Chaos


We borrow a working definition for chaos theory from Dr. Stephen Kellert: The Qualitative Study of Unstable Aperiodic Behavior in Deterministic Nonlinear dynamical systems. I should briefly dissect some of these terms to better describe what is and what is not chaotic in nature:

Chaos is qualitative in that it seeks to know the general character of a system's long-term behavior, rather than seeking numerical predictions about a future state. What characteristics will all solutions of a system exhibit? How does this system change from exhibiting one behavior to another?
Chaotic systems are unstable since they tend not to resist any outside disturbances but instead react in significant ways. In other words, they do not shrug off external influences but are partly navigated by them.

The variables describing the state of a system do not demonstrate a regular repetition of values and are therefore aperiodic. This unstable aperiodic behavior is highly complex since it never repeats and continues to show the effects of the disturbance(s).

These systems are deterministic because they are made up of few, simple differential equations, and make no references to implicit chance mechanisms. This is not to be completely equated with the metaphysical or philosophical idea of determinism (that human choices could be predetermined as well).

Finally, a dynamic system is a simplified model for the time-varying behavior of an actual system. These systems are described using differential equations specifying the rates of change for each variable.

Edward Lorenz would stretch the definition of chaos to include phenomena that are slightly random, provided that their much greater apparent randomness is not a by-product of their slight true randomness. In other words, real-world processes that appear to be behaving randomly - perhaps the falling leaf or the flapping flag - should be allowed to qualify as chaos, as long as they would continue to appear random even if any true randomness could somehow be eliminated.

What this means is when we make slight changes to a system at one time, and the later behavior of the system may soon become completely different. In Lorenz' meteorological computer modeling, he discovered the foundation of mainstream chaos: that simply-formulated systems with few variables could display highly complex behavior that was unpredictable and unforseeable. He saw that slight differences in one variable had profound effects on the outcome of the whole system. In Chaos parlance, this is referred to as sensitive dependence on initial conditions. In real weather situations, this could mean the development of a front or pressure-system where there never would have been one in previous models. In differential plotting this took on a new form called a strange attractor (see figure 1). Initial conditions need not be the ones that existed when a system was created, but may be the ones at the beginning of any stretch of time that interests an investigator.


Exhibits || Lexicon || Timeline

The Exploratorium, 1996

Chaos

Chaos


We borrow a working definition for chaos theory from Dr. Stephen Kellert: The Qualitative Study of Unstable Aperiodic Behavior in Deterministic Nonlinear dynamical systems. I should briefly dissect some of these terms to better describe what is and what is not chaotic in nature:

Chaos is qualitative in that it seeks to know the general character of a system's long-term behavior, rather than seeking numerical predictions about a future state. What characteristics will all solutions of a system exhibit? How does this system change from exhibiting one behavior to another?
Chaotic systems are unstable since they tend not to resist any outside disturbances but instead react in significant ways. In other words, they do not shrug off external influences but are partly navigated by them.

The variables describing the state of a system do not demonstrate a regular repetition of values and are therefore aperiodic. This unstable aperiodic behavior is highly complex since it never repeats and continues to show the effects of the disturbance(s).

These systems are deterministic because they are made up of few, simple differential equations, and make no references to implicit chance mechanisms. This is not to be completely equated with the metaphysical or philosophical idea of determinism (that human choices could be predetermined as well).

Finally, a dynamic system is a simplified model for the time-varying behavior of an actual system. These systems are described using differential equations specifying the rates of change for each variable.

Edward Lorenz would stretch the definition of chaos to include phenomena that are slightly random, provided that their much greater apparent randomness is not a by-product of their slight true randomness. In other words, real-world processes that appear to be behaving randomly - perhaps the falling leaf or the flapping flag - should be allowed to qualify as chaos, as long as they would continue to appear random even if any true randomness could somehow be eliminated.

What this means is when we make slight changes to a system at one time, and the later behavior of the system may soon become completely different. In Lorenz' meteorological computer modeling, he discovered the foundation of mainstream chaos: that simply-formulated systems with few variables could display highly complex behavior that was unpredictable and unforseeable. He saw that slight differences in one variable had profound effects on the outcome of the whole system. In Chaos parlance, this is referred to as sensitive dependence on initial conditions. In real weather situations, this could mean the development of a front or pressure-system where there never would have been one in previous models. In differential plotting this took on a new form called a strange attractor (see figure 1). Initial conditions need not be the ones that existed when a system was created, but may be the ones at the beginning of any stretch of time that interests an investigator.


Exhibits || Lexicon || Timeline

The Exploratorium, 1996