Lorenz Equations Fixed Point Analysis

The 3-D Lorenz system has equilibrium points whose stability is analyzed. The origin is always a fixed point and is a global attractor for r less than 1, proved via a Lyapunov function argument. A pitchfork bifurcation at r=1 creates two symmetric fixed points, which will become the middle of the two rotating patterns on the attractor. ► Next, deducing the Lorenz attractor https://youtu.be/N69uJMTObKY ► Lorenz equations Derivation and chaotic waterwheel experiment https://youtu.be/fIG2jtOhW0U Volume contraction and symmetry https://youtu.be/ivPG1gudCwM Lorenz' original 1963 paper (PDF) https://is.gd/lorenzpaper Simulate the 3D Lorenz equations: https://is.gd/Lorenz ► From 'Nonlinear Dynamics and Chaos' (online course). Playlist https://is.gd/NonlinearDynamics ► Lyapunov functions https://youtu.be/0uSTZFM0GdE ► Additional background on 2D dynamical systems Intro to 2D dynamical systems https://youtu.be/oNij9lns5RI Phase plane introduction https://youtu.be/U4IM7HFzcuY Classifying 2D fixed points https://youtu.be/7Ewe_tVa5Fs Gradient systems https://youtu.be/uGUzPZzvPWQ Index theory https://youtu.be/bO8FxxpocNQ Limit cycles https://youtu.be/9rVscJwDpBo Averaging theory https://youtu.be/UzQU1nyM-No ► Advanced lecture on the pitchfork bifurcation of the origin in the Lorenz system Part 1 https://youtu.be/twsO1e3Hqws Part 2 https://youtu.be/Fz-yj_qEvuA ► Dr. Shane Ross, Virginia Tech professor (Caltech PhD) Subscribe https://is.gd/RossLabSubscribe​ ► Follow me on Twitter https://twitter.com/RossDynamicsLab ► Course lecture notes (PDF) https://is.gd/NonlinearDynamicsNotes ► *Free online courses by Dr. Ross* 📚Nonlinear Dynamics & Chaos https://is.gd/NonlinearDynamics 📚Hamiltonian Dynamics https://is.gd/AdvancedDynamics 📚Lagrangian & 3D Rigid Body Dynamics https://is.gd/AnalyticalDynamics 📚Center Manifolds, Normal Forms, and Bifurcations https://is.gd/CenterManifolds 📚Attitude Dynamics & Control https://is.gd/SpaceVehicleDynamics 📚3-Body Problem Orbital Mechanics https://is.gd/3BodyProblem 📚Space Manifolds https://is.gd/SpaceManifolds References: Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 9: Lorenz Equations cardiac 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 nonlinear oscillation nerve cells driven current nonlinear circuit glycolysis biological chemical oscillation adenosine diphosphate ADP fructose Liapunov gradient systems passive dynamic biped walker Tacoma Narrows bridge collapse Charles Conley index theory gradient system autonomous on the plane phase plane are introduced 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 dynamics point vortices pendulum Newton's Second Law Conservation of Energy topology Verhulst Oscillators Synchrony Torus friends on track roller racer dynamics 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 #mathsmemes #math4life #mathstudents #mathematician #mathfacts #mathskills #mathtricks #KAMtori #Hamiltonian