A dynamical system consists of two ingredients: a rule or ``dynamic,'' which specifies how a system evolves, and an initial condition or ``state'' from which the system starts. The most successful class of rules for describing natural phenomena are differential equations. All the major theories of physics are stated in terms of differential equations. This observation led the mathematician V. I. Arnold to comment, ``consequently, differential equations lie at the basis of scientific mathematical philosophy,'' our scientific world view. This scientific philosophy began with the discovery of the calculus by Newton and Leibniz and continues to the present day.
Dynamical systems theory and nonlinear dynamics grew out of the qualitative study of differential equations, which in turn began as an attempt to understand and predict the motions that surround us: the orbits of the planets, the vibrations of a string, the ripples on the surface of a pond, the forever evolving patterns of the weather. The first two hundred years of this scientific philosophy, from Newton and Euler through to Hamilton and Maxwell, produced many stunning successes in formulating the ``rules of the world,'' but only limited results in finding their solutions. Some of the motions around us--such as the swinging of a clock pendulum--are regular and easily explained, while others--such as the shifting patterns of a waterfall--are irregular and initially appear to defy any rule.
The mathematician Henri Poincaré (1892) was the first to appreciate the true source of the problem: the difficulty lay not in the rules, but rather in specifying the initial conditions. At the beginning of this century, in his essay Science and Method, Poincaré wrote :
A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that that effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.Poincaré's discovery of sensitive dependence on initial conditions in what are now termed chaotic dynamical systems has only been fully appreciated by the larger scientific community during the past three decades. Mathematicians, physicists, chemists, biologists, engineers, meteorologists--indeed, individuals from all fields have, with the help of computer simulations and new experiments, discovered for themselves the cornucopia of chaotic phenomena existing in the simplest nonlinear systems.
Before we proceed, we should
distinguish nonlinear dynamics
from dynamical systems theory .{
The latter is a
well-defined branch of mathematics, while nonlinear dynamics
is an interdisciplinary field that draws on all the sciences,
especially mathematics and the physical sciences.
Scientists in all fields are united by their need to solve
nonlinear equations, and each different discipline has made
valuable contributions to the analysis of nonlinear systems.
A meteorologist discovered the first strange attractor in an
attempt to understand the
unpredictability of the weather.{
A biologist promoted the study of the quadratic map in
an attempt to understand population dynamics. And engineers, computer
scientists, and applied mathematicians gave
us a wealth of problems along with
the computers and programs needed
to bring nonlinear systems alive on our computer screens.
Nonlinear dynamics is
interdisciplinary, and nonlinear dynamicists rely on
their colleagues throughout all the sciences.
To define a nonlinear dynamical system we first look at an example of a linear dynamical system. A linear dynamical system is one in which the dynamic rule is linearly proportional to the system variables. Linear systems can be analyzed by breaking the problem into pieces and then adding these pieces together to build a complete solution. For example, consider the second-order linear differential equation
The dynamical system defined by this differential equation is linear because all the terms are linear functions of x. The second derivative of x (the acceleration) is proportional to -x. To solve this linear differential equation we must find some function x(t) with the following property: the second derivative of x (with respect to the independent variable t) is equal to -x. Two possible solutions immediately come to mind,
since
and
that is, both 
and 
satisfy the linear differential equation.
Because the differential equation is linear, the sum
of these two solutions defined by
is also a solution. This can be verified by
calculating
Any number of solutions can be added together in this way to form a new solution; this property of linear differential equations is called the principle of superposition . It is the cornerstone from which all linear theory is built.
Now let's see what happens when we apply the same method to a nonlinear system. For example, consider the second-order nonlinear differential equation
Let's assume we can find two different solutions to this
nonlinear differential equation, which we will again call
and 
. A quick calculation,
shows that the solutions of a nonlinear equation
cannot usually be added together to build a larger solution
because of the ``cross-terms'' ( 
).
The principle of superposition fails to hold for nonlinear systems.
Traditionally, a differential equation is ``solved'' by finding a function that satisfies the differential equation. A trajectory is then determined by starting the solution with a particular initial condition. For example, if we want to predict the position of a comet ten years from now we need to measure its current position and velocity, write down the differential equation for its motion, and then integrate the differential equation starting from the measured initial condition. The traditional view of a solution thus centers on finding an individual orbit or trajectory. That is, given the initial condition and the rule, we are asked to predict the future position of the comet. Before Poincaré's work it was thought that a nonlinear system would always have a solution; we just needed to be clever enough to find it.
Poincaré's discovery of chaotic behavior in the three-body problem showed that such a view is wrong. No matter how clever we are we won't be able to write down the equations that solve many nonlinear systems. This is not wholly unexpected. After all, in a (bounded) closed form solution we might expect that any small change in initial conditions should produce a proportional change in the predicted trajectories. But a chaotic system can produce large differences in the long-term trajectories even when two initial conditions are close. Poincaré realized the full implications of this simple discovery, and he immediately redefined the notion of a ``solution'' to a differential equation.
Poincaré was less interested in an individual orbit than in all possible orbits. He shifted the emphasis from a local solution--knowing the exact motion of an individual trajectory--to a global solution--knowing the qualitative behavior of all possible trajectories for a given class of systems. In our comet example, a qualitative solution for the differential equation governing the comet's trajectory might appear easier to achieve since it would not require us to integrate the equations of motion to find the exact future position of the comet. The qualitative solution is often difficult to completely specify, though, because it requires a global view of the dynamics, that is, the possible examination of a large number of related systems.
Finding individual solutions is the traditional approach
to solving a differential equation. In contrast,
recurrence is a key theme in Poincaré's quest for the
qualitative solution of a differential equation.
To understand the recurrence properties of a
dynamical system, we need to know what regions of space are visited and
how often the orbit returns to those regions. We can seek to statistically
characterize how often a region of space is visited; this leads to the
so-called ergodic
theory of dynamical systems. Additionally, we can try to
understand the geometric transformations undergone by a group of
trajectories; this leads to the so-called topological theory of
dynamical systems emphasized in this book.
There are many different levels of recurrence. For instance, the comet could crash into a planet. After that nothing much happens (unless you're a dinosaur). Another possibility is that the comet could go into an orbit about a star and from then on follow a periodic motion. In this case the comet will always return to the same points along the orbit. The recurrence is strictly periodic and easily predicted. But there are other possibilities. In particular, the comet could follow a chaotic path exhibiting a complex recurrence pattern, visiting and revisiting different regions of space in an erratic manner.
To summarize, Poincaré advocated the qualitative study of differential equations. We may lose sight of some specific details about any individual trajectory, but we want to sketch out the patterns formed by a large collection of different trajectories from related systems. This global view is motivated by the fact that it is nonsensical to study the orbit of a single trajectory in a chaotic dynamical system. To understand the motions that surround us, which are largely governed by nonlinear laws and interactions, requires the development of new qualitative techniques for analyzing the motions in nonlinear dynamical systems.