How does symbolic dynamics contribute to the study of dynamical systems?

Question

After checking a couple of resources about symbolic dynamics, I still can't see any application of this field to dynamical systems whose state evolution may be modelled as $\dot{x} = f(x)$ (or by a difference equation). Maybe symbolic dynamics has nothing to do with this application. Otherwise, can one cite a reference in which I can find such a usage?

Answer 1

The connection between continuous-time dynamical systems and symbolic dynamics is not immediately obvious because it usually relies on the intermediary step of connecting both to a discrete dynamical system, whose dynamics are topologically conjugate$^*$ to the symbolic system and also "representative" of the dynamics of the continuous system.

To illustrate; suppose you were attempting to demonstrate that a continuous-time dynamical system exhibited chaotic behavior. One way you could attempt to do this is by analyzing the dynamics of a Poincaré map associated with the continuous-time dynamical system, and trying to demonstrate chaotic behavior on that. You could directly demonstrate that such a thing occurs by showing that the dynamics on some subset of the phase space on the Poincaré map are topologically conjugate to some sufficiently complex symbolic dynamical system, which would imply chaos in the Poincaré map, which implies chaos in the continuous-time system. This is usually the central tool in results that identify chaos in continous-time systems such as the Smale-Birkhoff homoclinic theorem.

References that discuss this connection include Guckenheimer & Holmes's book and Wiggins's book.

$^*$Topological conjugacy between two discrete dynamical systems $(X_1,d_1,f_1)$ and $(X_2,d_2,f_2)$ implies that there exists a homeomorphism $h: X_1 \to X_2$ such that $h(f_1(x)) = f_2(h(x))$ for all $x \in X_1$. Many dynamical features are preserved by topological conjugacy; periodicity/eventual periodicity/aperiodicity are all preserved, as well as topological transitivity. Sensitivity to initial conditions is also preserved via topological conjugacy if the underlying system's metric space is compact.

Answer 2

Just to expand a little bit on the excellent answer by aghostinthefigures: you can look up the notion of "Markov partition", which gives you a very flexible and useful way to conjugate (discrete) dynamical systems with good properties (hyperbolic, transitive) to, say, subshift of finite type.

A very simple but common case is when you have an iterated function system (IFS) with the strong separation property, in which case the dynamics is conjugated to a full shift.

Applications include, for instance, the computations of invariants such as the topological or metric entropy.

Finally, in case you are interested in connections with physics, the Birkhoff-Smale's theorem mentionned by ghost applies to some simple configurations of the 3-body problem, which is arguably the first instance of chaotic system studied (by Poincaré).

Answer 3

First instance of symbolic dynamics is often attributed to Hadamard (https://eudml.org/doc/235168); he used it in the context of the geodesic flow on a negatively curved surface. Recall that the geodesic flow is the flow of a second order ODE. Ghys' talk "Hadamard and geodesics of negatively curved surfaces" (https://youtu.be/LdD78dv7EoI) includes a nice description of this (starting around 15:00): roughly speaking one considers geodesics that stay for all time in a pair of pants. Splitting the pair of pants into two hexagons along three curves $\alpha,\beta,\gamma$, Hadamard shows that given any sequence of $\alpha,\beta,\gamma$'s with no repetitions there is a unique geodesic $\phi$ that crosses the three curves in the predetermined order.