---
abstract: 'Two commonly used types of dynamics, chaotic and monotone, are compared.  It is shown that monotone maps in strongly ordered spaces do not have chaotic attracting sets.' 
author:
- |
    Morris W. Hirsch[^1]\
    Department of Mathematics\
    University of Wisconsin at Madison\
    University of California at Berkeley
title: '**On the nonchaotic nature of monotone dynamical systems**'
---

Introduction
============

This article contrasts two protean types of dynamical systems, [ *chaotic*]{} and [*monotone*]{}.  Both types occur frequently in mathematical models of applied fields, including Biology, Chemistry, Economics and Physics. 

Topological spaces, denoted by capital Roman letters, have metrics denoted by $d$. The interior of $P\subset Y$ is $\,{\ensuremath{\operatorname{\mathsf{Int}}}}\,P$. The distance from $y\in Y$ to $Q\subset Y$ is $${\ensuremath{\operatorname{\mathsf{dist}}}}\,(y,Q):=\inf\{d(y,q){\colon\thinspace}q\in Q\}.$$

Maps are assumed to be continuous.  The [*orbit*]{} of $y$ under $T{\colon\thinspace}Y\to Y$ is the set $\{y,Ty, T^2x, \dots\}$. If this orbit finite, $y$ and its orbit are [*periodic*]{}. 

A set $A\subset Y$ [*attracts*]{} the point $y \in Y$ if

$$\label{eq:1}
 \lim_{k\to\infty}d(T^ky, A) =0.$$

We call $A$ [*attracting*]{} for $T$ provided:

-   $TA\subset A$,

-   $A$ is compact and nonempty,

-   $A$ attracts every point in some open neighborhood $W$ of $A$ such that $TW\subset W$.

We call $A$ is an [*attractor*]{} for $T$ when the limit in (\[eq:1\]) is uniform in $W$. The union of the subbasins of $A$ is the [*basin*]{} of $A$, an open set invariant under $T$. An attractor is [*global*]{} if its basin is all of $Y$.

Chaotic dynamics. 
-----------------

The hallmark of a chaotic system is “sensitive dependence on initial conditions:” there exists ${\ensuremath{\delta}}>0$ such that for any two initial states $x(0), y(0)$, at some time $t>0$ the distance from $x(t)$ to $y(t)$ exceeds ${\ensuremath{\delta}}$. If a system with this feature models a natural system (e.g., weather, economy, ecology, gene system, a disease) then it cannot be used to make accurate long-term predictions. 

This was discovered by the meterologist [Edward Lorenz]{} in his seminal 1963 article, “Deterministic Non-periodic Flow” [@Lorenz63].  After drastically simplifying standard equations for fluid flow, Lorenz arrived at the system of differential equations

$$\begin{split}
  \dot x &= 10(y - z),\\
  \dot y &=  28x -y   - xz,\\
  \dot z &= xy - (8/3)z. 
\end{split}$$

Despite its simple algebraic form, Lorenz found a disturbing feature in his numerical solution:

> …two states differing by imperceptible amounts may eventually evolve into two considerably different states.  If, then, there is any error whatever in observing the present state— and in any real system such errors seem inevitable— an acceptable prediction of an instantaneous state in the distant future may well be impossible. 

Lorenz’s extensive computations convincingly illustrate this phenomenon— an unexpected problem for applied dynamical systems— there was no rigorous mathematical proof of his findings until the 1999 paper of [W. Tucker]{} [@Tucker99]. 

The term “chaos” is used in many ways in mathematical literature.  In a widely accepted definition, [ R. Devaney]{} [@Devaney86] defined a map $f{\colon\thinspace}X\to X$ to be chaotic if it has these three properties:

> [Dense periodic points:]{} [*Every nonempty open subset of $Y$ meets a periodic orbit of $f$.*]{}
>
> [Topological transitivity:]{} [*The orbit of some point of $Y$ is dense in $Y$.*]{}
>
> [Sensitivity to initial conditions:]{} [*There exists ${\ensuremath{\delta}}> 0$ such that if $x, y \in Y$ are distinct, then $\,d (f^kx - f^ky) \ge {\ensuremath{\delta}}$ for some $k>0$.*]{}

In order to avoid trivial situations we add a fourth property:[Suggested by Devaney [@Devaney19].]{} 

> [Nondiscreteness:]{} [*Y is not a finite set*]{},

When these hold I call $Y$ and $f|Y$ [*chaotic*]{}. 

[P. Touhey]{} [@Touhey98] proved that a remarkably simple condition is equivalent to Devaney’s definition:

> [Sharing of periodic orbits:]{} [*Every pair of nonempty open sets meet a common periodic orbit.*]{} 

See also [@Touhey97; @Touhey00]. 

Devaney [@DHS04 p. 324] validated Lorenz’s conclusions by constructing a Poincaré (or “first return” map) $f{\colon\thinspace}C\to   C$ for Lorenz’s differential equations, having the following properties:

-   $C\subset {\ensuremath{{\mathbb R}^{3}}}$ is an affine open 2-cell. 

-   $f$ has a chaotic global attractor. 

It is easy to see that every periodic orbit fulfills the definition of “chaotic,” but such orbits are not of much dynamical interest. 

Our main result, Theorem 1, shows that monotone maps in strongly ordered spaces have no other chaotic attracting sets. 

Monotone dynamics
-----------------

The state space of a monotone systems is a space $X$ endowed with a (partial) order, denoted by $\,\preceq$. The set $\{(x,y) \in X\times X{\colon\thinspace}x\preceq y\}$ is assumed to be closed. 

