Henon Map- Strange Attractor with Fractal Microstructure

Hénon wanted to see the infinite complex of surfaces suspected in the Lorenz attractor, so he devised a 2-D map with a strange attractor to illustrate the fractal microstructure; the first direct visualization of such a structure. We discuss the simple composition of the map--stretching, folding, and re-injection--as well as its properties, meant to mimic the Lorenz differential equations and create an attractor.

► Next, experimental strange attractors
Experimental strange attractors | for...

► Previously, Baker's map | simple 2D map with a fractal chaotic attractor
Baker's Map- Simple 2D Map with a Fra...

► Stretching and folding on the Rössler attractor
Geometry of strange attractors Geometry of Strange Attractors: Chaos...

► Additional background
Nonlinear dynamics & chaos intro Nonlinear Dynamics & Chaos Introducti...
1D ODE dynamical systems Graphical Analysis of 1D Nonlinear ODEs
Bifurcations Bifurcations Part 1, Saddle-Node Bifu...
Bead in a rotating hoop Bead in a Rotating Hoop, Part 1- Deri...
2D nonlinear systems 2D Nonlinear Systems Introduction- Be...
Limit cycles Limit Cycles, Part 1: Introduction & ...
3D Lorenz equations introduction 3D Systems, Lorenz Equations Derived,...
Discrete time maps introduction Maps, Discrete Time Dynamical Systems...
Self-similarity in bifurcation diagrams Logistic Map, Part 2: Bifurcation Dia...
Fractals Fractals: Koch Curve, Cantor Set, Non...
Geometry of strange attractors Geometry of Strange Attractors: Chaos...

► 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
Twitter: RossDynamicsLab

► Course lecture notes (PDF)
https://is.gd/NonlinearDynamicsNotes

► Henon map simulator
https://is.gd/henonmap

References:
Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 12: Strange Attractors

Hénon's 1976 paper: A two-dimensional mapping with a strange attractor
https://is.gd/henonpaper

Prof Ghrist Math, ADS : Vol 4 : Chapter 5.2 : The Henon Map
ADS : Vol 4 : Chapter 5.2 : The Henon...

Chapters:

0:00 Motivation for Hénon map
5:15 The map as a composition of simple operations
10:54 Properties of the Henon map
24:02 Henon attractor


Rossler attractor Mandelbrot set capacity self-similar dimension box-counting dimension correlation dimension period doubling bifurcation bifurcation discrete map analog logtisc equation Poincare map largest Liapunov exponent fractal dimension of lorenz attractor box-counting dimension crumpled paper unstable focus supercritical subcritical topological structural stability epsilon method of multiple scales Oscillator Duffing oscillator nonlinear oscillators nonlinear oscillation nerve cells driven current nonlinear circuit oscillation Liapunov gradient systems Conley index gradient system autonomous on the plane phase plane are introduced 2D ordinary differential equations bifurcation robustness fragility cusp unfolding perturbations structural stability emergence critical point critical supercritical bifurcation subcritical 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 dynamics Lorenz equations chaotic strange attractor convection chaos chaotic Michel Henon attractor

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