Lyapunov Exponents & Sensitive Dependence on Initial Conditions

One signature of chaos is sensitive dependence on initial conditions, quantified using Lyapunov exponents, which measure exponential divergence of nearby trajectories in phase space. When the largest Lyapunov exponent is positive, that is diagnostic of chaos (but care must taken).

When a system has a positive Liapunov exponent, there is a time horizon beyond which prediction breaks down, called the Lyapunov time. We use the weather as an example.

► Next, working definitions of 'chaos' and 'attractor'
   • Chaotic Attractors: a Working Definit...  

► Lorenz equations
Derivation and chaotic waterwheel    • 3D Systems, Lorenz Equations Derived,...  
Volume contraction and symmetry    • Lorenz Equations Properties: Volume C...  
Fixed point analysis    • Lorenz Equations Fixed Point Analysis  
Deducing the Lorenz attractor    • Lorenz Attractor- How It Was Found  

► Additional background
Pitchfork bifurcations of fixed points    • Bifurcations Part 3- Pitchfork Bifurc...  
Hopf bifurcations, unstable limit cycles    • Bifurcations in 2D, Part 2: Hopf Bifu...  
Quasiperiodic motion on a torus    • Coupled Oscillators, Quasiperiodicity...  
Trapping region, Poincaré-Bendixson    • Limit Cycles, Part 3: Poincare-Bendix...  

► Advanced lecture on the center manifold of the origin in the Lorenz system
   • Center Manifolds Depending on Paramet...  

► From 'Nonlinear Dynamics and Chaos' (online course).
Playlist https://is.gd/NonlinearDynamics

► Dr. Shane Ross, Chaotician, Virginia Tech professor (Caltech PhD)
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► Course lecture notes (PDF)
https://is.gd/NonlinearDynamicsNotes

References:
Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 9: Lorenz Equations

► Courses and Playlists by Dr. Ross

📚Attitude Dynamics and Control
https://is.gd/SpaceVehicleDynamics

📚Nonlinear Dynamics and Chaos
https://is.gd/NonlinearDynamics

📚Hamiltonian Dynamics
https://is.gd/AdvancedDynamics

📚Three-Body Problem Orbital Mechanics
https://is.gd/SpaceManifolds

📚Lagrangian and 3D Rigid Body Dynamics
https://is.gd/AnalyticalDynamics

📚Center Manifolds, Normal Forms, and Bifurcations
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