Topic 23

CHAOS - One Face of Nonlinear Dynamics

Dynamics of a Pendulum

Pendulums were the basis of the first good quality clocks. Passage of time was noted by the repetative swings of the pendulum. The time of swings is ac onstant even if distance travelled is not. One can plot speed of the pendulum versus position and define the dynamics of its motion in a phase space plot.

Phase space plots of the motion ofa pendulum display interesting characteristics. A limit cycle occurs when the pendulum swings back and forth without slowing down. This happens in clocks because there is a spring which gives the pendulum a small push on each swing. When the spring runs down, the pendulum starts to swing in smaller and smaller arcs and eventually just stops - a stable end point.

What happens to a pendulum if the forcing is stronger. The phase space diagram can become very complicated - onset of complexity and finally chaos. Sometimes the phase space paths circle around some imaginary point - termed a strange attractor. At other times, the motion may jump back and forth between two different strange attractors.

Logistic maps are another way to see chaotic motion. Logistic maps are an iterative tool for looking at the dynamic evolution of systems. Subtle changes in input parameters can strongly change the logistic map. Numerical feedbakc is a key element in explaining the patterns seen in logistic maps.

Feigenbaum Numbers

Geigenbaum found that patterns of change in bifurcation diagrams, logistic maps, and phase space diagrams all occured in unique intervals definable by a unique set of numbers - 4.66920......

These numbers imply that there is underlying universality and order to nonlinear dynamics independent of the individual sets of equations being studied.

That order can also be related to Fractals. Holes in bifurcation diagrams, logistic maps, and phase space diagrams are analogous to holes in cantor set. step changes related to feigenbaum numbers are fractal.

One characteristic of nonlinear systems is that they do not fill space. They have 'holes' whch are regions that are impossible values for a system. One can't do that with linear systems and normal statistics.

Chaos - A Non-Newtonian View

Chaos can be viewed as a paradigm in the same way as relativity and quantum mechanics.

Relativity expains the breakdown of Newton's Laws at high velocities.

Quantum mechanics descirbes the breakdown of Newton's Laws at small spatial scale.

Chaos is the breakdown of Newton's Laws for conditions of nonlinearity/feedback.

This can be stated in another way by contrasting Newtonian and Chaotic views of the universe:

Simple systems behave in simple ways. NO, simple systems may evolve into complex, chaotic systems

Complex behavior implies complex causes. NO, complex systems have underlying simple order.

Different systems behave differently. NO, universality of chaotic behavior.