Geometry of Strange Attractors Chaos From Stretching and Folding in Phase Space

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Strange attractors in dynamical systems are often due to a mechanism of repeated stretching and folding of the phase space. We discuss this chaos-generating mechanism abstractly in terms of kneading dough for a croissant: the pastry map. In the culinary spirit of the pastry map, Otto Rössler found inspiration in a taffy-pulling machine, creating the simplest 3D ODE generating chaos with a single nonlinear term, and we analyze the stretching and folding along the Rössler attractor. • ► Next, the baker's map, a 2D map with stretching folding •    • Baker's Map- Simple 2D Map with a Fra...   • ► Additional background • Nonlinear dynamics chaos intro    • Nonlinear Dynamics   Chaos Introducti...   • 1D ODE dynamical systems    • Graphical Analysis of 1D Nonlinear ODEs   • Bifurcations    • Bifurcations Part 1, Saddle-Node Bifu...   • Bead in a rotating hoop    • Bead in a Rotating Hoop, Part 1- Deri...   • 2D nonlinear systems    • 2D Nonlinear Systems Introduction- Be...   • Limit cycles    • Limit Cycles, Part 1: Introduction   ...   • 3D Lorenz equations introduction    • 3D Systems, Lorenz Equations Derived,...   • Discrete time maps introduction    • Maps, Discrete Time Dynamical Systems...   • Self-similarity in bifurcation diagrams    • Logistic Map, Part 2: Bifurcation Dia...   • Fractals    • Fractals: Koch Curve, Cantor Set, Non...   • ► 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 12: Strange Attractors • Otto Rossler, An equation for continuous chaos , Physics Letters, 1976 • https://is.gd/rosslerpaper • Ralph Abraham and Christopher Shaw, Dynamics: The Geometry Of Behavior , 1992 • https://www.amazon.com/dp/0201567172 • • • • Rossler attractor Mandelbrot set capacity self-similar dimension box-counting dimension correlation dimension 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 Cvitanovic 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 #StrangeAttractor #Fractals #Rossler #Mandelbrot #MandelbrotSet #Universality #Renormalization #Feigenbaum #PeriodDoubling #Bifurcation #LogisticMap #Cvitanovic #DifferenceEquation #PoincareMap #chaos #LorenzAttractor #ChaosTheory #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 #mathstudents #mathematician #mathfacts #mathskills #mathtricks #KAMtori #Hamiltonian

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