulfilled: D is normally ordered and D = BC ∪ AC; while (M) and (D∗) hold as noted above. Therefore Theorem 2.26 implies the conclusion.
In the special case that (4.2) is scalar (n = 1) we note that the set E of equilibria is totally ordered in Cr or C so the set of quasiconvergent points coincides with the set of convergent points: Q = C. The classical scalar delay differential equation (4.1) has been thoroughly investigated in the case of monotone delayed feedback (f (0) = 0 and f0 > 0) by Krisztin et al. [107]. They characterize the closure of the unstable manifold of the trivial solution in case it is three dimensional and determine in remarkable detail the dynamics on this invariant set.
Smith and Thieme [200, 201, 193] introduce an exponential ordering, not induced by a cone in IRn, that extends the scope of application of the theory described here.
One of the salient results from this work is that a scalar delay equation for which the product of the delay r and the Lipschitz constant of f is smaller than e−1 generates an eventually strongly monotone semiflow with respect to the exponential ordering and therefore the generic orbit converges to equilibrium: the dynamics mimics that of the associated ordinary differential equation obtained by ignoring the delay. See also work of Pituk [155].
We have considered only bounded delays. Systems of delay differential equations with unbounded and even infinite delay are also of interest. See Wu [235] for extensions to such systems. Wu and Freedman [235] and Krisztin and Wu [104, 105, 106] extend the theory to delay differential equations of neutral type.
5 Monotone Maps
5.1 Background and Motivating Examples
One of the chief motivations for the study of monotone maps is their importance in the study of periodic solutions to periodic quasimonotone systems of ordinary differential equations. See for example the monograph of Krasnoselskii [98], the much cited paper of de Mottoni and Schiaffino [41], Hale and Somolinos [57], Smith [188, 189], Liang and Jiang [118], and Wang and Jiang [228, 229, 230]. To fix ideas, let f : IR × IRn → IRn be a locally Lipschitz function and consider the ordinary differential equation
x0 = f (t, x) (5.1)
As usual, denote by x(t, t0, x0) the non-continuable solution of the initial value problem x(t0) = x0, which for simplicity we assumed is defined for all t. If f is periodic in t of period one: f (t + 1, x) = f (t, x) for all (t, x), then it is natural to consider the period map T : IRn→ IRn defined by
T (x0) = x(1, 0, x0). (5.2)
Its fixed points (periodic points) are in one-to-one correspondence with the periodic (subharmonic) solutions of (5.1). If K is a cone in IRn for which f satisfies the quasi-monotone condition (QM), then it follows from Theorem 3.2 that T is a quasi-monotone map:
x ≤ y implies T x ≤ T y. Moreover, T has the important property, not shared with general monotone maps, that it is an orientation-preserving homeomorphism.
In a similar way, periodic solutions for second order parabolic partial differential equations with time-periodic data can be analyzed by considering period maps in ap-propriate function spaces. Here monotonicity comes from classical maximum principles.
Hess [62] remains an up-to-date survey. See also Alikakos et al. [3] and Zhao [245]. Re-markable results are known for equations on a compact interval with standard boundary conditions. Chen and Matano [24] show that every forward (backward) bounded solu-tion is asymptotic to a periodic solusolu-tion; Brunovsky et al. [22] extend the result to more general equations. Chen et al. [25] give conditions for the period map to generate Morse-Smale dynamics and thus be structurally stable. Although monotonicity of the period map is an important consideration in these results, it is not the key tool. The fact that the number of zeros on the spatial interval of a solution of the linearized equation is non-increasing in time is far more important. See Hale [60] for a nice survey.
A different theme in order-preserving dynamics originates in the venerable subject of nonlinear elliptic and parabolic boundary value problems.
A different theme in order-preserving dynamics originates in the venerable subject of nonlinear elliptic and parabolic boundary value problems. The 1931 edition of Courant and Hilbert’s famous book [34] refers to a paper of Bieberbach in G¨ottingen Nachrichten, 1912 dealing with the elliptic boundary value problem ∆u = eu in Ω , u|∂Ω = f, in a planar region Ω. A solution is found by iterating a monotone map in a function space.
Courant and Hilbert extended this method to a broad class of such problems. Out of this technique grew the method of “upper and lower solutions” (or “supersolutions and subsolutions”) for solving, both theoretically and numerically, second order elliptic PDEs (see Amann [4], Keller and Cohen [92], Keller [93, 94], Sattinger [172]). Krasnoselskii
& Zabreiko [100] trace the use of positivity in functional analysis— closely related to monotone dynamics— to a 1924 paper by Uryson [221] on concave operators. The systematic use of positivity in PDEs was pioneered Krasnoselskii & Ladyshenskaya [99]
and Krasnoselskii [97] .
