Difference between revisions of "Chaos theory"
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| − | In [[mathematics]], ''chaos theory'' is the field which studies the behavior of [[dynamical systems]] that are highly sensitive to initial conditions -- a response popularly referred to as the [[butterfly effect]]. | + | In [[mathematics]], '''chaos theory''' is the field which studies the behavior of [[dynamical systems]] that are highly sensitive to initial conditions -- a response popularly referred to as the [[butterfly effect]]. |
== Description == | == Description == | ||
| Line 9: | Line 9: | ||
In other words: the deterministic nature of these systems does not make them predictable. | In other words: the deterministic nature of these systems does not make them predictable. | ||
| − | This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by [[Edward Lorenz]] as: | + | This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by [[Edward Norton Lorenz]] as: |
<blockquote>Chaos: When the present determines the future, but the approximate present does not approximately determine the future.</blockquote> | <blockquote>Chaos: When the present determines the future, but the approximate present does not approximately determine the future.</blockquote> | ||
| Line 19: | Line 19: | ||
== Analytical techniques == | == Analytical techniques == | ||
| − | This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. | + | This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as [[Recurrence plot|recurrence plots]] and [[Poincaré map|Poincaré maps]]. |
== Disciplines == | == Disciplines == | ||
| − | Chaos theory has applications in several disciplines, including | + | Chaos theory has applications in several disciplines, including: |
| + | * [[Computer science]] | ||
| + | * [[Database management]] | ||
* Meteorology | * Meteorology | ||
| − | * Sociology | + | * Sociology |
| + | * [[Physics]] | ||
| + | * [[Engineering]] | ||
| + | * Economics | ||
| + | * Biology | ||
| + | * [[Philosophy]] | ||
== See also == | == See also == | ||
| + | * [[Amplitude death]] | ||
| + | * [[Anosov diffeomorphism]] | ||
| + | * [[Bifurcation theory]] | ||
| + | * [[Catastrophe theory]] | ||
| + | * [[Chaos theory in organizational development]] | ||
| + | * [[Chaotic mixing]] | ||
| + | * [[Chaotic scattering]] | ||
| + | * [[Complex dynamics]] | ||
| + | * [[Complexity]] | ||
| + | * [[Control of chaos]] | ||
| + | * [[Deterministic Nonperiodic Flow]] - academic paper by [[Edward Norton Lorenz]] which was highly influential in the emerging field of chaos theory. | ||
* [[Deterministic system]] | * [[Deterministic system]] | ||
| + | * [[Dynamical systems]] | ||
| + | * [[Edge of chaos]] | ||
| + | * [[Emergence]] | ||
| + | * [[Excitable medium]] | ||
| + | * [[Fractal]] | ||
| + | * [[Horseshoe map]] | ||
| + | * [[John H. Hubbard]] | ||
| + | * [[Julia set]] | ||
| + | * [[Edward Norton Lorenz]] - author of [[Deterministic Nonperiodic Flow]]. | ||
| + | * [[Mandelbrot set]] | ||
| + | * [[Kolmogorov–Arnold–Moser theorem]] | ||
| + | * [[Ill-conditioning]] | ||
| + | * [[Ill-posedness]] | ||
| + | * [[Mandelbrot set]] | ||
| + | * [[Nonlinear system]] | ||
| + | * [[Numerical analysis]] - the study of [[Algorithm|algorithms]] that use numerical approximation (as opposed to general symbolic manipulations) for the problems of [[mathematical analysis]] (as distinguished from [[discrete mathematics]]). | ||
| + | * [[Numerical stability]] | ||
| + | * [[Patterns in nature]] | ||
| + | * [[Poincaré section]] | ||
| + | * [[Predictability]] | ||
| + | * [[Quantum chaos]] | ||
| + | * [[Recursion]] | ||
* [[Rounding errors]] | * [[Rounding errors]] | ||
| − | * [[ | + | * [[Santa Fe Institute]] |
* [[Strange attractor]] | * [[Strange attractor]] | ||
| + | * [[Synchronization of chaos]] | ||
| + | * [[Unintended consequence]] | ||
== External links == | == External links == | ||
* [https://en.wikipedia.org/wiki/Chaos_theory Chaos theory] @ Wikipedia | * [https://en.wikipedia.org/wiki/Chaos_theory Chaos theory] @ Wikipedia | ||
| + | |||
| + | [[Category:Chaos theory]] | ||
| + | [[Category:Fractals]] | ||
| + | [[Category:Mathematics]] | ||
| + | [[Category:Physics]] | ||
Latest revision as of 11:11, 19 August 2016
In mathematics, chaos theory is the field which studies the behavior of dynamical systems that are highly sensitive to initial conditions -- a response popularly referred to as the butterfly effect.
Contents
Description
Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general.
This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved.
In other words: the deterministic nature of these systems does not make them predictable.
This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Norton Lorenz as:
Chaos: When the present determines the future, but the approximate present does not approximately determine the future.
Chaos in natural systems
Chaotic behavior exists in many natural systems, such as weather and climate.
Analytical techniques
This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps.
Disciplines
Chaos theory has applications in several disciplines, including:
- Computer science
- Database management
- Meteorology
- Sociology
- Physics
- Engineering
- Economics
- Biology
- Philosophy
See also
- Amplitude death
- Anosov diffeomorphism
- Bifurcation theory
- Catastrophe theory
- Chaos theory in organizational development
- Chaotic mixing
- Chaotic scattering
- Complex dynamics
- Complexity
- Control of chaos
- Deterministic Nonperiodic Flow - academic paper by Edward Norton Lorenz which was highly influential in the emerging field of chaos theory.
- Deterministic system
- Dynamical systems
- Edge of chaos
- Emergence
- Excitable medium
- Fractal
- Horseshoe map
- John H. Hubbard
- Julia set
- Edward Norton Lorenz - author of Deterministic Nonperiodic Flow.
- Mandelbrot set
- Kolmogorov–Arnold–Moser theorem
- Ill-conditioning
- Ill-posedness
- Mandelbrot set
- Nonlinear system
- Numerical analysis - the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).
- Numerical stability
- Patterns in nature
- Poincaré section
- Predictability
- Quantum chaos
- Recursion
- Rounding errors
- Santa Fe Institute
- Strange attractor
- Synchronization of chaos
- Unintended consequence
External links
- Chaos theory @ Wikipedia
