Topic 21

Introduction to Chaos Theory - A paradigm for the 21st Century

 

What is Chaos?

Qualitatively, chaos theory explains systems with unpredictable behavior.

Chaos theory is built on the concepts of nonlinearity and feedback.

In its most complete form, Chaos theory is embodied within the mathematical field of Nonlinear Dynamical Systems.

 

Before Chaos

To understand the significance of chaos theory, we ned to go back to Newton and explore the nature of quantitative measurements.

There are three interelated themes to such measurements: randomness, discreteness, dynamism.

Randomness: whenever a measurement is made, there will be some amount of error involved - statistical analysis and probability were evolved in part to characterize such errors. In some cases, observed processes are 'too complex' to measure in detail - statistical analysis helps to show average tendencies. Examples: statistical mechanics for describing behavior of atoms in gases, probability theory for prediction of events.

Discreteness: most processes acting in the world today are continuous - always active. However, we normally measure such systems only at discrete times. We have incomplete knowledge of what is happening to the systems between measurements. Such measured systems are not continuous - just a series of events.

Dynamism: Most of the physics problems we have considered before are static - they exist in time but do not change continuously. But there are many systems and problems where change is a key element in the evolution of the processes. For such systems, a critical issue is predictability.

Linear Systems Analysis: most of the problems we have considered before, including Newton's Laws, statistics, and probability theory are examples of linear systems. A key feature of linear systems is that the functions are addative. Tiny errors in one system component will translate linearly into tiny errors in later system measurements.

During the 19th Century, linear systems and careful measurements gave the impression that we could predict the future - scientific determinism. Linear equations should always be able to give an exact and predictable answer. Problems of friction, air resistence, and fluid motion were ignored or treated as problems that have average linear behavior.

 

Nonlinearity is the Road to Chaos

Nonlinearity in mathematics means tha the future values of a process depend on past values of t he same process. This is referred to as feedback. Positive feedback means that, if a process goes one direction at one time setep, that nonlinear feedback will make it go the same way att he next time step, maybe even at an accelerated rate. Negative feedback means the opposite.

Edward Lorentz was one of the first to systematically investigate nonlinear systems and their feedback. He was particularily interested in climate and its predictability. He used computers and simple models to try and predict weather. he found that he could not predict weather for even the simplest models even though the equations were exactly known. When he made small changes in theinput parameters of his model, the results would quickly diverge from previous values. This effect became known as the Butterfly Effect - a butterfly in the USA may cause changes in air turbulance that could significantly effect weather in Asia (now thatnonlinearity!).

Nonlinear dynamical sytems are groups of equations that encompass chaos. They incorporate nonlinearity and feedback. They may behave like linear systems under certain circumstances, but with small changes in input parameters, they may become truly chaotic in behavior.

Evidence for chaotic behavior is measured in several ways. The first is sensitivity of the equationsto small changes in input parameters. This variabiltiy can be plotted using bifurcation diagrams. One can also plot the dynamic behavior of systems useing phase space diagrams.

Other concepts that are related to chaos theory and have grown up alongside it include: slef-similairty, fractals, non-integer power-law relationships, and red spectra. we sill consider several of these in our next two lectures.

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