Maps, Discrete Time Dynamical Systems - Introduction and Use

We introduce a new class of dynamical systems in which time is discrete, rather than continuous. These systems are known as difference equations, recursion relations, iterated maps, discrete maps or simply maps. Some phenomena are better modeled mathematically by discrete time maps, rather than differential equations. Maps are the more general class of dynamical system, since the numerical algorithms which solve differential equations are actually discrete time-step maps which are just a small subset of all possible maps. Other uses of maps are to simplify the study of phenomena that are otherwise complicated in a continuous setting. We are focusing on maps because one-dimensional maps can show chaos, whereas differential equations, like the Lorenz equations, require three dimensions for chaos. ► Next, the logistic map, an example map with rich behavior https://youtu.be/PVo1mHnU7WU ► Additional background 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 course of mine https://youtu.be/NYoA5B2qsdc References: Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 10: One-Dimensional Maps 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 #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

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