A dynamical system is the description of how a system evolves in time, under a fixed rule applied to a point in a state space. The branch studies the long-run behaviour of such systems, including their trajectories, stability, periodicity, and chaos, with applications across mathematics, physics, biology, chemistry, engineering, economics, history, and medicine. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionGiven a fixed rule for how a system's state changes from moment to moment, what can be said about its long-run behaviour, stability, periodicity, or unpredictability? 1 Key DebateA fully deterministic dynamical system can still be practically unpredictable. Edward Lorenz summarized the tension this creates for the field: chaos is when the present determines the future but the approximate present does not approximately determine the future. 2 Classification
Pure or AppliedBoth / Interdisciplinary 1 Dynamical Systems
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Associated With
Source Dynamical System (Wikipedia)
Source Henri Poincare (Wikipedia)
Includes
Source Wikipedia: Baker's map
Source Julia Set (Wikipedia)
Source Lorenz System (Wikipedia)
Source Mandelbrot Set (Wikipedia)
Source Poincare recurrence theorem (Wikipedia)
Source Poincare-Bendixson Theorem (Wikipedia)
Source Van der Pol oscillator (Wikipedia)
Sources
1. Dynamical System (Wikipedia)
Wikimedia FoundationLead paragraphQuote, Lead paragraph
has applications to a wide variety of fields such as mathematics, physics, biology, chemistry, engineering, economics, history, and medicine
2. Chaos Theory (Wikipedia)
Wikimedia Foundationlead, definition of chaosQuote, lead, definition of chaos
Chaos: When the present determines the future but the approximate present does not approximately determine the future.
Henri Poincare (Wikipedia)
Wikimedia FoundationAssociated With: Henri Poincare, lead paragraphQuote, Associated With: Henri Poincare, lead paragraph
In his research on the three-body problem, Poincare became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory.
Mandelbrot Set (Wikipedia)
Julia Set (Wikipedia)
Lorenz System (Wikipedia)
Wikipedia: Baker's map
Includes: Baker's Map, Lead sentenceQuote, Includes: Baker's Map, Lead sentence
In dynamical systems theory, the baker's map is a chaotic map from the unit square into itself.
Van der Pol oscillator (Wikipedia)
Includes: Van der Pol Oscillator, Lead sentenceQuote, Includes: Van der Pol Oscillator, Lead sentence
In the study of dynamical systems, the van der Pol oscillator (named for Dutch physicist Balthasar van der Pol) is a non-conservat
Poincare-Bendixson Theorem (Wikipedia)
Wikimedia FoundationIncludes: Poincare-Bendixson Theorem, Lead sentenceQuote, Includes: Poincare-Bendixson Theorem, Lead sentence
t the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere.
Poincare recurrence theorem (Wikipedia)
Includes: Poincare Recurrence Theorem, Lead sentenceQuote, Includes: Poincare Recurrence Theorem, Lead sentence
currence Theorem is a foundational result in both dynamical systems and statistical mechanics.
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