Amann [5] showed how a sequence {un} of approximate solutions to an elliptic prob-lem can be viewed as the trajectory {Tnu0} of u0 under a certain monotone map T in a suitable function space incorporating the boundary conditions, with fixed points of T being solutions of the elliptic equation. The dynamics of T can therefore be used to investigate the equation. Thus when T is globally asymptotically stable, there is a unique solution; while if T has two asymptotically stable fixed points, in many cases degree theory yields a third fixed point. As Amann [6] emphasized, a few key properties of T — continuity, monotonicity and some form of compactness— allow the theory to be efficiently formulated in terms of monotone maps in ordered Banach spaces.
Many questions in differential equations are framed in terms of eigenvectors of linear and nonlinear operators on Banach spaces. The usefulness of operators that are positive in some sense stems from the theorem of Perron [154] and Frobenius [48], now almost a century old, asserting that for a linear operator on IRn represented by a matrix with positive entries, the spectral radius is a simple eigenvalue having a positive eigenvector, and all other eigenvalues have smaller absolute value and only nonpositive eigenvectors.
In 1912 Jentsch [84] proved the existence of a positive eigenfunction with a positive eigenvalue for a homogeneous Fredholm integral equation with a continuous positive kernel.
In 1935 the topologists Alexandroff and Hopf [2] reproved the Perron-Frobenius theorem by applying Brouwer’s fixed-point theorem to the action of a positive n × n matrix on the space of lines through the origin in IRn+. This was perhaps the first explicit use of the dynamics of operators on a cone to solve an eigenvalue problem. In 1940 Rutman [169] continued in this vein by reproving Jentsch’s theorem by means of Schauder’s fixed-point theorem, also obtaining an infinite-dimensional analog of Perron-Frobenius, known today as the Krein-Rutman theorem [103, 213]. In 1957 G. Birkhoff [20] initiated the dynamical use of Hilbert’s projective metric for such questions.
The dynamics of cone-preserving operators continues to play an important role in functional analysis; for a survey, see Nussbaum [145, 146]. One outgrowth of this work has been a focus on purely dynamical questions about such operators; some of these results are presented below. Polyhedral cones in Euclidean spaces have lead to inter-esting quantitative results, including a priori bounds on the number of periodic orbits.
For recent work see Lemmens et al. [116], Nussbaum [147], Krause and Nussbaum [102], and references therein.
Monotone maps frequently arise as mathematical models. For example, the discrete
Lotka-Volterra competition model (see May and Oster [133]):
(un+1, vn+1) = T (un, vn) := (unexp[r(1 − un− bvn)], vnexp[s(1 − cun− vn)]) generates a monotone dynamical system relative to the fourth-quadrant cone only when the intrinsic rate of increase of each population is not too large (r, s ≤ 1) and then only on the order interval [0, r−1]×[0, s−1] (Smith [196]). Fortunately in this case, every point in the first quadrant enters and remains in this order interval after one iteration. As is typical in ecological models, the Lotka-Volterra map is neither injective nor orientation-preserving or orientation-reversing. For monotone maps as models for the spread of a gene or an epidemic through a population, see Thieme [218], Selgrade and Ziehe [177], Weinberger [231], Liu [120] and the references therein.
5.2 Definitions and Basic Results
A continuous map T : X → X on the ordered metric space X is monotone if x ≤ y ⇒ T x ≤ T y, strictly monotone if x < y ⇒ T x < T y, strongly monotone if x < y ⇒ T x T y, and eventually strongly monotone if whenever x < y, there exists n0 ≥ 1 such that Tnx Tny. We call T strongly order-preserving (SOP) if T is monotone, and whenever x < y there exist respective neighborhoods U, V of x, y and n0 ≥ 1 such that n ≥ n0 ⇒ TnU ≤ TnV .1 As with semiflows, eventual strong monotonicity implies the strong order preserving property.
The orbit of x is O(x) := {Tnx}n≥0, and the omega limit set of x is ω(x) :=
T
k≥0O(Tkx). If O(x) has compact closure, ω(x) is nonempty, compact, invariant (that is, T ω(x) = ω(x)) and invariantly connected. The latter means that ω(x) is not the disjoint union of two closed invariant sets [115].
If T (x) = x then x is a fixed point or equilibrium. E denotes the set of fixed points.
More generally, if Tkx = x for some k ≥ 1 we call x periodic, or k-periodic. The minimal such k is called the period of x (and O(x)).
Let Y denotes an ordered Banach space with order cone Y+. A linear operator A ∈ L(Y ) is called positive if A(Y+) ⊂ Y+ (equivalently, A is a monotone map) and strongly positive if A(Y+\ {0}) ⊂ Int Y+) (equivalently, A is a strongly monotone map).
