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Chaos theory is a branch of mathematics and an interdisciplinary area of scientific study.

It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. The theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning there is sensitive dependence on initial conditions). This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution and is fully determined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as: Chaos: When the present determines the future but the approximate present does not approximately determine the future. This behavior can be studied through the analysis of a chaotic mathematical model or through analytical techniques such as recurrence plots and Poincaré maps. Chaos theory has applications in a variety of disciplines, including meteorology, anthropology, sociology, environmental science, computer science, engineering, economics, ecology, and pandemic crisis management. The theory formed the basis for such fields of study as complex dynamical systems, edge of chaos theory and self-assembly processes. Chaos theory concerns deterministic systems which are predictable for some amount of time and then appear to become random. The amount of time for which the behavior of a chaotic system can be effectively predicted depends on three things: How much uncertainty can be tolerated in the forecast, how accurately its current state can be measured, and a time scale depending on the dynamics of the system, called the Lyapunov time. Some examples of Lyapunov times are: Chaotic electrical circuits, about 1 millisecond; weather systems, a few days (but unproven); the inner solar system, 4 to 5 million years. In chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time.

However, in chaos theory, the term is defined more precisely.

As suggested in Lorenz's book entitled The Essence of Chaos, published in 1993, "sensitive dependence can serve as an acceptable definition of chaos".

Topological mixing is often omitted from popular accounts of chaos, which equate chaos with only sensitivity to initial conditions.

In 1982, Mandelbrot published The Fractal Geometry of Nature, which became a classic of chaos theory. Thus Feigenbaum (1975) and Coullet & Tresser (1978) discovered the universality in chaos, permitting the application of chaos theory to many different phenomena. Also in 1987 James Gleick published Chaos: Making a New Science, which became a best-seller and introduced the general principles of chaos theory as well as its history to the broad public. Currently, chaos theory remains an active area of research, involving many different disciplines such as mathematics, topology, physics, social systems, population modeling, biology, meteorology, astrophysics, information theory, computational neuroscience, pandemic crisis management, etc.

Chaos theory has been used for many years in cryptography.

While a chaotic model for hydrology has its shortcomings, there is still much to learn from looking at the data through the lens of chaos theory. There is always potential difficulty in distinguishing real chaos from chaos that is only in the model.

The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory. John Briggs and David Peat, Turbulent Mirror: : An Illustrated Guide to Chaos Theory and the Science of Wholeness, Harper Perennial 1990, 224 pp.

The chaos theory of evolution – article published in Newscientist featuring similarities of evolution and non-linear systems including fractal nature of life and chaos.

Quick Facts

• This behavior is known as deterministic chaos, or simply chaos.
• The theory formed the basis for such fields of study as complex dynamical systems, edge of chaos theory and self-assembly processes.
• Chaos theory concerns deterministic systems which are predictable for some amount of time and then appear to become random.
• Chaos theory has applications in a variety of disciplines, including meteorology, anthropology, sociology, environmental science, computer science, engineering, economics, ecology, and pandemic crisis management.
• The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning there is sensitive dependence on initial conditions).

Source material: Wikipedia - "Chaos theory". Adapted and summarized for DiscoverScroll. Original contributors are credited through the linked Wikipedia article. Read original on Wikipedia. CC BY-SA 4.0. Changes were made from the original.