Baker's Map- Simple 2D Map with a Fractal Chaotic Attractor

The baker's map, inspired by kneading bread, exhibits sensitive dependence on initial conditions, thanks to stretching, cutting, and stacking (which take the place of folding). It's orbits show chaos and it has a strange attractor with a Cantor-like cross section. We also discuss the importance of dissipation, or phase space volume shrinking. It leads to attractors, but one can have chaos without attractors, in systems that preserve volume, like fluids and other mechanical systems.

► Next, the Henon map, a 2D map with visible layers to its attractor
https://youtu.be/ZgYfzSiuIK0

► Previously, geometry of strange attractors
https://youtu.be/Y-xqI3P-OxE

► 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
Fractals https://youtu.be/Zt93gdydmbM

► 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

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

► Chapters:

0:00 Baker's map
10:41 Importance of dissipation for creating attractors. One can have chaos without attractors and vice versa
Rossler attractor 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 Liapunov exponent fractal dimension of lorenz attractor box-counting dimension crumpled paper stable focus unstable focus supercritical subcritical topological Cvitanovic equivalence genetic switch structural stability Andronov-Hopf Andronov-Poincare-Hopf small epsilon method of multiple scales two-timing Van der Pol Oscillator 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 fragility cusp unfolding perturbations structural stability emergence critical point critical slowing down supercritical bifurcation subcritical bifurcations buckling beam model change 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 Torus friends on track roller racer dynamics on torus Lorenz equations chaotic strange attractor convection chaos chaotic

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Видео Baker's Map- Simple 2D Map with a Fractal Chaotic Attractor канала Dr. Shane Ross