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Horseshoe Map - Essence of Chaos, Symbolic Dynamics, and the Shift Map

A 2D map with the essential ingredients of stretching, folding, and re-injection that give rise to chaos--the Smale horseshoe map. It arises in applications; we motivate it via a homoclinic tangle, but it is more general. It has an invariant set, a Cantor set or "cloud of points", each point having a unique "name" or address which gives the point's past and future whereabouts under the map. The key is that the horseshoe map is "modeled" by something called the shift map on the space of all possible addresses (or bi-infinite sequences of 0's and 1's); an approach called symbolic dynamics. The horseshoe map gives rise to a very precise sense of chaos, in that finite precision of initial conditions leads to unpredictability in a short time; the essence of "sensitive dependence on initial conditions". It also has a countable infinity of periodic orbits and an uncountable infinity of nonperiodic orbits. As the system considered is conservative (phase space volume preserving), not dissipative, we show that we can have chaos without an attractor. ► New topics posted regularly Be notified. Subscribe https://is.gd/RossLabSubscribe​ ► Previously, Separatrix splitting leads to chaos via homoclinic tangle    • Separatrix Splitting Leads to Chaos v...   ► Want more advanced study of dynamical systems? Center manifolds, normal forms, bifurcations https://is.gd/CenterManifolds Hamiltonian & advanced mechanical systems https://is.gd/AdvancedDynamics Lagrangian & 3D rigid body dynamics https://is.gd/AnalyticalDynamics ► 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...   Geometry of strange attractors    • Geometry of Strange Attractors: Chaos...   ► 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 The horseshoe arises naturally in the analysis of transient chaos in differential equations. Roughly speaking, the Poincaré map of such systems can often be approximated by the horseshoe. During the time the orbit remains in a certain region corresponding to the square above, the stretching and folding of the map causes chaos. However, almost all orbits get mapped out of this region eventually (into the “overhang”), and then they escape to some distant part of phase space; this is why the chaos is only transient. References by the "Three Steves": Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 12: Strange Attractors Stephen Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos", 2003, Second Edition, Springer. https://www.springer.com/gp/book/9780... Stephen Smale, "Differentiable dynamical systems", Bull. Amer. Math. Soc. 73(6): 747-817 (1967). PDF: https://is.gd/smalepaper ► Advanced playlist 📚Center Manifolds, Normal Forms, and Bifurcations https://is.gd/CenterManifolds Stephen Smale Horseshoe map Horseshoe-type chaos forced damped double-well Duffing oscillator system ball in cart Moon's beam Rossler attractor equation Poincare Lyapunov exponent lorenz unstable supercritical topological structural stability oscillators driven nonlinear oscillation autonomous on the plane phase are introduced 2D ordinary differential equations bifurcation dynamics dynamical systems differential space Strogatz graphical method Fixed Point Equilibria Stability Functions Hamilton topology Synchrony Lorenz equations chaotic strange Henon #NonlinearDynamics #DynamicalSystems #SymbolicDynamics #Fractal #StrangeAttractor #DuffingSystem #Duffing #nonautonomous #HorseshoeMap #forcing #MoonsBeam #HenonMap #Bifurcation #DifferenceEquation #PoincareMap #chaos #LorenzAttractor #ChaosTheory #LyapunovExponent #Lyapunov #Oscillators #NonlinearOscillators #LimitCycle #Oscillations #VanDerPol #VectorFields #topology #geometry #IndexTheory #EnergyConservation #Hamiltonian #FixedPoint #SaddleNode #Eigenvalues #HyperbolicPoints #DifferentialEquations #dynamics #PhaseSpace #PhasePortrait #PhasePlane #Poincare #Strogatz #Wiggins #Lorenz #VectorField #GraphicalMethod #FixedPoints #EquilibriumPoints #Stability #NonlinearODEs #StablePoint #UnstablePoint #Stability #LinearStability #LinearStabilityAnalysis #StabilityAnalysis #VectorField #TwoDimensional #PopulationDynamics #Population #Logistic #Pendulum #Newton #dynamics #Poincare​ #mathematicians #maths #mathematician #KAMtori #Hamiltonian