Chaos in cosmological Hamiltonians

This paper summarizes a numerical investigation which aimed to identify and characterize regular and chaotic behavior in time-dependent Hamiltonians H(r, p, t) = p(2)/2 + V(r, t), with V = R(t)V0(r) or V = V0[R(t)r], where V0 is a polynomial in x, y, and/or z and R(t) proportional to tP is a time-dependent scale factor. When p is not too negative, one can distinguish between regular and chaotic behavior by determining whether an orbit segment exhibits a sensitive dependence on initial conditions. However, chaotic segments in these potentials differ from chaotic segments in time-independent potentials in that a small initial perturbation will usually exhibit a sub- or superexponential growth in time. Although not periodic, regular segments typically exhibit simpler shapes, topologies, and Fourier spectra than do chaotic segments. This distinction between regular and chaotic behavior is not absolute since a single orbit segment can seemingly change from regular to chaotic and vice versa. All these observed phenomena can be understood in terms of a simple theoretical model.