- Classical chaotic scattering experiments were performed on ten different
four hill potentials, unique under x-axis reflection, given by:
V(x,y) = x²y²e⁻⁽ˣ² ⁺ ʸ²⁾ σᵢ where σᵢ was ±1 depending upon quadrant. Our work
was based on a paper (9) that only studied the case where σᵢ =1 in all quadrants.
The system used was a time independent Hamiltonian with energy conservation.
Chaos manifested itself in our experiment in two ways. First, below some
critical energy, particles would become almost trapped by the system. Second,
above the critical energy, the particles directly strike the saddle point.
Trajectories were integrated using a standard 4ᵗʰ order Runge-Kutta ODE
solver. Scattering angle and time delay were measured as a function of the impact
parameter. Chaos was seen in these scattering functions as discontinuities. The
Lyapunov exponent was measured for a number of energies using a brute force
method (separation of nearby trajectories). The capacity dimension was measured
using the uncertainty exponent method (13).
The critical energy was measured using both capacity dimension and
Lyapunov exponent data. Five of the potentials had a critical energy within error
bounds of E0.039: V₂c, V₃, V₄ₐ, V₄b, V₆. Three were with error bounds of
Ec=0.135: V₁, V₂ₐ, V₅b. Potential V₂b had a value of 0.071 and V₅ₐ had 0.092.
All bifurcations in this system were abrupt. Three of the potentials were
generic abrupt: V₁, V₂ₐ, V₅b. Six of the potentials had abrupt crisis bifurcations:
V₂b, V₂, V₃, V₄ₐ, V₄b, V₆. Potential V5A had abrupt nonhyperbolic bifurcations.
The Lyapunov exponent for all ten potentials had a linear high energy
behavior that had not been seen before. The systems showed chaos and the largest
Lyapunov exponents for them were measured (below the critical energy) to be:
V₁=0.446 + 0.013, V₂ₐ=0.472 ± 0.007, V₂b=0.663 ± 0.007, V₂c=0.683 + 0.010,
V₃=0.655 ± 0.011, V₄ₐ=0.7l4 + 0.010, V₄b=0.743 + 0.003, V₅ₐ=0.74l ± 0.009,
V₅b=0.46l ± 0.006, V₆=0.697 ± 0.008.
We then concluded that systems that contained attractors were more chaotic
than the one that did not. In addition, we saw that the sign of the hills on the left
side of the system governed the shape of the scattering functions but not all of the