Fractal Dimension - Box-Counting & Correlation Dimension скачать в хорошем качестве

Fractal Dimension - Box-Counting & Correlation Dimension

Fractals found in nature or in strange attractors from dynamics often require different notions of fractal dimension, like the box-counting and correlation dimension, which are easier to compute, and applicable to shapes that are not necessarily self-similar. We consider some fractals like the coastline of Great Britain, the Koch snowflake, the Sierpinski triangle, and even the fractal cow. We measure the dimension of the strange attractors we've seen, like the Lorenz attractor and the chaotic attractor for the logistic map. Fractal basin boundaries in the double pendulum are shown, as well as self-similarity in an area-preserving map from a mechanical system. ► Next, geometry of strange attractors    • Geometry of Strange Attractors: Chaos From...   ► Previously, an introduction to fractals    • Fractals: Koch Curve, Cantor Set, Non-Inte...   ► Additional background Nonlinear dynamics & chaos intro    • Nonlinear Dynamics & Chaos Introduction- L...   1D ODE dynamical systems    • Graphical Analysis of 1D Nonlinear ODEs   Bifurcations    • Bifurcations Part 1, Saddle-Node Bifurcation   Bead in a rotating hoop    • Bead on a Rotating Hoop: Deriving the Equa...   2D nonlinear systems    • 2D Nonlinear Systems Introduction- Bead in...   Limit cycles    • Limit Cycles, Part 1: Introduction & Examples   3D Lorenz equations introduction    • 3D Systems, Lorenz Equations Derived, Chao...   Discrete time maps introduction    • Maps, Discrete Time Dynamical Systems - In...   Self-similarity in bifurcation diagrams    • Logistic Map, Part 2: Bifurcation Diagram ...   ► 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 ► Fractal structure of island in Hamiltonian systems The paper I show is from James Meiss of the University of Colorado, 'Thirty Years of Turnstiles and Transport', Chaos (2015) https://doi.org/10.1063/1.4915831 But I also like an earlier paper of Prof. Meiss which taught me a lot : 'Symplectic maps, variational principles, and transport', Reviews of Modern Physics (1992) https://doi.org/10.1103/RevModPhys.64... I also have some video lectures on tori in Hamiltonian systems at https://is.gd/AdvancedDynamics References: Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 11: Fractals 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 Lyapunov exponent fractal dimension of lorenz attractor box-counting dimension crumpled paper stable focus unstable focus supercritical subcritical topological equivalence structural stability 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 f 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 #Fractals #StrangeAttractor #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