Slide 1
Cover page.
Slide 2
Here is an image which gives a basic direct description about chaos theory. You might not be able to understand what it means, but don’t worry, I am going to explain it and talk about what chaos theory is later. Just keep it in mind and remember how it looks like.
Slide 3
This picture is a simplified picture that is as the same as the first slide’s one. Without going into the technical details, I am going to try to give an idea of this picture’s discovery. “When one changes the forces acting on a physical dynamical system, one often sees period doubling, as showed in the picture. A periodic orbit is replaced by another one close to it, but one in which you have to make two turns before coming back exactly at the point of departure. The time it takes to come back – called the period – has therefore about doubled. The period doubling is observed in certain convection experiments: a fluid heated from below undergoes some periodic motion; changing the heat setting produces another type of periodic motion with a period twice as long. Period doubling has also been observed in a periodically dripping tap: as the tap is opened more the period doubles (under certain conditions). This picture actually gives a basic concept of chaos theory from a mathematic perspective.” (Ruelle, 1991: 67- 68).
Slide 4
So here the first question comes out. What is the definition of chaos theory?
In order to know what the chaos is, we need to know some background knowledge about chaos theory’s history.
The name “chaos theory” comes from the fact that the system described by theory as an apparently disordered, but chaos theory is really about figuring out the underlying order from apparently random data.
The first true experimenter in chaos was a meteorologist whose name is Edward Lorenz in 1960. He was working on one of problems of weather prediction with using a set of computer and the experiment had a set of twelve equations to model the weather. The result of experiment didn’t predict the weather itself; however the experiment brought another undesired achievement that the computer could theoretically predict weather. In 1963, Lorenz published his achievement about his discovery including the unpredictability of the weather and types of equations that are relevant to those behaviors. Unfortunately, his contribution was not attracted a lot because he was a meteorologist, which meant his achievements were not paid attention by other mathematician and physicists. Therefore, people would not learn Lorenz’s discoveries until years later.
Chaos Theory, as a theory, was formally introduced to the public by James Alan Yorke (1975) and his partners in 1975 from the mathematic perspective. He defined the chaos theory as a “time evolution with sensitive dependence on initial condition. The motion on a strange attractor is thus chaotic. One also speaks of deterministic noise when the irregular oscillations that are observed appear noisy, but the mechanism that produces them is deterministic” (Ruelle, 1991).
Dr. Kellert (1993) defines Chaos Theory as a qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems (p.2). In this definition, the system is dynamic and unstable; the values of the variables in the system are changes with time. Term “unstable” means a system always resists by small disturbances so that the system will never settle into a form of behavior. Term “aperiodic” means the system does not repeat itself (Tsoukas, 1998).
The rules that specify how the system changes that are usually presented by differential equations that show the changes of variables’ rate (Tsoukas, 1998). Tsoukas (1998) concludes differential equations as “allowing one to calculate the state of the system at other times, given its state at one point in time. The rate of change of each variable is expressed in either linear or nonlinear terms. Linearity means that a unit change in variable x will always cause a specific change in variable y. by contrast, nonlinearity means that the change in variable y brought about by a unit change in variable x will depend on the magnitude of variable x” (p.298).
Slide 5
“Chaos theory highlights the impossibility of long-term prediction for nonlinear systems, since the task of prediction would require knowledge of initial conditions of impossibly high accuracy” (Tsoukas, 1998).
“… the mathematics of chaos privileges a qualitative approach to the understanding of chaotic systems by seeking to provide an analysis of the general pattern of a system’s behavior rather than the precise value of its variables at a certain point in time” (Tsoukas, 1998: 303; Hayek, 1989)
Slide 6
Werndl (2009) gives the concept of dynamical system in his article: “a dynamical system is a mathematical model consisting of a phase space X, the set of all possible states of the system, and evolution equations that describe how solutions evolve in phase space”; it is usually used to describe the time dependence points in a geometrical space. Therefore, Chaos Theory can be examined in dynamical systems theory.
Researchers have been using dynamical models for the study of species populations for more than a century (Davies, 1999), which predicts trends in populations under some external influences. Moreover, non-linear dynamical models and chaos have the power on economics, financial markets and investment management (William, 1989).
“Imagine a bargaining model … in which each party has been instructed by higher headquarters to respond to each new offer by her opposite number with a counter-offer that is to be calculated from a simple reaction function … if the perfectly deterministic sequence of offers and counter-offers that must emerge from these simple rules were to begin to oscillate wildly and apparently at random, the negotiations could easily break down as each party … came to suspect the other side of duplicity and sabotage. Yet all that may be involved, as we will see, is the phenomenon referred to as chaos…” (Davies, 1999: 4; William, 1989)
“Newton’s theory gave a satisfactory account of a mass of observations, which had been reduced to three laws by Johannes Kepler. Kepler’s laws are
- Planetary orbits are plane ellipses with the sun at one focus;
- a line joining a planet with the sun sweeps out area at a rate constant in time;
- the square of the periods of the orbits are proportional to the cubes of the mean radii. ” (Davies, 1999: 5)
Slide 7
Edward Lorenz is the first person who terms chaos; therefore, he discoveries and presents a number of concepts or theories about chaos theory. Other scientists do the further researches on chaos theory after his work.
In Edward Lorenz’s system, some objects are called strange attractors. Some non-periodic solutions are all attracted to some region of state space whose dimension is not an integer. Therefore, if the chaos solutions are existed in regardless of the initial conditions, these systems are called as strange attractors (Davies, 1999).
To quote Davies’s (1999) statement: “deterministic dynamical systems may exhibit regular behavior for some values of their control parameters and irregular behavior for others. One speaks of regular and chaotic behavior in such as a system.” (p.11)
Kellert (1993) summarized two features of strange attractor: “nearby points evolve to opposite sides of the attractor, yet the trajectories are confined to a region of phase space with a particular shape” (p. 14-15).
Slide 8
This term was first described by Lorenz at the December of 1972. With a sequence of changes in some conditions, flapping wings by a little butterfly could make a hurricane. Gleick (1987) gives the Butterfly Effect a technical name: sensitive dependence on initial conditions. It presents that a small change at one place can result in large differences to a later stage in a nonlinear system.
Chaos theory brings a revolution of science. In the past, “physicists assumed that very small changes would cause only very small differences in the numbers, not qualitative changes in behavior” (Gleick, 1987: 47). Chaos theory introduces a new concept of system’s stability. Both stable and unstable behaviors are supposed in a system. A chaotic system could be stable when its irregularity is not disturbed by small changes in a dynamical system.
Slide 9
The difference between random data and chaotic data can be figured out by the data’s elements. In a dynamical system, random data present non-deterministic, which is the most obvious characteristic of a random data. In the same dynamical system, a chaotic data should have following three characters: no asymptotically periodic, no lyapunov exponent vanishes, and the largest lyapunov exponent is strictly positive.
Slide 10
Chaos Theory covers a lot of fields: as showed on the PowerPoint.
career development, geology, mathematics, microbiology, biology, computer science, economics, engineering, finance, hydrology, medicine, meteorology, planning, philosophy, physics, politics, population dynamics, psychology, robotics, and tourism. (Abraham, 1995; Bloch, 2005; Cartwright, 1991; Kiel, 1997; Harney, 2009; Hristu-Varsakelis, 2008; Leonidas, 1996; Mckercher, 1998; Peters, 1994; Sivakumar, 2000)
Chaos Theory 