Howard University, 16 February 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 $X$

$x(t) = F^t(x_0), t\in\Bbb R$,

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. $X=C^2(M)$).

$$\partial_t u = \partial_{xx}u + f(u)$$

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

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

Our goal

Each Dynamical System has some number of
"maximal limit sets"

(i.e. limit sets not contained in a larger limit set).

These can be

attractors, repellors and saddles.

Often the dynamics on them is chaotic.

Our goal

Our main goal is to develop a theory
to study how these limit sets
are related to each other.

This will allow us to give
a complete description of
the qualitative dynamics of the system.

Ultimately, we will encode the full qualitative behavior of the system into a directed graph.

Limit sets

Maximal limit sets
The red loop is the limit set of the inner spiral.
(notice that the red loop is not a single orbit!)
Also point $f$ is a limit set.
The red loop is a maximal limit set but $f$ is not.

Node = A Maximal limit set
We create graphs whose nodes are
maximal limit sets.

Nodes are a generalization of
periodic orbits.

Trajectory graph

Smale's graph

In Differential Dynamical Systems, Bull. of AMS (1967),
Smale defined a graph

encoding the dynamics of Axiom-A systems.

What to do when a system is not Axiom-A?

trajectory graph

What are the nodes and edges?

trajectory graph

Node: here, each fixed point

and each periodic orbit is a node in itself.

trajectory graph

Edge: there is an edge $N_i\mapsto N_j$ if there is $x$
with $\omega(x)\subset N_j$ and $\alpha(x)\subset N_i$.

$\omega(x) = \{y: F^{t_n}(x)\to y\text{ for some }t_n\to\infty\}$
$\alpha(x) = \{y: F^{t_n}(x)\to y\text{ for some }t_n\to-\infty\}$

trajectory graph

Edge point: a point $x$ such that
$\omega(x)\subset N_j$ and $\alpha(x)\subset N_i$, $i\neq j$,
is an edge point.

Examples
of graphs

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)

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!

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 trajectory graph $\Gamma$ of a semi-flow 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 trajectory graph of $F$ is connected.

An application

Newhouse showed that, in dimension 2 and higher,
there are maps having robustly infinitely many sinks
on a compact set.

The graph of these discrete-time semi-flows
must be connected by our theorem.

Hence, these maps must have also at least a repellor or a saddle node.

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)