Chaos in cosmological Hamiltonians

Area of Science:

• Physics
• Applied Mathematics
• Dynamical Systems

Background:

• Time-dependent Hamiltonians exhibit complex dynamics.
• Distinguishing regular from chaotic behavior is crucial for understanding dynamical systems.

Purpose of the Study:

• To numerically identify and characterize regular and chaotic behavior in specific time-dependent Hamiltonians.
• To differentiate chaotic dynamics in time-dependent potentials from those in time-independent potentials.

Main Methods:

• Numerical investigation of Hamiltonians of the form H(r, p, t) = p(2)/2 + V(r, t).
• Analysis of potential forms V = R(t)V0(r) or V = V0[R(t)r], where R(t) is a time-dependent scale factor.
• Assessment of sensitive dependence on initial conditions to distinguish regular and chaotic segments.

Main Results:

• Chaotic segments in these potentials display sub- or superexponential growth of initial perturbations, unlike time-independent potentials.
• Regular segments, though not periodic, show simpler shapes, topologies, and Fourier spectra.
• A transition between regular and chaotic behavior within a single orbit segment was observed.

Conclusions:

• The observed phenomena of regular and chaotic behavior in time-dependent Hamiltonians can be explained by a simple theoretical model.
• Sensitive dependence on initial conditions, with specific growth rates, serves as a key differentiator for chaotic behavior.
• The distinction between regular and chaotic behavior is fluid, with potential for transitions within orbits.