EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 4, 2019, 1350-1359
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Monotone flows with dense periodic orbits
Morris W. Hirsch
Department of Mathematics, University of Wisconsin at Madison, WI 53706, USA

Abstract. The main result is Theorem 1: A flow on a connected open set X ⊂ Rd is globally periodic
provided (i) periodic points are dense in X, and (ii) at all positive times the flow preserves the partial order
defined by a closed convex cone that has nonempty interior and contains no straight line. The proof uses the
analog for homeomorphisms due to B. Lemmens et al. [27], a classical theorem of D. Montgomery [31, 32],
and a sufficient condition for the nonstationary periodic points in a closed order interval to have rationally
related periods (Theorem 2).

2010 Mathematics Subject Classifications: 37C65, 37C25, 57Sxx

Key Words and Phrases: Monotone dynamical systems, Periodic points, Topological transformation
groups

1. Introduction

Many dynamical systems (ϕ, X), especially those used as models in applied fields, are mono-
tone: The state space X has a nontrivial (partial) order which the dynamic ϕ := {ϕt}t∈R preserves
in positive time:

y � x =⇒ ϕty � ϕt x, (t ≥ 0).

The great virtue of monotone systems is that long-term behavior of trajectories tends to be com-
paratively simple. While there can be exotic invariant sets, it is commonly the case that there are
large sets of initial states x(0) for which x(t) approaches the fixed point set as t becomes infinite.
This holds for cooperative systems (1) when the inequalities (2) are strict (Hirsch [16]).

Example. Consider a population divided into n groups labeled i = 1, . . . , n. At time t the state of
the system is characterized by a vector x(t) ∈ Rn whose i’th component xi(t) is the size (or density,
concentration, probability, etc.) of group i. The growth rate of the groups is governed by a system
of differential equations in the positive orthant Rn

+ = [0,∞)n, having Kolmogorov form ([23, 39]):

dxi

dt
= xi

n∑
j=1

gi j(x1, . . . , xn), xi ≥ 0 (i = 1, . . . , n). (1)

DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3534

Email address: mwhirsch@chorus.net (M. W. Hirsch)

http://www.ejpam.com 1350 c© 2019 EJPAM All rights reserved.



M. W. Hirsch / Eur. J. Pure Appl. Math, 12 (4) (2019), 1350-1359 1351

The system is cooperative (or “mutualist”) if the growth rate of population i tends to increase with
the size of each population j , i, modeled by

∂gi j

∂x j
≥ 0, i , j. (2)

When the functions gi j are continuously differentiable, this assumption makes the positive-time
solution process preserve the vector order on Rn determined by the cone Rn

+ := [0,∞)n:

s ≤ t =⇒ xi(s) ≤ xi(t), (i = 1, . . . , n).

If the inequality on partial derivatives in Equation (2) is reversed, the system is called competitive.
Another common dynamical property is dense periodicity: the set of periodic points is dense.

Often considered typical of chaotic dynamics, this condition is closely connected to many other
important dynamical topics: structural stability, ergodic theory, Hamiltonian mechanics, smooth-
ness and so forth. But in contrast to monotonicity, dense periodicity is usually demonstrated only
in certain compact sets.

The goal of this article is to show that flows that are both monotonic and densely periodic are
rare, because they are globally periodic: Our main result, Theorem 1, implies that such a flow
factors through an action of the circle group.

For a sampling of the large literature on monotone dynamics, consult the following works and
references therein: [1–4, 6–9, 12, 15, 20, 25, 28–30, 34, 36, 38, 40, 44–46]. Surveys of order-preserving
dynamical systems are given in [19, 26, 41, 42].

1.1. Terminology

Let Z denote the integers, N the nonnegative integers, N+ the positive integers, R the reals, and
Q+ the positive rationals. Rd is d-dimensional Euclidean space.

A subset S of a topological space Y is given the induced topology. When Y has been specified,
the closure of S is denoted by S .

Maps between topological spaces are always assumed continuous.
Let X be an ordered space: a topological space endowed with a partial order relation generally

symbolized by �, and denoted formally as (X,�). If (X′,�′) is also an ordered space, a map
T : X → X′ is monotone provided x � y =⇒ T x �′ Ty.

We write x � y as a synonym for y � x. If y � x and y , x, we write y � x, x ≺ y. For sets
A, B ⊂ X, the notation A � B means a � b for all a ∈ A, b ∈ B; and similarly for � and so forth.

The partial order is always assumed to be closed as a binary relation: The sets

{(x, y) ∈ X × X : x � y}, {(x, y) ∈ X × X : x � y}

are closed in X × X. Consequently:

• For all p ∈ X, the sets {x ∈ X : x � p} and {y ∈ X : y � p} are closed in X.

• If limi ai = a, limi bi = b, and ai � bi, then a � b.



