Experiments with a Malkus–Lorenz water wheel: Chaos and Synchronization
(Color online) Experimental implementation of the Malkus–Lorenz water wheel, constructed using a bicycle wheel, syringes, and rare earth magnets forming a magnetic brake. The schematics on the right show a side view (bottom) and a top view (top) of the wheel. (See text for details.)
(Color online) Measurements of angle versus time for an empty wheel are compared with fits that take into account damping due to viscous friction (two-parameter fit) and a combination of viscous and kinetic friction (three-parameter fit). (a) Angle-data and fits. (b) Residuals of fits (triangles and squares) and angular velocity ω as determined from the two-parameter fit (solid line).
Measurements of the water volume in the syringe as a function of time and the two-parameter fit of Eq. (10). The dotted line indicates the value of Vb (see text). (a) Syringe without needle (V off = 4 cm3). Fit-result: V 0 = 29 cm3, k = 0.10 s−1 (b) Syringe with needles (V off = 18 cm3) of 16 gauge (Fit-result: V 0 = 29 cm3, k = 0.017 s−1) and 18 gauge (Fit-result: V 0 = 29 cm3, k = 0.006 s−1). (c) Geometry of the syringe, consisting of a cylindrical main body (radius rc , height h), a tapered section, and either a nozzle or a needle as exit “pipe” (radius rp , height zp ). The total height of the tapered section and exit “pipe” is ℓ.
Numerically determined largest Lyapunov exponent as a function of ρ and σ. White corresponds to a zero exponent implying periodic oscillation of ω. Red colors correspond to positive and blue colors to negative exponents, indicating, respectively, chaos and steady state evolution of ω. The approximate upper boundary of the experimentally accessible region is shown by the solid black line. The yellow squares indicate parameter values corresponding to the chaotic and periodic experimental time series in Fig. 5.
(Color online) (a) and (b) The experimental time series of the rescaled angular velocity x = ω/k as a function of the dimensionless time s = kt is shown as the upper trace, and the corresponding numerical solution x num as the lower trace. The parameters are (a) σ ≈ 3.6 and ρ ≈ 140, resulting in periodic oscillations, and (b) σ ≈ 2.5 and ρ ≈ 66, resulting in chaotic oscillations. A corresponding two dimensional time-delay embedding of the experimental data is shown in (c) for the periodic and (d) for the chaotic case.
(a) Experimental time-series (solid line) and model output (crosses). (b) Difference between model and experiment.
Geometric illustration of the QR decomposition scheme for computing Lyapunov exponents.
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