We follow the logic that led Lorenz to deduce the existence of the strange attractor in his equations. From phase space volume contraction, the absence of stable fixed points, limit cycles, or quasiperiodic orbits, and the absence of escape to infinity, the only possible attractor remaining is a 2-dimensional-ish object which trajectories wander around on in a seemingly erratic, aperiodic manner. This is the famous Lorenz attractor, the first known strange or chaotic attractor.
As Sherlock Holmes said in The Sign of Four, "When you have eliminated the impossible, whatever remains, however improbable, must be the truth."
The main parameter, r, is proportional to the Rayleigh number (Ra), a dimensionless number associated with buoyancy-driven flow, or convection, when Ra is above a critical value, Rc.
► Chapters
0:00 Lorenz system
0:37 What happens for r less than 1?
1:20 What happens for r greater than 1? Bifurcation diagram.
3:51 Is there a stable limit cycle?
5:59 Do trajectories escape to infinity?
8:00 Could trajectories settle into quasiperiodic behavior?
10:38 What are we left with? The strange attractor.
14:11 3D visualization of motion on the Lorenz attractor
15:51 Sketch of Lorenz attractor
► Next, quantifying chaos via Lyapunov exponents
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► Lorenz equations
Derivation [ Ссылка ]
Volume contraction & symmetry [ Ссылка ]
Fixed point analysis [ Ссылка ]
Lorenz' 1963 paper [ Ссылка ]
Simulate 3D Lorenz equations: [ Ссылка ]
► Additional background
Pitchfork bifurcations of fixed points [ Ссылка ]
Hopf bifurcations, unstable limit cycles [ Ссылка ]
Quasiperiodic motion [ Ссылка ]
Trapping region, Poincaré-Bendixson [ Ссылка ]
► Advanced lecture on the center manifold of the origin in the Lorenz system, the Lorenz manifold
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► From 'Nonlinear Dynamics and Chaos' (online course).
Playlist [ Ссылка ]
► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
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► Follow me on Twitter
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► Course lecture notes (PDF)
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References:
Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 9 Lorenz Equations
Ralph Abraham & Christopher Shaw, "Dynamics: The Geometry Of Behavior", 2nd Edition
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