Oscillator forced double-well

Plate 4 Maps of the short-term behavior of the periodically forced double-well oscillator. Equations, parameters, and color code as in Plate 3. However, instead of showing the system s asymptotic behavior, these plates show the color-coded value ofx(z) after only 1, 2, 3, and 4 drive cycles, respectively. The red and blue regions correspond to initial conditions that converge rapidly to one of the two attractors. A rainbow of colors is found near the basin boundary, because those initial conditions lead to trajectories that linger far from either attractor during the time shown. 

This section provides a glimpse of some of the phenomena that arise in a particular forced oscillator, the driven double-well oscillator studied by Francis Moon... 

Plate 3 Fractal basin boundaries for the periodically forced double-well oscillator... 

Fig. 8. (A) Measured forces between two charged mica surfaces in 10" M KCl, where beyond 30 A (and out to 500 A) the repulsion is well described by conventional electrostatic double-layer force theory. Below 30 A there is an additional hydration repulsion, with oscillations superimposed below 15 A. (B) Forces between two uncharged lecithin bilayers in the fluid state in water. At long range there is an attractive van der Waals force, and at short range (i.e., below 25 A) there is a monotonically repulsive steric hydration force. (C) Forces between two hydrophobized mica surfaces in water where the hydrophobic interaction is much stronger than could be expected from van der Waals forces alone. From Israelachvili and Marra (1986).

There also appears a fundamental force component which is proportional to the surface potential difference between the sample and the tip. We scan the tip over the surface typically at the rate of 0.1 Hz/line, while the tip-surface distance is controlled to give a constant F2a- The actual SMM system consists of a commercial AFM (Nanoscope, Digital Instruments) and a double-lock-in amplifier to selectively detect the oscillating electric force signals. We used AFM cantilevers with sharpened tips purchased from Olympus (OMCL-RC-800-PSA) with a spring constant of k=037 N/m. The AC voltage was 3 Vpp at 7.2 kHz, which should be chosen well below the resonance frequency of the cantilever. The resultant oscillation was about 2 nm or so at the closest approach of the tip to the surface. The lateral resolution was 10 nm with the potential sensitively of 1 mV. 

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