M. W. Hirsch / Eur. J. Pure Appl. Math, 12 (4) (2019), 1350-1359 1352

The order interval [a, b] is the closed set {x ∈ X : a � x � b}; its interior is the open order
interval [[a, b]]. We write a � b to indicate [[a, b]] , ∅.

Let f : Y → Z be a map. For k ∈ N, the k’th iterate f k : y→ yk is the map defined recursively
by:

y0 = y, yk = f (yk−1) if yk−1 ∈ Y.

The fixed point set of f is
F ( f ) := {x : f (x) = x}

and the periodic set is
P( f ) :=

⋃
k F ( f k).

A flow on X is an indexed family ψ :=
{
ψt}

t∈R of homeomorphisms ψt : X ≈ X such that

ψr ◦ ψs = ψr+s, (r, s ∈ R), (3)

and the evaluation map
evψ : R × X → X, (t, x) 7→ ψt x (4)

is continuous. When X is ordered, ψ is monotone provided the maps ψt, t ≥ 0 are monotone.

Example. A flow on Rn defined by a cooperative system of differential equations (1), (2) is
monotone for the vector order

x � y ⇐⇒ xi ≥ yi, (i = 1, . . . , n).

A set Y ⊂ X is invariant under ψ if ψtY = Y for all t ∈ R, and ψ
∣∣∣Y denotes the flow in Y whose

evaluation map (4) is
evψ|Y : (t, y) 7→ ψty, (t, y) ∈ R × Y.

The orbit of x is
O(x) := {ψt x : t ∈ R},

and the orbit of a set S ⊂ X is
O(S ) :=

⋃
x∈S

O(x).

Since orbits are invariant and the flow on an orbit is transitive, distinct orbits are disjoint.
The periodic set of ψ is

P = P(ψ) :=
⋃
t>0

P(ψt)

and the equilibrium set is
E = E(ψ) :=

⋂
t∈R

F
(
ψt).

If p ∈ P \ E, its period is

per (p) = per (p, ψ) := min
{
t > 0: ψt p = p

}
> 0,



M. W. Hirsch / Eur. J. Pure Appl. Math, 12 (4) (2019), 1350-1359 1353

and O(p) is a cycle. If per(p) = r > 0, the flow ψ
∣∣∣O(p) is topologically conjugate to the flow on

the topological circle R/rZ covered by the translational flow on R. Since the flows on cycles are
transitive, every cycle is unordered: no two points are related by �.

The flow is called:

• densely periodic if P is dense in X,

• pointwise periodic if P = X,

• globally periodic if ψt is the identity map of X for some t > 0,

• monotone if ψt is monotone for all t ≥ 0.

This is the chief result:

Theorem 1 (Main). Assume:

(H1) X is a connected open set in d-dimensional Euclidean space Rd.

(H2) K ⊂ Rd is a closed cone with nonempty interior that is convex (contains the line seg-
ment joining any two of its points), solid (has nonempty interior), and pointed (contains no
straight line).

(H3) The order on X is defined as: x � y ⇐⇒ x − y ∈ K.

(H4) ϕ is a monotone flow on X.

(H5) ϕ is densely periodic.

Then ϕ is globally periodic.

2. Resonant flows

Let ψ denote a monotone flow on an arbitrary ordered space X.

Definition. A set S ⊂ X is resonant and ψ is resonant in S , provided:

a, b ∈ S ∩ P \ E =⇒ per (a)/per (b) ∈ Q+.

It is easy to see that:

• If S is resonant, so is its orbit and every subset.

• The intersection of resonant sets is resonant.

• The union of resonant sets is resonant if their intersection meets a cycle,

Theorem 2 (Resonance Criterion). Assume

p, q ∈ P \ E, [p, q] ⊂ X, p ≺ q, O(p) ⊀ O(q). (5)

Then [p, q] is resonant.



M. W. Hirsch / Eur. J. Pure Appl. Math, 12 (4) (2019), 1350-1359 1354

Proof. Set
per (p) = r > 0, per (q) = s > 0. (6)

I claim:
r/s is rational. (7)

This is trivial if r = s. To fix ideas, assume assume r < s, the case r > s being similar. Set
ξ := r/s. For all n,m ∈ Z:

p =
(
ψr)n p = ψnr p, q =

(
ψs)mq = ψmsq.

Monotonicity implies
p = ψnr p ≺ ψnrq = ψnr+msq = ψ(nξ+m)sq,

whence
p ≺ ψ(nξ+m)sq, (n,m ∈ Z). (8)

Assume per contra that ξ is irrational. Then

Λ :=
{
(nξ + m)s : n,m ∈ Z

}
.

is dense in R,∗ whence
Γ :=

{
Φtq : t ∈ Λ

}
is dense in O(q). As the partial order relation is closed, (8) implies

p � Γ ⊂ Γ = O(q). (9)

Invariance of cycles implies O(p) � O(q) by (9), monotonicity of ψ, and transitivity of ψ|O(p).
Therefore disjointness of O(p) and O(q) implies O(p) ≺ O(q). Since this contradicts the hypothe-
sis, (7) is proved.