The following result is useful for proving smooth maps monotone or strongly mono-tone:
Lemma 5.1 Let X ⊂ Y be a p-convex set and f : X → Y a locally C1 map with quasiderivative h : U → L(Y ) defined on an open set U ⊂ Y . If the linear maps h(x) ∈ L(Y ) are positive (respectively, strongly positive) for all x ∈ U , then f is monotone (respectively, strongly monotone).
1Our use of “strongly order-preserving” conflicts with Dancer & Hess [37], who use these words to mean what we have defined as “strongly monotone”. Our usage is consistent with that of several authors. Tak´aˇc [207, 208] uses “strongly increasing” for our SOP.
Proof By p-convexity it suffices to prove that every p ∈ X has a neighborhood N such that f |N ∩ X is monotone (respectively, strongly monotone). We take N to be an open ball in U centered at p. Suppose p + z ∈ X ∩ N, z > 0. By p-convexity, X ∩ N contains the line segment from p to p + z. The definition (above Lemma 2.15) of locally C1 implies that the map g : [0, 1] → Y, t 7→ f (p + tz) is C1 with g0(t) = h(tz)z. Therefore
f (p + z) − f (z) = g(0) − g(1) = Z 1
0
g0(t)dt = Z 1
0
h(tz)zdt
Because h(tz) ∈ L(Y ) is positive and z > 0, we have h(tz)z ∈ Y+, and therefore f (p + z) − f (p) ≥ 0. If the operators h(tz) are strongly positive, f (p + z) − f (p) 0.
Proposition 5.2 (Nonordering of Periodic Orbits) A periodic orbit of a mon-otone map is unordered.
Proof: If not, there exists x in the orbit such that Tk(x) > x for some k > 0. Induction on n shows that Tnk(x) > x for all n > 0. But if x has period m > 0, induction on k proves that Tmk(x) = x.
Lemma 5.3 (Monotone Convergence Criterion) Assume T is monotone and O(z) has compact closure. If m ≥ 1 is such that Tmz < z or Tmz > z then ω(z) is an m-periodic orbit.
Proof: Consider first the case m = 1. Compactness of O(z) implies the decreasing sequence {Tkz} converges to a point p = ω(x). Now suppose m > 1. Applying the case just proved to the map Tm, we conclude that {Tkmz} converges to a point p = Tm(p).
It follows that ω(z) = {p, T p, T2p, · · · , Tm−1p}.
Lemma 5.3 yields information on one-sided stability of compact limit sets when T is SOP; see Hirsch [69].
In order to state the following lemma succinctly, we call a set J ⊂ N an interval if it is nonempty and contains all integers between any two of its members. For a, b ∈ N we set [a, b] = {j ∈N : a ≤ j ≤ b} (there will be no confusion with real intervals). Two intervals overlap if they have more than one point in common.
Let J ⊂ N be an interval and f : J → X be a map. A subinterval [a, b] ⊂ J, a < b is rising if f (a) < f (b), and falling if f (b) < f (a).
Theorem 5.4 A trajectory of a monotone map cannot have both a rising interval and a falling interval.
Proof: Follows from Theorem 1.6.
Lemma 5.5 If T is monotone, ω(z) cannot contain distinct points having respective neighborhoods U, V such that Tr(U ) ≤ Tr(V ) for some r ≥ 0.
Proof: Follows from Theorem 5.4 (see proof of Lemma 1.7).
The next result is fundamental to the theory of monotone maps:
Theorem 5.6 (Nonordering Principle) Let ω(z) be an omega limit set for a mono-tone map T .
(i) No points of ω(z) are related by .
(ii) If ω(z) is a periodic orbit or T is SOP, no points of ω(z) are related by <.
Proof: Follows from Proposition 5.2 and Lemma 5.5 (see the proof of Theorem 1.8).
Call x convergent if ω(x) is a fixed point, and quasiconvergent if ω(x) ⊂ E. Just as for semiflows, Proposition 5.6 leads immediately to a convergence criterion:
Corollary 5.7 Assume Φ is SOP.
(i) If an omega limit set has a supremum or infimum, it reduces to a single fixed point.
(ii) If the fixed point set is totally ordered, every quasiconvergent point with compact orbit closure is convergent.
Proof: Part (i) follows from Theorem 5.6(ii), since the supremum or infimum, if it exists, belongs to the limit set. Part (ii) is a consequence (i).
Failure of the Limit Set Dichotomy We now point out a significant difference between strongly monotone maps and semiflows:
The Limit Set Dichotomy fails for strongly monotone maps.
Recall that for an SOP semiflow with compact orbit closures, the dichotomy (Theo-rem 1.16) states:
If a < b, either ω(a) < ω(b) or ω(a) = ω(b) ⊂ E.
