Скачать с ютуб Lorenz System Bifurcation Diagram- Exploring Parameter Space в хорошем качестве

Lorenz System Bifurcation Diagram- Exploring Parameter Space

What happens if we change the parameters? We take a tour, finding pairs of limit cycles linked through each other, transient chaos, noisy periodicity, as well as strange attractors, and windows of periodicity intermingled within chaos. The Lorenz manifold, the stable manifold of the origin, is also significant. We show some art inspired by the Lorenz system. ► Next, the dynamics on the Lorenz strange attractor    • Dynamics on Lorenz Attractor | Lorenz...   ► Lorenz equations Simulate it! https://is.gd/Lorenz 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   Lorenz' original 1963 paper (PDF) https://is.gd/lorenzpaper ► Additional background Definitions of chaos and attractor    • Chaotic Attractors: a Working Definit...   Lyapunov exponents to quantify chaos    • Lyapunov Exponents & Sensitive Depend...   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, 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 Doedel, Eusebius J., Bernd Krauskopf, and Hinke M. Osinga. "Global organization of phase space in the transition to chaos in the Lorenz system." Nonlinearity 28, no. 11 (2015): R113. 📕PDF of paper: https://iopscience.iop.org/article/10... 📕PDF of preprint: https://web.archive.org/web/201602111... largest Liapunov exponent 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 nonlinear oscillation nerve cells driven current nonlinear circuit glycolysis biological chemical oscillation Liapunov gradient systems Conley index theory gradient system autonomous on the plane phase plane are introduced 2D ordinary differential equations 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 topology Verhulst Oscillators Synchrony Torus friends on track roller racer dynamics on torus Lorenz equations chaotic strange attractor convection chaos chaotic #NonlinearDynamics #DynamicalSystems #Bifurcation #LyapunovExponent #Lyapunov #LorenzAttractor #chaos #Oscillators #Synchrony #Torus #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