Fractal Dimension - Box-Counting & Correlation Dimension

Описание: 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 and Chaos — Introductio...  
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





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