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 At...
► Previously, an introduction to fractals
• Fractals: Koch Curve, ...
► Additional background
Nonlinear dynamics & chaos intro • Nonlinear Dynamics & C...
1D ODE dynamical systems • Graphical Analysis of ...
Bifurcations • Bifurcations Part 1, S...
Bead in a rotating hoop • Bead in a Rotating Hoo...
2D nonlinear systems • 2D Nonlinear Systems I...
Limit cycles • Limit Cycles, Part 1: ...
3D Lorenz equations introduction • 3D Systems, Lorenz Equ...
Discrete time maps introduction • Maps, Discrete Time Dy...
Self-similarity in bifurcation diagrams • Logistic Map, Part 2: ...
► From 'Nonlinear Dynamics and Chaos' (online course).
Playlist is.gd/Nonlinea...
► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
Subscribe is.gd/RossLabS...
► Follow me on Twitter
/ rossdynamicslab
► Course lecture notes (PDF)
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► 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)
doi.org/10.106...
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)
doi.org/10.110...
I also have some video lectures on tori in Hamiltonian systems at
is.gd/Advanced...
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
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[@jhupiterz]

Great video, thanks. I am looking into using the correlation dimension to map the "shape compelxity" of 3D objects, as represented by clouds of nodes. Each node would then be attributed a dimension, thus creating a map of my object. I haven't found any paper that tried that, is there a reason for this? Maybe the correlation dimension is not appropriate for 3D objects? Thanks!

[@ProfessorRoss]

I don't know why that is. I think the correlation dimension should be general for n-D objects. But I'm not an expert in it.

[@dietbohe8604]

Would the fractal dimension vary if the coastlines all had the same length but were in different shapes?

[@ProfessorRoss]

Good question. One of the features of a curve (like a coastline) that's also a fractal is that they have infinite length. So while it seems non-intuitive, the coastline of, for example., Australia is also infinite, even though it encloses a different area than Great Britain's infinite coastline. Let me know if this answer helps.

[@Yangilly]

I am wondering can this method be used for A high-resolution image? Look forward to your reply, thanks!

[@robfrost1]

What would a "box" be in a totally disconnected space such as a subset of the 2-dic numbers? This must have an answer as you referenced it can be done for the Cantor set. Or was the Cantor set considered as a subset of the Real number line topology?

[@jarekk.8247]

The universe is probably a fractal on the largest scale with the number of dimensions equal to π.

[@ProfessorRoss]

I have not heard that before. It’s worth considering. As a first step, see if you can design a set in 4 dimensions with a fractal dimension of pi.

[@jarekk.8247]

@@ProfessorRoss As proof, I can provide a formula for calculating the fine structure constant containing mathematical constants, including the PI number as the number of dimensions on the largest scale. The result agrees with current experimental data within the statistical error. α = 1/[(2^4+5^4)^φ*e^5]^(1/π) α = 1/(641^φ*e^5)^(1/π) = 0,007297352568 φ = 1,6180339887... golden ratio, e = 2,7182818284... (Napier's constant, Euler's number) or α = 1/[5164926^(1/π)] = 0,007297352564

[@jarekk.8247]

@@ProfessorRoss This formula shows that the higher dimensions, i.e. the fourth and fifth, influence the strength of particle interaction by determining the alpha value of the fine structure constant. But the basis for all this is the fractal nature of our reality in the PI dimension.

[@ogunstega7348]

HI Dr Ross, I am wondering how do you make the plot of those highlands and Surrounding islands when zoomed? can it be done with MATLAB?

[@whooshie5172]

You are really smart dude!

[@luccalus]

Hi Dr Ross. The Fractals on the Taurus were v cool. About the islands in the Hamiltonian system - any references?

[@ProfessorRoss]

Hi Ahmed, regarding the 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) doi.org/10.1063/1.4915831 But I also like an earlier paper of his which taught me a lot : 'Symplectic maps, variational principles, and transport' (1992) drive.google.com/file/d/1huX7JqfiCEZVo8VvFatqn0D4uhhfa50u/view?usp=sharing I also have some video lectures on tori in Hamiltonian systems at is.gd/AdvancedDynamics

[@brendawilliams8062]

111….( x) 3. Eleven times. Will change your direction downward by ( x) 3

[@brendawilliams8062]

Anyways boxes are cool