Unlocking Chaos: Exploring the Logistic Map's Bifurcation Diagram & Self-Similarity 🔍

The logistic map has an iconic bifurcation diagram, showing chaotic attractors intermingled with periodic windows, the largest being the period-3 window. We numerically explore trajectories via cobweb diagrams and properties of the bifurcation diagram, like self-similarity, and hint at how it arises. ► Next, the bifurcation that leads to period doubling Be notified. Subscribe https://is.gd/RossLabSubscribe​ ► Logistic map introduction https://youtu.be/PVo1mHnU7WU ► 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 Twitter https://twitter.com/RossDynamicsLab ► Course lecture notes (PDF) https://is.gd/NonlinearDynamicsNotes ► Advanced lecture on maps from another of my courses https://youtu.be/NYoA5B2qsdc ► Courses and Playlists by Dr. Ross 📚Attitude Dynamics and Control https://is.gd/SpaceVehicleDynamics 📚Nonlinear Dynamics and Chaos https://is.gd/NonlinearDynamics 📚Hamiltonian Dynamics https://is.gd/AdvancedDynamics 📚Three-Body Problem Orbital Mechanics https://is.gd/SpaceManifolds 📚Lagrangian and 3D Rigid Body Dynamics https://is.gd/AnalyticalDynamics 📚Center Manifolds, Normal Forms, and Bifurcations https://is.gd/CenterManifolds References: Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 10: One-Dimensional Maps 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 chaotic #NonlinearDynamics #DynamicalSystems #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 #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