Dynamcial systems, part 1: Introduction and definitions
This post is the first in a series on dynamical systems aiming towards explaining the contents of my PhD thesis. This first part will cover the basic definitions, objects of interest and important results on dynamical systems in general. The reader will be assumed to be familiar with the basics of calculus, topology and manfolds.
Introduction
Dynamical systems describe how objects and their properties change with respect to time. For example it can describe the motion of a particle under the influence of a force such as gravity, how the concentrations of various substances change during a chemical reaction or how the number of fish in a lake changes from year to year.
Mathematically a dynamical system consists of a space \(M\) called the phase space and a function \(F\), commonly called the evolution function, taking as its arguments a time \(t\) and a point \(x\) of phase space and outputs another point \(y=F(t,x)\) of phase space, the point to which \(x\) has moved after time \(t\). We therefore also require that \(F(0,x)=x\) for every point \(x\), i.e. if no time passes then the points do not move. Although in general the time parameter \(t\) can belong to any given monoid \(\mathcal{T}\) in practice either the real numbers \(\mathbb{R}\) or the integers \(\mathbb{Z}\) are used, possibly restricted to their nonnegative parts. Therefore one usually also talks about continuous time dynamical systems and discrete time dynamical systems.
Discrete time dynamical systems are usually modeled simply as a function \(f\colon M\to M\) taking points of phase space to other points of phase space and the time \(t\) gives the number of iterations of this function. If the point \(x\) is mapped to the point \(y\) after time \(t=n\) we write \(y=f^{n}(x)\) where \(f^{n}\) denotes \(n\) iterations of \(f\). In other words, in terms of an evolution function \(F\) we have \(F(n,x)=f^{n}(x)\).
On the other hand, continuous time dynamical systems commonly arise as the flow of a set of differential equations where time is modeled as a subset of the real numbers \(\mathbb{R}\). Typically these differential equations are a set of ordinary differential equations with a finite dimensional phase space but they could also be partial differential equations with an infinite dimensional phase space.
For simplicity we will only be concerned with phase spaces which are finitedimensional smooth manifolds and discrete time dynamical systems which are diffeomorphisms of their underlying phase space in the remainder of this post. Many concepts and results generalize to other settings but some require technicalities that are not suited for a blog post but better left to a proper textbook on dynamical systems. You can find some suggesitons in the "Further Reading" section below.
Fixed points and other invariant sets
The goal of studying dynamical systems is often not to find a complete description of the orbit of every point, such as finding a closed expression for the solution of a system of ordinary differential equations, since such a description may be impossible to find. Instead one often focuses on finding a more qualitative description of a subset of the orbits. This can be achieved by studying certain special subsets of phase space that limit the behavior of orbits at or around it.
The simplest such subset is the fixed point. A fixed point of a dynamical system \(f\) is a point \(x\in M\) such that \(f(x) = x\), i.e., a point that doesn't move around but stays fixed in time.^{1} Fixed points are of interest not only because they are the simples example of an invariant set but mainly because it can often be used to limit the behavior of nearby points as well. For example, consider a dynamical system \(f\) with a fixed point \(x_{0}\) such that all eigenvalues of \(D_{x_{0}}f\) are contained inside the unit circle, i.e., have absolute value less than one. There will then be some neighborhood \(U\) of \(x_{0}\) in which the dynamics is welldescribed by the linear map \(D_{x_{0}}f\) and, as a result, there is some constant \(C < 1\) such that
\[ d(f(x), x_{0})\leq C d(x, x_{0}) < d(x, x_{0}). \]
This means that the point \(x_{0}\) will attract all points in its vicinity. The fixed point is therefore called an attracting fixed point. Conversely, if all eigenvalues of \(D_{x_{0}}f\) ar outside the unit circle the fixed point will instead repel all nearby points and is therefore called a repellng fixed point. Assuming that \(f\) is invertible we have that any attracting fixed point of \(f\) is also a repelling fixed point of \(f^{1}\) and vice versa.
