Howard University, Apr 8, 2024

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The Graph of a
Dynamical System

Joint work with Jim Yorke

Roberto De Leo

roberto.deleo@howard.edu

Howard University
Washington, DC (USA)

Acknowledgment: This material is based upon work supported by the National Science Foundation under Grant DMS-2308225.
Disclaimer: Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors
and do not necessarily reflect the views of the National Science Foundation.

Setup
&
Goals

Semi-flows

By semi-flow $F$ on a metric space $X$

we mean a continuous map

$$ F: \Bbb T\times X\to X, $$

$$ \text{where }\;\Bbb T=0,1,2,\dots\text{ (discrete time)} $$

$$ \text{or }\Bbb T=[0,\infty)\text{ (continuous time)}, $$ such that:

  1. $F^t(F^s(x)) = F^{t+s}(x)$;
    2.
  2. $F^0(x) = x$.
    3.

Dynamical Systems

By Dynamical System

we mean the action of a semi-flow $F$

on a metric space $X$.

Examples

Discrete-time

Iterations of a map $f:X\to X$

on a metric space $X$

$$F^n(x) = \underbrace{f\circ\dots\circ f}_{n}(x)$$

$n=0,1,2,\dots$

Examples

Continuous-time, finite dimension

Flow of an ODE $\dot x=v(x)$ on a manifold $X$

$$\frac{d}{dt}F^t(x_0)\Bigg|_{t=t_0} = v(F^{t_0}(x_0))$$

i.e.

$x(t)=F^t(x_0)$

is the unique solution of the ODE

such that $x(0)=x_0$

Examples

Continuous-time, infinite dimension

Semi-flow of a reaction-diffusion PDE

on a Banach space $X$ (e.g. $X=C^0(M)$).

$$\frac{d}{dt}F^t(u_0)\Bigg|_{t=t_0} = \partial_{xx}F^{t_0}(u_0) + f(F^{t_0}(u_0))$$ i.e.

$u(t) = F^t(u_0), t\in[0,\infty)$,

is the unique solution of the PDE with $u(0)=u_0$

Recurrence

Periodic orbits are the simplest type
of compact invariant set.

Recurrent points are a first generalization of periodic points: $x$ is recurrent if there is a $t_n\to\infty$ such that $F^{t_n}x\to x$.

Recurrence has been generalized in several ways: non-wandering points (Birkhoff), generalized recurrence (Auslander), chain recurrence (Conley), strong chain recurrence (Easton) and more.

Our goals

  1. Shifting the focus from asking
    "is point $x$ recurrent under $F$?"
    to asking
    "is point $y$ downstream from point $x$ under $F$?"

  2. Showing how the "being downstream" relation can be encoded into a graph. Hence, this graph will contain all qualitative info on the system.

  3. Proving some important general property about these graphs.

Streams

The orbit relation ${\cal O}_F$

It is natural to say that

$y$ is downstream from $x$ under $F$

when $y$ is in the orbit of $x$ under $F$,

i.e. $y=F^t(x)$ for some $t\geq0$.

Symbolically, we write $x\succcurlyeq_{F}y$.

We set

$$ {\cal O}_F = \{(x,y)\,|\,x\succcurlyeq_{F}y\}. $$

Periodic points

${\cal O}_F$ is a quasi order on $X$,

i.e. a binary relation that is transitive and reflexive:

  1. if $x\succcurlyeq_{F}y$ and $y\succcurlyeq_{F}z$, then $x\succcurlyeq_{F}z$;

  2. $x\succcurlyeq_{F}x.$

Periodic points under $F$ satisfy an interesting property:

$x$ is periodic iff

either $x$ is fixed or there is $y\neq x$ s.t.

$x\succcurlyeq_{F}y$ and $y\succcurlyeq_{F}x$.

The relation $\overline{{\cal O}_F}$

Closedness is a desirable property
for a "being downstream" relation:

if $x_n\to x_\infty$, $y_n\to y_\infty$ and $x_n\succcurlyeq_{F}y_n$ for all $n$,

we would like that

$x_\infty\succcurlyeq_{F}y_\infty$.

A natural way to get this

is to consider the relation $\overline{{\cal O}_F}$.

