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Lorenz Attractor- How It Was Found

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    • Lyapunov Exponents & Sensitive Dependence ...   ► Lorenz equations Derivation    • 3D Systems, Lorenz Equations Derived, Chao...   Volume contraction & symmetry    • Lorenz Equations Properties: Volume Contra...   Fixed point analysis    • Lorenz Equations Fixed Point Analysis   Lorenz' 1963 paper https://is.gd/lorenzpaper Simulate 3D Lorenz equations: https://is.gd/Lorenz ► Additional background Pitchfork bifurcations of fixed points    • Bifurcations Part 3- Pitchfork Bifurcation   Hopf bifurcations, unstable limit cycles    • Bifurcations in 2D, Part 2: Hopf Bifurcati...   Quasiperiodic motion    • Coupled Oscillators, Quasiperiodicity, Syn...   Trapping region, Poincaré-Bendixson    • Limit Cycles, Part 3: Poincare-Bendixson W...   ► Advanced lecture on the center manifold of the origin in the Lorenz system, the Lorenz manifold    • Center Manifolds Depending on Parameters- ...   ► From 'Nonlinear Dynamics and Chaos' (online course). Playlist https://is.gd/NonlinearDynamics ► Dr. Shane Ross, Virginia Tech professor (Caltech PhD) Subscribe https://is.gd/RossLabSubscribe​ ► Follow me on Twitter   / rossdynamicslab   ► Course lecture notes (PDF) https://is.gd/NonlinearDynamicsNotes References: Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 9 Lorenz Equations Ralph Abraham & Christopher Shaw, "Dynamics: The Geometry Of Behavior", 2nd Edition fractal dimension of lorenz attractor box-counting dimension crumpled paper stable focus unstable focus supercritical subcritical topological equivalence genetic switch structural stability Andronov-Hopf Andronov-Poincare-Hopf small epsilon method of multiple scales two-timing Van der Pol Oscillator Duffing oscillator nonlinear oscillators nerve cells driven current nonlinear circuit glycolysis biological chemical oscillation Liapunov gradient systems Conley index theory gradient system autonomous phase plane 2D ordinary differential equations 2d ODE vector field topology cylinder bifurcation robustness fragility cusp unfolding perturbations structural stability emergence critical point critical slowing down supercritical bifurcation subcritical bifurcations buckling beam model change of stability nonlinear dynamics dynamical systems differential equations dimensions phase space Poincare Strogatz graphical method Fixed Point Equilibrium Equilibria Stability Stable Point Unstable Point Linear Stability Analysis Vector Field Two-Dimensional 2-dimensional Functions Hamiltonian Hamilton streamlines weather vortex point vortices pendulum Newton's Second Law Conservation of Energy topology Verhulst Oscillators Synchrony Torus on torus Lorenz equations chaotic strange attractor convection chaos chaotic #NonlinearDynamics #DynamicalSystems #LorenzAttractor #chaos #Oscillators #Synchrony #Torus #Bifurcation #Hopf #HopfBifurcation #NonlinearOscillators #AveragingTheory #LimitCycle #Oscillations #nullclines #RelaxationOscillations #VanDerPol #VanDerPolOscillator #LimitCycles #VectorFields #topology #geometry #IndexTheory #EnergyConservation #Hamiltonian #Streamfunction #Streamlines #Vortex #SkewGradient #Gradient #PopulationBiology #FixedPoint #DifferentialEquations #SaddleNode #Eigenvalues #HyperbolicPoints #NonHyperbolicPoint #CuspBifurcation #CriticalPoint #buckling #PitchforkBifurcation #robust #StructuralStability #DifferentialEquations #dynamics #dimensions #PhaseSpace #PhasePortrait #PhasePlane #Poincare #Strogatz #Wiggins #Lorenz #VectorField #GraphicalMethod #FixedPoints #EquilibriumPoints #Stability #NonlinearODEs #StablePoint #UnstablePoint #Stability #LinearStability #LinearStabilityAnalysis #StabilityAnalysis #VectorField #TwoDimensional #Functions #PopulationGrowth #PopulationDynamics #Population #Logistic #GradientSystem #GradientVectorField #Cylinder #Pendulum #Newton #LawOfMotion #dynamics #Poincare​ #mathematicians #maths #mathstudents #mathematician #KAMtori #Hamiltonian