Understanding the Logistic Map: The Path to Chaos Through Period Doubling 📈

The logistic map is a simple discrete model of population growth with very complicated dynamics. It depends on a growth rate parameter r. We consider the dynamics at various values of the parameter and find that there’s a branch of stable fixed points which bifurcates into stable attractor cycles of period 2, 4, 8, 16, .... The period-doubling cascade. The bifurcation diagram shows chaos intermingled with periodic windows. ► Next, the bifurcation diagram and self-similarity https://youtu.be/2nEBSyMsQE8 ► Additional background Introduction to mappings https://youtu.be/-vV5A4HullY Logistic equation (1D ODE) https://youtu.be/iOumaIR5gzA Lorenz map on strange attractor https://youtu.be/P4tjxOFnGNo Lorenz equations introduction https://youtu.be/fIG2jtOhW0U Definitions of chaos and attractor https://youtu.be/uDpYU01dhk0 Lyapunov exponents to quantify chaos https://youtu.be/22VVVn1zPdM ► Robert May's 1976 article introducing the logistic map (PDF) https://is.gd/logisticmappaper ► 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 X https://x.com/RossDynamicsLab ► Course lecture notes (PDF) https://is.gd/NonlinearDynamicsNotes ► Advanced lecture on maps from another course of mine https://youtu.be/NYoA5B2qsdc References: Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 10: One-Dimensional Maps ► *Related Courses and Series Playlists 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, & Bifurcations https://is.gd/CenterManifolds 📚3-Body Problem Orbital Dynamics https://is.gd/3BodyProblem 📚Space Manifolds https://is.gd/SpaceManifolds 📚Space Vehicle Dynamics https://is.gd/SpaceVehicleDynamics period doubling cascade period-doubling bifurcation flip bifurcation discrete map analog of logistic equation Ecological Forecasting 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 chaotic #NonlinearDynamics #DynamicalSystems #PopulationGrowth #EcologicalForecasting #LogisticMap #PeriodDoubling #DifferenceEquation #PoincareMap #chaos #LorenzAttractor #LyapunovExponent #Lyapunov #Liapunov #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