Chaos Theory
(Lorenz Poincaré)

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Studying complex and dynamic systems to reveal patterns of order (non-chaos) out of seemingly chaotic behaviors. Explanation of Chaos Theory of Lorenz ('60) and Poincaré. (ca. 1900)

Contributed by: Amita Paul



Fractal (image applying Chaos Theory)What is Chaos Theory? Description

The Chaos Theory method from Lorenz and Poincaré is a technique that can be used for studying complex and dynamic systems to reveal patterns of order (non-chaos) out of seemingly chaotic behaviors.

"Chaos Theory is the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems" (Kellert, 1993, p. 2). Aperiodic behavior is observed when there is no variable, describing the state of the system, that undergoes a regular repetition of values. Unstable aperiodic behavior is highly complex: it never repeats and it continues to manifest the effects of any small perturbation.

As per the current mathematical theory a chaotic system is defined as showing "sensitivity to initial conditions". In other words, to predict the future state of a system with certainty, you need to know the initial conditions with infinite accuracy, since errors increase rapidly with even the slightest inaccuracy.

This is why the weather is so difficult to forecast. The theory also has been applied to business cycles, and dynamics of animal populations, as well as in fluid motion, planetary orbits, electrical currents in semi-conductors, medical conditions (like epileptic seizures), and the modeling of arms races.

During the 1960s Edward Lorenz, a meteorologist at MIT,  worked on a project to simulate weather patterns on a computer. He accidentally stumbled upon the butterfly effect after deviations in calculations off by thousandths greatly changed the simulations. The Butterfly Effect reflects how changes on the small scale, can influence things on the large scale. It is the classic example of chaos, where small changes may cause large changes. A butterfly, flapping its wings in Hong Kong, may change tornado patterns in Texas.

Chaos Theory regards organizations/businesses as complex, dynamic, non-linear, co-creative and far-from-equilibrium systems. Their future performance cannot be predicted by past and present events and actions. In a state of chaos, organizations behave in ways which are simultaneously both unpredictable (chaotic) and patterned (orderly).

Origin of Chaos Theory. History

Another chaos theory example (fractal)Ilya Prigogine, Nobel laureate, showed that complex structures could result from simpler ones. This is like order coming from chaos. Henry Adams previously described this with his quote "Chaos often breeds life, when order breeds habit". Henri Poincaré was really the "Father of Chaos [Theory]," however. The planet Neptune was discovered in 1846 and had been predicted from the observation of deviations in Uranus' orbit. King Oscar II of Norway was willing to give a prize to anyone who could prove or disprove that the solar system was stable. Poincaré offered his solution, but when a friend found an error in his calculations, the prize was taken away until he could come up with a new solution that worked. He found that there was no solution. Not even the laws of Sir Isaac Newton provided a solution to this huge problem. Poincaré had been trying to find order in a system where there was none to be found. Chaos theory was formulated during the 1960s. Significant and more practical work was done by Edward Lorenz in the 1960s. The name chaos was coined by Jim Yorke, an applied mathematician at the University of Maryland (Ruelle, 1991).

Calculation of the Chaos Theory? Formula

To apply Chaos Theory, a single measured variable x(n) = x(t0 + nt) with a starting time, t0, and a lead time, t, provides an n-dimensional space, or phase space, that represents the full multivariate state space of the system; up to 4 dimensions may be required to represent the phase space for a chaotic system. Thus, over a long period of time, an observed system will develop patterns within a nonlinear time series that can be used to predict future states (Solomatine et al, 2001).

Usage of Chaos Theory. Applications

The principles of Chaos Theory have been successfully used to describe and explain diverse natural and artificial phenomena. Such as:

  • Predicting epileptic seizures.
  • Predicting financial markets.
  • Modeling of manufacturing systems.
  • Making weather forecasts.
  • Creating Fractals. Computer-generated images applying Chaos Theory principles. (See figures on this page.)

In a scenario where businesses operate in a turbulent, complex and unpredictable environment, the tenets of Chaos Theory can be extremely valuable. Application areas can include:

Steps in Chaos Theory. Process

To control chaos, the system or process of chaos has to be controlled. To control a system, what is needed is:

  1. A target, objective or goal which the system should reach. For a system with predictable behavior (deterministic) this may be a particular state of the system.
  2. A system capable of reaching the target or goal.
  3. Some means of influencing the system behavior. These are the control inputs (decisions, decision rules, or initial states).

Strengths of Chaos Theory. Benefits

Chaos theory has wide applicability in modern science and technology era. Communication and management may see a paradigm shift, as will several other business areas. Research and study in this area by academics can be extremely useful for the business and financial world.

Limitations of Chaos Theory. Disadvantages

The limitations of applying Chaos Theory are in due mostly from choosing the input parameters. The methods chosen to compute these parameters depend on the dynamics underlying the data and on the kind of analysis intended, which is in most cases highly complex and not always accurate.

Chaos theory is not as simplistic to find an immediate and direct application in the business environment, but mapping of the business environment using the knowledge of chaos definitely is worthwhile studying.

Assumptions of Chaos Theory. Conditions

  • Small actions produce rather large consequences, creating a chaotic atmosphere.

Book: James Geick - Chaos-Making a new Science -

Book: Ali Bulent Cambel - Applied Chaos Theory : A Paradigm for Complexity -

Book: Richard Tiplady - World of Difference -

Book: Garnett P. Williams - Chaos Theory Tamed -

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