Next we prove:

u ∈ [p, q] ∩ P \ E =⇒
per (u)
per (q)

∈ Q+. (10)

This is trivial if u ∈ E or u = q, so we assume u < E and p � u ≺ q. Note that O(u) ⊀ O(q) because
otherwise O(p) ≺ O(q), contrary to hypothesis. Therefore (10) follows from (7).

Resonance of [u, v] now follows: If u, v ∈ [p, q]∩P \ E, then applying the claim to both u and
v gives:

per (u)
per (v)

=
per (u)
per (q)

·
per (q)
per (v)

∈ Q+.

Proposition 1. Let p, q ∈ P(ψ) \ E(ψ) satisfy (5). If P(ψ) is dense in [p, q], there exists l > 0 such
that:

∗Equivalently: The orbit of a rotation of the circle S1 through an irrational multiple of π is dense in S1. This result
is ancient, going back to Nicole Oresme in the 14th century! See Grant [11], Kar [21]. A short proof based on the
pigeon-hole principle is in Speyer [43]. Stronger density theorems are in Bohr [5], Kronecker [24], Weyl [47, 48].



M. W. Hirsch / Eur. J. Pure Appl. Math, 12 (4) (2019), 1350-1359 1355

(a) P(ψ) ∩ [p, q] = P(ψl) ∩ [p, q],

(b) P(ψl) is dense in [p, q],

(c) ψl[p, q] = [p, q].

Proof. Let per (p) = r > 0. Because [p, q] is resonant (Theorem 2), if z ∈ [p, q] ∩ P(ψ) \ E(ψ)
there exists r ∈ N+ such that ψrz = z.

Since the periods of all points in [p, q] ∩ P(ψ) \ E(ψ) are rational multiples of r, there exist
m, n ∈ N+ such that:

(ψr)m p = p, (ψr)nq = q, and P(ψ) ∩ [p, q] = P(ψr) ∩ [p, q],

validating (a) and (b) for l := mnr. Monotonicity implies ψl[p, q] ⊂ [p, q], so (c) follows from (b)
and continuity of ψl.

2.1. Proof of Theorem 1

Recall the hypotheses, assumed henceforth:

(H1) X is a connected open set in d-dimensional Euclidean space Rd.

(H2) K ⊂ Rd is a closed convex cone that has nonempty interior and contains no straight line.

(H3) The partial order relation on X is determined by K:

x � y ⇐⇒ x − y ∈ K.

(H4) ϕ is a monotone flow on X.

(H5) ϕ is densely periodic.

The conclusion is: ϕ is globally periodic.

Definition. A homeomorphism T : X ≈ X is:

• densely periodic if P(T ) is dense in X,

• pointwise periodic if P(T ) = X,

• globally periodic if T k is the identity map of X for some k ∈ N+.

A crucial ingredient in the proof of Theorem 1 is the recently proved analog for monotone
homeomorphisms:

Theorem 3 (B. Lemmens et al. [27]). Assume (H1), (H2), (H3). Then a monotone homeomorphism
T : X ≈ X is globally periodic provided it is densely periodic.†

†Conjectured in M. Hirsch [18], and proved for polyhedral cones K.



REFERENCES 1356

We will also use an elegant result from the early days of transformation groups:

Theorem 4 (D. Montgomery [31, 32]). A homeomorphism of a connected topological manifold
is globally periodic provided it is pointwise periodic.‡

Proposition 2. Let x ∈ X \ E be arbitrary. There is an open neighborhood

Wx ⊂ X \ E

of x, and a real number l := lx > 0, such that:

ϕl|Wx is a globally periodic homeomorphism of Wx.

Proof. Every x ∈ X \ E has an open neighborhood Vx ⊂ X \ E that contains no orbit. If not,
there is a sequence {xk} in converging in X to x such that

t ∈ R =⇒ lim
k→∞
‖ϕt xk − xk‖ = 0,

whence t ∈ R =⇒ ϕt x = x. But this gives the contradiction x ∈ E.
As P(ϕ) is dense, there exist periodic points px, qx such that:

px � x � qx, [px, qx] ⊂ Vx, O(px) ⊀ O(qx).

Define Wx to be the open order interval [[px, qx]]. By Proposition 1 there exists l > 0 such that
ϕl|Wx is densely periodic. Therefore Theorem 3 implies ϕl|Wx is globally periodic.

To finish the proof of Theorem 1, observe that ϕ is pointwise periodic by Proposition 2. There-
fore Theorem 4 implies ϕ is globally periodic.

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