Oscillator double-well

This has the form of a double-well oscillator coupled to a transverse harmonic mode. The adiabatic approximation was discussed in great detail from a number of quantum-mechanical calculations, and it was shown how the two-dimensional problem could be reduced to a one-dimensional model with an effective potential where the barrier top is lowered and a third well is created at the center as more energy is pumped into the transverse mode. From this change in the reactive potential follows a marked increase in the reaction rate. Classical trajectory calculations were also performed to identify certain specifically quanta effects. For the higher energies, both classical and quantum calculations give parallel results. 

Basins for damped double-well oscillator) Suppose we add a small amount of damping to the double-well oscillator of Example 6.5.2. The new system is X = y, y = -by + x-x, where 0 < b 1. Sketch the basin of attraction for the stable fixed point (x, y ) =(1,0). Make the picture large enough so that the global structure of the basin is clearly indicated. 

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... 

Basins for tne unforced oscillator) Sketch the basins for the weakly damped double-well oscillator (12.5.1) in the unforced case when F-Q. How does their shape depend on the size of the damping What happens to the basins as the damping tends to zero What implications does this have for the predictability of the unforced system ... 

Coexisting chaos and limit cycle) Consider the double-well oscillator (12.5,1) with parameters S = 0.15, F =0.3, and (O — 1. Show numerically that the system has at least two coexisting attractors a large limit cycle and a smaller strange attractor. Plot both in a Poincare section,... 

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

Figure Al.1.4. Waveflinctions for the four lowest states of the double-well oscillator. The ground-state wavefunction is at the bottom and the others are ordered from bottom to top in terms of increasing energy. 

Figure Al.1.5. Ground state wavefunction of the double-well oscillator, as obtained in a variational calculation using eight basis functions centred at the origin. Note the spurious oscillatory behaviour near the origin and the location of the peak maxima, both of which are well inside the potential minima.

This approach gives very good results for the quartic anharmonic double well oscillator V x) = —5x + x (i.e. ground and first excited states), and the quartic potential V(x) = x. The first has all real turning points whereas the second has real and pure imaginary turning points. The results of this analysis are given in Tables 8 and 9, respectively (Handy and Brooks (2000)). 

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