Takac [210], Theorem 3.10, gives conditions on strongly monotone maps under which a < b implies that either ω(a)∩ω(b) = ∅ or ω(a) = ω(b). He also gives a counterexample showing that ω(a) ∩ ω(b) = ∅ does not imply ω(a) < ω(b), nor does ω(a) = ω(b) imply that these limit sets consist of fixed points (they are period-two orbits in his example).
However, the mapping in his example is defined on a disconnected space.
For any map T in a Banach space, having an asymptotically stable periodic point p of period > 1, the Limit Set Dichotomy as formulated above must fail: take a point q > p so near to p that O(p) = ω(p) = ω(q). Clearly ω(p), being a nontrivial periodic orbit, contains no fixed points. Thus the second assertion of the Limit Set Dichotomy fails in this case.
Dancer and Hess [37] gave a simple example in IRkfor prime k of a strongly monotone map with an asymptotically stable periodic point of period k which we describe below.
Therefore the second alternative of the Limit Set Dichotomy can be no stronger than that ω(a) = ω(b) is a periodic orbit.
The Limit Set Dichotomy fails even for strictly monotone maps in IR2. Let f (x) = 2 arctan(x), let a > 0 be its unique positive fixed point, and note that 0 < f0(a) < 1.
Define T0 : IR2 → IR2 by T0(x, y) := (f (y), f (x)). Then E = {(−a, −a), (0, 0), (a, a)}
since f has no points of period 2. The fixed points of T02 are the nine points obtained by taking all pairings of −a, 0, a. An easy calculation shows that {(−a, a), (a, −a)} is an asymptotically stable period-two orbit of T0 because the Jacobian matrix of T02 is f0(a)2 times the identity matrix. T0 is strictly monotone but not strongly monotone.
Now consider the perturbations T(x, y) := T0(x, y) + (x, y). It is easy to see that T
is strongly monotone for > 0; and by the implicit function theorem, for small > 0, T has an asymptotically stable period-two orbit O(p) with p near (−a, a). As noted in [37], this example can be generalized to IRk for prime k.
Tak´aˇc [211] shows that linearly stable periodic points can arise for the period map associated with monotone systems of ordinary and partial differential equations. Other counterexamples for low dimensional monotone maps can be found in Smith [196, 195].
As we have shown, asymptotically stable periodic orbits that are not singletons can exist for monotone, even strongly monotone maps. Later we will show that the generic orbit of a smooth, dissipative, strongly monotone map converges to a periodic orbit.
Here, we show that every attractor contains a stable periodic orbit.
Recall that a point p is wandering if there exists a neighborhood U of p and a positive integer n0 such that Tn(U ) ∩ U = ∅ for n > n0. The nonwandering set Ω, consisting of all points q that are not wandering, contains all limit sets. In the following, we assume that X is an open subset of the strongly ordered Banach space Y and T : X → X is monotone with compact orbit closures. The following result is adapted from Hirsch [70], Theorems 4.1, 6.3.
Theorem 5.8 If T is strongly monotone and K is a compact attractor, then K contains a stable periodic orbit.
The proof relies on the following result that does not use strong monotonicity nor that K attracts uniformly:
Theorem 5.9 Let p ∈ K be a maximal (resp., minimal) nonwandering point. Then p is periodic, and every neighborhood of p contains an open set W p (resp., W p) such that ω(x) = O(p) for all x ∈ W .
Proof Suppose K attracts the open neighborhood U of K and fix y p, y ∈ U . Since p is nonwandering there exists a convergent sequence xi → p and a sequence ni → ∞ such that Tnixi → p. For all large i, xi ≤ y. Passing to a subsequence, we assume that Tniy → q. By monotonicity and xi ≤ y for large i, we have q ≥ p. But q ∈ K ∩ Ω and the maximality of p requires q = p. Since p y and Tniy → p it follows that Tmy y for some m. Lemma 5.3 implies that ω(y) is an m-periodic orbit containing p. As this holds for every y p, the result follows.
Lemma 5.10 Let p, q ∈ K be fixed points such that p q, p is order stable from below, and q is order stable from above. Then K ∩ [p, q] contains a stable equilibrium.
Proof Let R be a maximal totally ordered set of fixed points in K ∩ [p, q]. An argument similar to the one in the proof of Theorem 1.30 shows that the fixed point
e := inf{z ∈ E ∩ R : z is order stable from above}
is order stable. That e is stable follows from the analog of Proposition 1.28 .
Proof of Theorem 5.8 Theorem 5.9 shows that some iterate Tn, n ≥ 0 has fixed points p, q as in Lemma 5.10, which result therefore implies Theorem 5.8,
Jiang and Yu, Theorem 2, [89] implies that if T is analytic, order compact with strongly positive derivative, then K must contain an asymptotically stable periodic orbit.