The main insight that Poincaré 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 of the allowed states. 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.
In constrast to this notion of apparent difference, Poincaré’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. Poincaré’s approach gives a precise way for us to say how two systems are qualitatively the same.
Roughly speaking, a bifurcation is a qualitative change in an attractor’s structure as a control parameter is smoothly varied. For example, a simple equilibrium, or fixed point attractor, might give way to a periodic oscillation as the stress on a system increases. Similarly, a periodic attractor might become unstable and be replaced by a chaotic attractor.
In Benard convection, to take a real world example, heat from the surface of the earth simply conducts its way to the top of the atmosphere until the rate of heat generation at the surface of the earth gets too high. At this point heat conduction breaks down and bodily motion of the air (wind!) sets in. The atmosphere develops pairs of convection cells, one rotating left and the other rotating right. In a dripping faucet at low pressure, drops come off the faucet with equal timing between them. As the pressure is increased the drops begin to fall with two drops falling close together, then a longer wait, then two drops falling close together again. In this case, a simple periodic process has given way to a periodic process with twice the period, a process described as “period doubling”. If the flow rate of water through the faucet is increased further, often an irregular dripping is found and the behavior can become chaotic.