The relation $\overline{{\cal O}_F}$
$(x,y)\in\overline{{\cal O}_F}$

if and only if

for each $\varepsilon$, there is an orbit of $F$ starting within $\varepsilon$ from $x$ that ends up in finite time within $\varepsilon$ from $y$.

Non-wandering points

Problem: $\overline{{\cal O}_F}$ is not transitive!

Let us write $x\succcurlyeq_{NW_F}y$ for $(x,y)\in\overline{{\cal O}_F}$.

What are the "$\overline{{\cal O}_F}$-recurrent" points?
These would be the $x$ such that either $x$ is fixed or
there is a $y\neq x$ s.t. $x\succcurlyeq_{NW_F}y$ and $y\succcurlyeq_{NW_F}x$.

Since $\overline{{\cal O}_F}$ is not transitive, though, we need to add a natural condition: for every $\varepsilon>0$, the two orbits segments must be part of the same orbit.

Non-wandering points

Problem: $\overline{{\cal O}_F}$ is not transitive!

Let us write $x\succcurlyeq_{NW_F}y$ for $(x,y)\in\overline{{\cal O}_F}$.

What are the "$\overline{{\cal O}_F}$-recurrent" points?
Either $x$ is fixed or, for every $\varepsilon>0$, there is an orbit starting within $\varepsilon$ from $x$
that comes back within $\varepsilon$ from $x$.

Hence, the set of "$\overline{{\cal O}_F}$-recurrent" points
coincides with the Non-Wandering set of $F$.

Non-wandering points

The "$\overline{{\cal O}_F}$-periodic orbits"
are the sets of all points that "belong to the same pond".

These are precisely the indecomposable closed transitive invariant subsets that Smale introduced in his study of the qualitative dynamics of Axiom-A diffeomorphisms.

Non-wandering points
In general, these sets can have non-empty intersection.

This happens for instance in case of the logistic map at the right endpoint of each periodic window.

In that case, a repelling invariant Cantor set intersects the map's chaotic attractor.

Streams

Transitivity is a desirable property
for a "being downstream" relation.
Definition (Yorke & DL).
A $F$-stream is
a closed quasi-order on $X$ that contains ${\cal O}_F$.

 

 

 

Given a $F$-stream $S$,

we write $x\succcurlyeq_{S}y$ for $(x,y)\in S$.

Note that $x\succcurlyeq_{F}y$ implies $x\succcurlyeq_{S}y$.

Streams

Easy to check:
  1. $X\times X$ is a $F$-stream for any $F$;
  2. the intersection of $F$-streams is a $F$-stream.
Corollary 1.
Given any $R\subset X\times X$, there is a smallest $F$-stream containing $R$.
In particular, every $F$ has a smallest $F$-stream.
Corollary 2.
Every $F$-stream contains $\overline{{\cal O}_F}$.

Streams' nodes

Definition (Yorke & DL).
Let $S$ be a stream. A point $x$ is $S$-recurrent if either
$x$ is fixed or
there is a $y\neq x$ s.t. $x\succcurlyeq_{S}y$ and $y\succcurlyeq_{S}x$.

 

 

 

We denote by ${\cal R}_S$ the set of all recurrent pts of $S$.
We call node each maximal set of points
that are downstream and upstream from each other.
So, nodes of a stream
are a natural generalization of periodic orbits.
Like distinct periodic orbits,
distinct nodes cannot have points in common.

Graph of a stream

Definition (Yorke & DL).
Let $S$ be a stream. The graph $\Gamma_S$ of $S$ is the directed graph whose nodes are the nodes of $S$ and such that there is an edge from node $A$ to node $B$ if and only if $A\succcurlyeq_{S}B$.

Examples
of streams
and their graphs

Chain-recurrence

Definition (Bowen).
A $\varepsilon$-chain from $x$ to $y$ is a finite sequence $x_1,\dots,x_n$ such that:
  1. $x_1=x$;
  2. $x_n=y$;
  3. $d(x_i,x_{i+1})<\varepsilon$.

 

 

 

 

Write $x\succcurlyeq_{C}y$ if, for every $\varepsilon>0$,
there exists a $\varepsilon$-chain from $x$ to $y$.

This relation is a stream
that we denote $C$ after Charles Conley.

