Introduction to Dynamical Systems

Welcome

Welcome to my free interactive Dynamical Systems text.

A short history of Dynamical Systems

Isaac Newton began developing calculus and differential equations in the mid 1600s, at least in part to predict the motion of the moon. This could be considered as one of the key beginnings to the study of dynamical systems, (although several ancient cultures such as the Babylonians tried to predict the moon's motion). However, the modern conception of qualitative dynamical systems took quite a lot longer to develop, with a relatively rapid expansion of concepts occuring in the late 1800s.

In 1870, Ernst Schroeder was perhaps the first person to explicitly call for the study of iterated functions - to me, at least, a surprisingly late date for what now seems like such a basic idea. Probably because of the novelty of this idea, it took a little while for his paper to make an impact.

Although not immediately connected to dynamical systems, in 1872 Karl Weierstrass constructed a continuous function that is nowhere differentiable:

$\sum_{n=0}^{\infty} a^n \cos(b^n \pi x) , \ \ \ \ ab > 1 + 3 \pi/2$

It is one of the first examples of fractal object, which arise frequently in dynamical systems. Other early occurrences of fractals appeared in other works of real analysis by Henry Smith (1875), Vito Volterra (1881), Georg Cantor (1882), Helge von Koch (1904), Giuseppe Peano (1890) and David Hilbert (1891).

Another important step towards modern dynamical systems was taken by Arthur Cayley in 1879, who investigated the iteration of Newton's root-finding method for polynomials in the complex plane. The study of complex dynamics was subsequently largely developed in France, by Koenigs, Fatou, and Julia from the 1880s up until World War I. Partly because of the almost unbelievably complicated fractal objects that arise in this subject, progress slowed until computers could illuminate more of the details.

The study of fractals - sets of fractional dimension - was at first undertaken separately from concerns of dynamical systems.

While thousands of mathematicians have contributed to dynamical systems, the works of Henri Poincaré may be the most influential of any from the modern era. His work on the three-body problem of celestial mechanics introduced a slew of new techniques and concepts that dramatically advanced the qualitative theory of differential equations and dynamical systems.

In 1901 the Swedish mathematician Ivar Bendixson published an important paper on limit sets of planar differential equations, including some work on what is now called the Poincare-Bendixson theorem.

The American mathematican George David Birkhoff extended several of Poincare's ideas in the early part of the 20th century, in particular on maps of annuli. His book Dynamical Systems (1927) is perhaps largely responsible for the subject having that name, and the beginning of the subject having its own identity within mathematics. Shortly after its publication, in 1931 and 1932 respectively, Andrei A. Markov (in Russia) and Hassler Whitney (in America) independently published the first abstract definitions of dynamical systems. This Markov was the son of the Markov you might know from linear algebra or probability (e.g. Markov processes, Markov chains, stochastic matrices). This almost simultaneous, parallel development of various parts of dynamical systems in Russia and the "west" (the North America and Western Europe primarily) would continue until the 1990s.

The two world wars both disrupted the development of dynamical systems. In World War I, many French mathematicians were sent to the front and either killed or injured, including several leading figures in the study of complex maps such as Gaston Julia. Julia and Pierre Fatou had glimpsed some of the immense complexity possible in planar maps, but it was not until the wide availability of computers that this could be fully appreciated. World War II was perhaps even more disruptive of mathematics in general, with many mathematicians and physicists turning to more applied problems because of the war.

Huge progress in dynamical systems was made in the 1960s by many people. Stephen Smale developed a powerful template for analyzing chaotic maps - the horseshoe map, while on the more analytic side Kolmogorov, Arnold, and Moser developed KAM theory. The amazing theorem of the Ukrainian Sharkovsky for one-dimensional maps, which we will examine later, was published in the 1960s but not fully appreciated in western countries until the 1970s. Chaotic dynamics for a simplified model of atmospheric physics was discovered by Lorenz in 1963.

By the 1970s dynamical systems was becoming an extremely hot topic in mathematics, and many advances in both theory and applications were developed. The topologist René Thom developed the study of generic bifurcations of dynamical systems into what he called "Catastrophe Theory", which attracted considerable popular attention at the time (somewhat similar to the attention given to "Chaos Theory" in the 1980s and 1990s).

In the 1980s, perhaps because of the spectacular computer-generated graphics of fractal structures such as the Mandelbrot set, dynamical systems began to become much more popularly known.

Currently the subject still has enormous numbers of open problems and applications, and remains a significant area of mathematics research.

Mandelbrot zoom gif — Zooming in on the Mandelbrot set, from wikipedia's Fractal Curve entry.

Continuous versus discrete dynamics

Both because of time and the interests of the authors, most texts on this subject focus more on either discrete or continuous dynamical systems. In this text we begin with discrete dynamical systems, and then later include some of the most interesting cases of continuous systems.

Example

As a simple example of how the two areas relate, lets review the elementary ordinary differential equation (ODE)

$x' = c x$

(where $c$ is a real parameter) and see how it can be related to discrete maps.

The ODE is linear and separable, so several ways of solving it are possible. The solution is simply multiples of the exponential function, $x = C_0 e^{c t}$ where $C_0 = x(0)$ is a parameter. So if $c$ is positive, the solutions increase exponentially in magnitude, and if $c$ is negative every solution will exponentially decay to $0$ .

If we uniformly sample the solution $x(t)$ at multiples of a time $h$ (i.e. at $0$ , $h$ , $2h$ , $3h$ , etc), we get the values $C_0$ , $e^{ch} C_0$ , $c^{2ch} C_0$ , $\ldots$ . Each one is a factor $e^{ch}$ larger or smaller than the last, so we could write these as solutions of the discrete system

$x_{i+1} = k x_i$

where $k = e^{ch}$ . This example generalizes to higher-dimensional systems, in which the eigenvalues of a linearized differential equation are related through exponentiation to the eigenvalues of a discretized version.

Overview of topics

After a short section on some useful mathematical preliminaries from real analysis and topology, we proceed by working from the simplest dynamical systems in one real dimension all the way to examples of complicated three-dimensional flows. Here is brief summary outline:

Useful preliminaries (mostly point set and metric topology).
One-dimensional autonomous ODEs. The dimension is so low that the dynamics are always qualitatively relatively simple.
One-dimensional real maps. These can already be surprisingly complex under iterations. Our main examples will be the quadratic/logistic map $\lambda x (1-x)$ and maps from Newton's method for low-order polynomials.
One-dimensional complex maps. When the quadratic/logistic map is extended to the complex numbers, we get bedazzling objects such as the Mandelbrot and Julia sets. In order to understand some of the geometry of these, we will cover some basics of fractal geometry.
Two-dimensional autonomous ODEs (2-d flows). The main example is Lienard systems. Poincare maps of 2-d flows result in 1-d maps.
Two-dimensional maps. Main examples will be the Henon map and toral automorphisms.
Three-dimensional flows. We will look at some famous examples that have chaotic strange attractor sets, such as the Lorenz equations, Chua's circuit, and the Rossler attractor.