Slightly more complicated are periodic points. These are points \(x_{0}\) for which \(f^{n}(x_{0}) = x_{0}\), i.e., they are fixed points of some iteration \(f^{n}\) of ((f)). The smallest such \(n\) is called the period of the periodic point. Note that if \(x_{0}\) is a periodic point then so are all points \(f^{i}(x_{0})\) for \(i = 0,1,\ldots,n1\). Just as for fixed points, the orbit
\[ \mathcal{O}^{f}(x_{0}) = \{f^{i}(x_{0})\colon i\geq 0\} \]
is called attracting respectively repelling if it is attracting respectively repelling as a fixed point of \(f^{n}\).
In full generality an invariant set is a subset \(U\subset M\) such that for every \(x\in U\) we have \(f^{i}(x)\in U\) for all \(i/geq 0\). An attractor is then an invariant subset \(U\) such that there is a neighborhood \(B(U)\) of \(U\), called the basin of attraction of \(U\), with the property that for each \(b\in B(U)\) and any neighborhood \(V\) of \(U\) there is an \(N\geq 0\) such that \(f^{n}(b)\in V\) for all \(n\geq N\).
So far so good but the eigenvalues of the differential at a fixed point are not always all inside the unit circle or all outside the unit circle. What happens when this condition fails? We will explore the situation when the eigenvalues can be mixed but none of them are on the unit circle. In such cases, which include attracting and repelling fixed points, the fixed point is called hyperbolic. The same terminology also applies to periodic points: a periodic point \(x_{0} = f^{n}(x_{0})\) is called hyperbolic if it is a hyperbolic fixed point of \(f^{n}\).
When we have a hyperbolic fixed point the tangent space at the fixed point is also split into two parts: a stable subspace \(E^{s}\) corresponding to the eigenvalues inside the unit circle and an unstable subspace \(E^{u}\) corresponding to the eigenvalues outside the unit circle. If \(x_{0}\) is a hyperbolic fixed point we can therefore write
\[ T_{x_{0}}M = E_{s}\oplus E_{u}. \]
Furthermore, the stable and unstable subspaces will "integrate" into the stable manifold, denoted \(W^{s}(x_{0})\), and unstable manifold, denoted \(W^{u}(x_{0})\), respectively.
The idea of the tangent space being split into a direct sum of a stable subspace and an unstable subspace naturally generalizes to larger invariant sets. Let \(U\subset M\) be a compact invariant set and suppose there are constants \(\lambda\in (0,1)\) and \(C > 0\) along with a splitting of the tangent space \(TM = E^{s}\oplus E^{u}\) over \(U\) such that for every \(p\in U\) we have:
 The splitting is invariant under \(f\), i.e., \[ D_{p}f(E^{s}(p))\subset E^{s}(f(p))\textrm{ and } D_{p}f(E^{u}(p))\subset E^{u}(f(p)), \]
 \(\Vert D_{p}f^{n}v\Vert \leq C\lambda^{n}\Vert v\Vert\) for all \(v\in E^{s}(p)\) and \(n\geq 0\),
 \(\Vert D_{p}f^{n}v\Vert \leq C\lambda^{n}\Vert v\Vert\) for all \(v\in E^{u}(p)\) and \(n\geq 0\).
We then call \(U\) a hyperbolic set. Hyperbolic fixed points are the simplest example of hyperbolic sets and just as for fixed points this invariant splitting "integrates" into invariant manifolds. This is the content of one of the most important theorems about hyperbolic sets and perhaps dynamical systems in general: the stable manifold theorem, sometimes called the stableunstable manifold theorem.^{2} We state it here in its full generality in terms of the local stable and unstable manifolds but its proof is well beyond the scope of a simple blog post. The interested reader can find it in any of the textbooks suggested as further reading at the end of this post.
