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 https://youtu.be/Y-xqI3P-OxE ► Previously, an introduction to fractals https://youtu.be/Zt93gdydmbM ► Additional background Nonlinear dynamics & chaos intro https://youtu.be/bOpxQ7hGpmM 1D ODE dynamical systems https://youtu.be/Mcqrn9V7_YI Bifurcations https://youtu.be/BBd68_q3Dgg Bead in a rotating hoop https://youtu.be/pzhZ3IM9l38 2D nonlinear systems https://youtu.be/oNij9lns5RI Limit cycles https://youtu.be/9rVscJwDpBo 3D Lorenz equations introduction https://youtu.be/fIG2jtOhW0U Discrete time maps introduction https://youtu.be/-vV5A4HullY Self-similarity in bifurcation diagrams https://youtu.be/2nEBSyMsQE8 ► 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 ► 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.64.795 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