Summary of Unit Eight
Summary of Unit Eight
Strange Attractors Introduction to Dynamical Systems and Chaos http://www.complexityexplorer.org
The Henon Map ●
●
Chaotic behavior is observed for a=1.4, b=0.3
David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
The Henon Attractor ●
Plot X and Y against each other
●
The result is a complicated, structured attractor
David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
A Strange Attractor
The Henon attractor is a strange attractor: ● It is an attractor: nearby orbits get pulled into it. It is stable. ● Motion on the attractor is chaotic: orbits are aperiodic and have sensitive dependence on initial conditions. David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
The Lorenz Equations ●
David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
The Lorenz Attractor
●
The orbits lie on a strange attractor.
David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
The Rossler Equations ●
David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
The Rossler Attractor
Image source: Rossler attractor. Made by User:Wofl. http://en.wikipedia.org/wiki/R%C3%B6ssler_attractor. cc-attribution-share alike 2.5 generic.
David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
Stretching and Folding ●
●
●
●
●
The key geometric ingredients of chaos Stretching pulls nearby orbits apart, leading to sensitive dependence on initial conditions Folding takes far apart orbits and moves them closer together, keeping orbits bounded. Stretching and folding occurs in 1D maps as well as higher-dimensional phase space. This explains how 1D maps can capture some features of higher-dimensional systems.
David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
Strange Attractors ●
Complex structures arising from simple dynamical systems.
●
The motion on the attractor is chaotic.
●
But all orbits get pulled to the attractor.
●
Combine elements of order and disorder.
●
Motion is locally unstable, globally stable.
David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
Strange Attractors Introduction to Dynamical Systems and Chaos http://www.complexityexplorer.org
The Henon Map ●
●
Chaotic behavior is observed for a=1.4, b=0.3
David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
The Henon Attractor ●
Plot X and Y against each other
●
The result is a complicated, structured attractor
David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
A Strange Attractor
The Henon attractor is a strange attractor: ● It is an attractor: nearby orbits get pulled into it. It is stable. ● Motion on the attractor is chaotic: orbits are aperiodic and have sensitive dependence on initial conditions. David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
The Lorenz Equations ●
David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
The Lorenz Attractor
●
The orbits lie on a strange attractor.
David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
The Rossler Equations ●
David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
The Rossler Attractor
Image source: Rossler attractor. Made by User:Wofl. http://en.wikipedia.org/wiki/R%C3%B6ssler_attractor. cc-attribution-share alike 2.5 generic.
David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
Stretching and Folding ●
●
●
●
●
The key geometric ingredients of chaos Stretching pulls nearby orbits apart, leading to sensitive dependence on initial conditions Folding takes far apart orbits and moves them closer together, keeping orbits bounded. Stretching and folding occurs in 1D maps as well as higher-dimensional phase space. This explains how 1D maps can capture some features of higher-dimensional systems.
David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
Strange Attractors ●
Complex structures arising from simple dynamical systems.
●
The motion on the attractor is chaotic.
●
But all orbits get pulled to the attractor.
●
Combine elements of order and disorder.
●
Motion is locally unstable, globally stable.
David P. Feldman
Introduction to Dynamical Systems and Chaos
http://www.complexityexplorer.org
