Experimental strange attractors | forced double-well oscillator

A ball rolling on a 'W'-shaped track can be shaken, leading to an experimentally verifiable strange attractor. As a time-dependent system (i.e., nonautonomous system), the dynamics can be viewed in 3 variables--position, velocity, and time (or phase of the forcing)--enough for chaos. Both this and an magnetoelastic system studied by Moon and Holmes, "Moon's beam", are described by the Duffing equation. Even the fractal dimension of the attractor can be measured as a function of the amount of mechanical damping.
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► 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: ...
Fractals • Fractals: Koch Curve, ...
Geometry of strange attractors • Geometry of Strange At...
► From 'Nonlinear Dynamics and Chaos' (online course).
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► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
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References:
Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 12: Strange Attractors
F.C. Moon and P.J. Holmes, "A magnetoelastic strange attractor", Journal of Sound and Vibration (1979) 65(2), 275-296.
doi.org/10.101...
F.C. Moon and G.-X. Li, "The fractal dimension of the two-well potential strange attractor", Physica D: Nonlinear Phenonena (1985) 17(1), 99-108.
doi.org/10.101...
J.A. Gottwald, L.N. Virgin, E.H. Dowell, "Experimental mimicry of Duffing's equation", Journal of Sound and Vibration (1992) 158(3), 447-467.
doi.org/10.101...
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 driven nonlinear oscillation Liapunov 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 nonlinear dynamics dynamical systems differential equations 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 topology Verhulst Oscillators Synchrony dynamics Lorenz equations chaotic strange attractor convection chaos chaotic Michel Henon attractor
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[@JBroadway]

Thank you for sharing your knowledge. These videos are fascinating!

[@francineochoaf.160]

Loved the video! Thanks!

[@ProfessorRoss]

You are so welcome!

[@esquire9445]

How is this different than an inverted pendulum on a cart?