We write $x\prec y$ if $x\preceq y$ and $x\ne y$. If $A$ and $B$ are sets, $$A\prec B {\ensuremath{\:\Longleftrightarrow\:}}a\prec b, \quad (a\in A,\, b\in B).$$ When $A$ is a singleton $\{a\}$ we also write $a \prec B$.

In the main result the ordered space $X$ is [*strongly ordered*]{}: If $W\subset X$ is a neighborhood of $x$, there are nonempty open sets $U, V \subset W$ such that $U\prec  x \prec V$.

[*Examples.*]{}  Euclidean space ${\ensuremath{{\mathbb R}^{n}}}$ is strongly ordered by the classical [*vector order*]{}: $$x\preceq y {\ensuremath{\:\Longleftrightarrow\:}}x_j\le y_j, \ (j=1,\dots,n).$$ Many Banach spaces of real-valued functions are strongly ordered by the functional order, $f_1 \preceq f_2$ iff $f_1 x\le  f_2 x$ for all $x$ in the domain.  These spaces include the spaces of $C^k$ functions on compact manifolds. 

For $p >0$, ${{\mathcal {L}}}_p$ spaces of real-valued functions have the order $f_1\preceq f_2$ iff $f_1 \le f_2$ almost everywhere.  These spaces are rarely strongly ordered. 

A map $T$ between ordered spaces is [*monotone*]{} if $$x\preceq y  \implies Tx\preceq Ty.$$

In dynamical models from several scientific fields, including biology, chemistry, physics, economics, the set of states is an ordered space $S$, with the order reflecting the relative “size” of states— density, population, etc.  The evolution of states over time is modeled by a dynamical system on $S$— a family $F=\{F_t\}$ of maps between subsets of $S$, closed under composition.  The time variable $t$ varies over either real numbers or integers. 

Monotonicity means that the different species cooperate: an increase in the growth rate of one tends to increase the sizes of the others.  In many real-world settings this is plausible.  For example: sheep and grass cooperate, in that grass feeds the sheep and sheep fertilize grass. 

In many cases the maps $F_t$ are monotone for $t\ge 0$, and $\Phi$ is called a monotone system.  A typical example is a system of differential equations in the positive orthant $\Rnp:= [0,\infty)^n$: $${\ensuremath{\frac{dx_i}{dt}}} = x_iG_i (x_1,\dots,x_n), x_i \ge 0, \quad
i=1,\dots,n,$$ modeling an ecology of $n$ interacting species.  Here $x_i$ and $G_i$ are proxies for the size and the [*per capita*]{} growth rate of species $i$. The state space is $\Rnp$ with the vector order: $$x \preceq y {\ensuremath{\:\Longleftrightarrow\:}}x_i\le y_i \quad \text{\em for all $i$}.$$ Monotonicity is readily established when the partial derivatives of the $G_i$ are continuous, and $$i \ne j \implies  {\ensuremath{\frac{\partial G_i}{\partial x_j}}} \ge 0.$$ If each species reproduces only once a year, the ecology is modeled by a map $T{\colon\thinspace}\Rnp\to \Rnp$, and the dynamics is monotone provided the partial derivatives of $T$ are continuous and nonnegative. 

Monotone dynamical systems often permit reliable predictions of behavior.  In many case it can be proved that typical trajectories have predictable fates, such tending toward fixed points or periodic orbits.  See for example references [@BasLemmens10; @DELEENHEER17; @DIRR; @ENCISOSONTAG06; @Hirsch82; @Hirsch83; @Hirsch88; @Hirsch17; @HS05; @LajmanovichYorke; @Landsberg96; @Potsche15; @Selgrade80; @Smale76; @Smith95; @Zhao94].  The recent survey by [H. Smith]{} [@Smith17] has an extensive bibliography. 

Monotonicity and chaos play quite different roles in applied dynamics.  Monotonicity is sometimes deliberately postulated, and can ordinarily be deduced from the form of defining equations without extensive computations.  Monotonicity is useful because it usually leads to predictable long-term behavior. 

But chaos is undesirable: it makes accurate long-term prediction is impossible, and is quite difficult to either prove or disprove.  But as Lorenz discovered, simple models of realistic systems can be chaotic. 

Results
-------

Our main result is very simply stated:

> [*A monotone map in a strongly ordered space cannot have a chaotic attractor.*]{} 

In fact slightly more is true:

Assume $X$ is strongly ordered, $\,T{\colon\thinspace}X\to X$ is monotone, $\,A\subset X$ is attracting for $T$, and one of the following conditions is satisfied:

(a)

:   Periodic points are dense in $A$. 

(b)

:   Some orbit is dense in $A$. 

Then $A$ is a periodic orbit, and therefore not chaotic. 

We rely on a [*deus ex machina*]{}, [@Hirsch85 Theorem 4.1]:

> [ *Some periodic orbit $O\subset A$ is attracting for $T{\colon\thinspace}X\to X$.*]{}

We show that $A=O$. Assume [*per contra*]{} $A\ne O$. Let $W\subset X$ be the attractor basin of $O$. Then $$\label{eq:2}
 x\in W \implies \lim_{k\to\infty}{\ensuremath{\operatorname{\mathsf{dist}}}}\,(T^kx, O) =0.$$ But (\[eq:2\]) cannot be true if $A\ne O$: There exists $x\in W\cap
A\,\verb=\=O$ whose orbit is periodic when (a) holds, or dense in the nonempty open set $A{\ensuremath{\setminus O}}$ when (b) holds. 

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[^1]: I am grateful to Professor Robert Devaney for helpful information and comments.