Stream vs recurrent set

While it is important to know which points are ($S$-)recurrent, that is not the whole story.
Take these two semiflows on the interval $[0,1]$:
  1. the identity map;
  2. the logistic map $f(x)=4x(1-x)$;

In both cases, every point is chain-recurrent.
Nodes, though, are very different.

In case (1), each pt is fixed and so each pt is a node.

The graph is the interval $[0,1]$ itself.

Stream vs recurrent set

While it is important to know which points are ($S$-)recurrent, that is not the whole story.
Take these two semiflows on the interval $[0,1]$:
  1. the identity map;
  2. the logistic map $f(x)=4x(1-x)$;

In both cases, every point is chain-recurrent.
Nodes, though, are very different.

In case (2), each point is chain-downstream from every other point and so there is a single node.

The graph here consists of a single point.

Stream vs recurrent set

While it is important to know which points are ($S$-)recurrent, that is not the whole story.
Take these two semiflows on the interval $[0,1]$:
  1. the identity map;
  2. the logistic map $f(x)=4x(1-x)$;

In both cases, every point is chain-recurrent.
Nodes, though, are very different.

In both cases, there are no edges.

Logistic map

Definition (Yorke & DL).
A graph $\Gamma$ is a tower if $\Gamma$ has no loops and there is an edge between each pair of nodes.
Theorem (Yorke & DL).
The graph of the logistic map is a tower.
Remark. This tower has a countable infinity of nodes for uncountably many parameter values.
This is the most complicated graph of a dynamical system in literature to date -- because the logistic map is simple enough to analyze!

Chafee Infante PDE

(toy model for reaction-diffusion PDEs)
$$\partial_t u = \partial_{xx}u + \lambda u(1-u^2),$$

$$\lambda\geq0$$

$$u=u(t,x),$$

$$x\in[0,\pi], t\geq0$$

$$u(t,0)=u(t,\pi)=0,$$

$$\;\;u|_{t=0}=g(x)$$

Fiedler & Rocha,
Heteroclinic orbits of semilinear parabolic equations,
J. of Diff. Eqs. (1996)
Chafee & Infante,
A bifurcation problem for a nonlinear parabolic PDE,
Applicable An. (1974)

Semiflows
with
"compact dynamics"
have a
connected graph

Compact dynamics

A semi-flow $F$ on a metric space $X$

has compact dynamics

if there is a compact $Q\subset X$ such that:

  1. all limit points of each $x\in X$ are in $Q$,
    namely $\omega_F(x)\subset Q$ for all $x\in X$;
    2.
  2. $Q$ attracts uniformly some of its neighborhoods,
    namely, for $\varepsilon>0$ small enough, there is $T>0$ s.t. $$F^T(Nbhd(Q,\varepsilon))\subset Nhbd(Q,\varepsilon/2).$$
    3.

Examples

  1. If $X$ is compact, every semi-flow on $X$ has compact dynamics (e.g. the logistic map).

  2. The Lorenz system.

  3. A large class of reaction-diffusion parabolic PDEs appearing in many natural sciences applications.

Connected graph

Definition (Yorke & DL).
The graph $\Gamma$ of a stream is connected if, given any subdivision
of the nodes of $\Gamma$ into two disjoint and closed collections $\cal A$ and $\cal B$,
there is an edge from $\cal A$ to $\cal B$ or viceversa.
Theorem (Yorke & DL).
Let $F$ be a semi-flow with compact dynamics.
Then the graph of any $F$-stream is connected.

The Smallest stream

Theorem (Yorke & DL).
Let $F$ be a semi-flow with compact dynamics.
Assume that the smallest $F$-stream has countably many nodes. Then the Chain stream of $F$ is the smallest $F$-stream.

 

 

 

In different terminology and setting,
the smallest stream was studied first by Joe Auslander.

Its recurrent points are called
generalized recurrent points.

Thank you!

Jim Yorke, RdL, The graph of the logistic map is a tower,
Discrete & Continuous Dynamical Systems, 41:11 (2021)

Jim Yorke, RdL, Infinite towers in the graphs of many DSs,
Nonlinear Dynamics, 105 (2021)

RdL, Backward asymptotics in S-unimodal maps, International J. of Bifurcations and Chaos, 32:6 (2022)

Ana Anusic, RdL, Graph and backward asymptotics of tent-like unimodal maps (2023)

Jim Yorke, RdL, Streams and Graphs of DSs (2024)