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Period-Doubling Route to Chaos | Universality, Experiments, ODEs, and Maps
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The bifurcation pattern seen in the logistic map turns out to be universal across a wide range of dynamic systems, including systems of differential equations, and more importantly, experiments. It seems the logistic map was just the first example of a large class of systems, which led to the discovery of new constants in the universe.
► Next, Feigenbaum's renormalization analysis of period-doubling, demonstrating where the universal constants come from
► Logistic map
► Additional background
► Ghosts and bottlenecks
► From 'Nonlinear Dynamics and Chaos' (online course).
► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
► Follow me on Twitter
► Course lecture notes (PDF)
► Advanced lecture on maps from another of my courses
► Robert May's 1976 article introducing the logistic map (PDF)
References:
Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 10: One-Dimensional Maps
► *Free online courses by Dr. Ross*
📚Nonlinear Dynamics & Chaos
📚Hamiltonian Dynamics
📚Lagrangian & 3D Rigid Body Dynamics
📚Center Manifolds, Normal Forms, and Bifurcations
📚Attitude Dynamics & Control
📚3-Body Problem Orbital Mechanics
📚Space Manifolds
intermittent period doubling cascade period-doubling bifurcation flip bifurcation discrete map analog of logistic equation Poincare map 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
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► Next, Feigenbaum's renormalization analysis of period-doubling, demonstrating where the universal constants come from
► Logistic map
► Additional background
► Ghosts and bottlenecks
► From 'Nonlinear Dynamics and Chaos' (online course).
► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
► Follow me on Twitter
► Course lecture notes (PDF)
► Advanced lecture on maps from another of my courses
► Robert May's 1976 article introducing the logistic map (PDF)
References:
Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 10: One-Dimensional Maps
► *Free online courses by Dr. Ross*
📚Nonlinear Dynamics & Chaos
📚Hamiltonian Dynamics
📚Lagrangian & 3D Rigid Body Dynamics
📚Center Manifolds, Normal Forms, and Bifurcations
📚Attitude Dynamics & Control
📚3-Body Problem Orbital Mechanics
📚Space Manifolds
intermittent period doubling cascade period-doubling bifurcation flip bifurcation discrete map analog of logistic equation Poincare map 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
#NonlinearDynamics #DynamicalSystems #ChaosTheory #Bifurcation #LogisticMap #PeriodDoubling #DifferenceEquation #PoincareMap #chaos #LorenzAttractor #LyapunovExponent #Lyapunov #Liapunov #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 #PitchforkBifurcation #robust #StructuralStability #DifferentialEquations #dynamics #dimensions #PhaseSpace #PhasePortrait #PhasePlane #Poincare #Strogatz #Lorenz #VectorField #GraphicalMethod #FixedPoints #EquilibriumPoints #Stability #NonlinearODEs #StablePoint #UnstablePoint #LinearStability #StabilityAnalysis #VectorField #TwoDimensional #Functions #PopulationGrowth #PopulationDynamics #Population #Logistic #GradientSystem #GradientVectorField #Cylinder #Pendulum #Newton #LawOfMotion #dynamics #Poincare #mathematicians #maths #mathstudents #mathematician #mathfacts #mathskills #mathtricks #KAMtori #Hamiltonian
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