Qualitative Dynamics

and the

Mathematical Theory of Dynamical Systems

Henri Poincare was the grandfather of the geometric approach to understanding the behavior of mechanical systems, systems such as a compound pendulum or the solar system. The main insight that he brought to mechanics was to view the temporal behavior of a system as a succession of configurations in a state space. The most important consequence was his focus on the geometric and topological structure (shape!) of the allowed states. Since the turn of the century, when Poincare lived, his approach has blossomed into the modern theory of dynamical systems.

Due to its geometric character, the approach he introduced has a kind of universality built in. Previously one would say that two systems are obviously different because their behavior is governed by different physical forces and constraints and because they are composed of different materials. Moreover, if their equations of motion, summarizing how the systems react and change state over time, are different, then their behavior is different.

To be concrete let's take a driven pendulum and a superconducting Josephson junction in a microwave field. These are physical systems that are different in just these ways. One is made out of a stiff wood rod and a heavy weight, say; the other consists of a loop of superconducting metal and operates near absolute zero temperature. The pendulum's state is given by the position and velocity of the weight; the Josephson junction's state is determined by the flow of tunneling quantum mechanical electrons. You don't have to know quantum mechanics to appreciate that these two systems seem quite different.

In constrast to this notion of apparent difference, Poincare's view ignores the particular form of the governing equations, even forgets what the underlying variables mean, and instead just looks at the set of states and how a system moves through them. In this view, two systems, like the pendulum and Josephson junction, are the same if they have the same geometric structures in their state spaces. In fact, the pendulum and Josephson junction both exhibit the period-doubling route to chaos and so are very, very similar systems despite their initial superficial dissimilarity. In particular, the mechanisms that produce the period-doubling behavior and eventual deterministic chaos are the same in both.

This type of universality allows one to understand the behavior and dynamics of systems in very many different branches of science within a unified framework. Poincare's approach gives a precise way for us to say how two systems are qualitatively the same.

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