Stable manifold theorem: Let \(f\) be a \(C^{1}\)diffeomorphism of \(M\) and let \(U\) be a hyperbolic set for \(f\). Then there exists an \(\varepsilon > 0\) such that for every \(x\in U\)

The sets
\[W^{s}_{\varepsilon}(x) = \{y\in M\colon \textrm{dist}(f^{n}(x), f^{n}(y)) < \varepsilon\quad\forall n\geq 0\},\]
\[W^{u}_{\varepsilon}(x) = \{y\in M\colon \textrm{dist}(f^{n}(x), f^{n}(y)) < \varepsilon\quad\forall n\geq 0\},\]
called the local stable manifold of \(x\) and the local unstable manifold of \(x\) respectively, are \(C^{1}\)embedded balls,

Something
Furthermore, it is also possible to define the global stable and unstable manifolds of a point \(x\) in a hyperbolic set, simply denoted \(W^{s}(x)\) and \(W^{u}(x)\).
The existence of the stable and unstable manifolds on a hyperbolic set can go a long way towards getting a description of the possible dynamics of \(f\). Consider for example the concept of homoclinic connections and homoclinic intersections. Let \(x_{0}\) be a hyperbolic fixed point. A homoclinic connection is then a connected invariant manifold \(N\subset M\) such that \(N\subset W^{s}(x_{0})\cap W^{u}(x_{0})\). Imagine a dynamical system \(f\) on \(\mathbb{R}^{2}\) with a hyperbolic fixed point \(x_{0}\) having a \(1\)dimensional stable manifold and a \(1\)dimensional unstable manifold, both of which are curves in the plane intersecting transversally at \(x_{0}\). One can then imagine one part of the "outgoing" unstable manifold turning back in to the fixed point to become one part of the stable manifold, see the below image.
Image goes here.
A homoclinic intersection is a different situation where the stable and unstable manifolds intersect but do not form a connection, for example if they intersect transversaly. It is then clear that if \(y\in W^{s}(x_{0})\cap W^{u}(x_{0})\) then, since both \(W^{s}(x_{0})\) and \(W^{u}(x_{0})\) are invariant, every point \(f^{n}(y)\) must also be a point of intersection between \(W^{s}(x_{0})\) and \(W^{u}(x_{0})\). This situation creates something called a homoclinic tangle, a complicated structure with complicated dynamics.
There are also heteroclinic conections and heteroclinic intersections. Like their homoclinic counterparts they signify situations where stable and unstable manifolds overlap as a connected manfold or intersect but originate from different hyperbolic fixed points, see images below for helpful illustrations in the case of a \(2\)dimensional dynamical system.
We will return to the homoclinic and heteroclinic connections and intersections in the next part of this series: Chaos. For now, we will briefly explore another way in which hyperblic sets can influence not only the dynamics on the phase space but their mere existence can sometimes restrict the possible topology of the phase space itself.
Mesures and ergodicity
While hyperbolic sets, stable manifolds and homoclinic tangles are an important part of the analytic aspect of dynamical systems, another very important aspect of dynamics are the probabilistic properties. Where the analytic aspects use the language of derivatives and smooth manifolds, the probabilistic aspect use the lanugage of measures and integrals.
Definition of invariant measure
Poncare recurrence theprem
Existence of invariant measures
Ergodic measures
Birkhoff ergodic theorem
Multiplicative ergodic theorem and existence of Lyapunov exponents.
Further Reading
Here are some links to free online resources for those that are interested in reading more about the topics covered in this blog post.
 Dynamical systems
 Hyperbolic sets
 Ergodic theory
 Invariant sets
 Stable manifold
 Poincare recurrence
 Birkhoff ergodic theorem
 Lyapunov exponents
Here are also some recommended textbooks for those wishing to learn dynamical systems.
While the book Katok and Hasselblatt does cover continuous time dynamical systems in some detail it can be quite advanced at tiems. A better introduction is the following book which covers both ordinary differential equations and flows.

For a continuous time dynamical system a fixed point would be a point \(x\) such that \(F(t, x) = x\) for all \(t\). ↩

It can be considered a special case of the HadamardPerron theorem but stating it in such generality quickly becomes unwieldy for a small blog post. Consult a proper textbook on dynamical systems to enjoy the full statement of this important theorem. ↩