Holographic functional methods are introduced as probes of discrete time-stepped maps that lead to chaotic behavior. The methods provide continuous time interpolation between the time steps, thereby revealing the maps to be quasi-Hamiltonian systems underlain by novel potentials that govern the motion of a perceived point particle. Between turning points, the particle is strictly driven by Hamiltonian dynamics, but at each encounter with a turning point the potential changes abruptly, loosely analogous to the switchbacks on a mountain road. A sequence of successively deepening switchback potentials explains, in physical terms, the frequency cascade and trajectory folding that occur on the particular route to chaos revealed by the logistic map.
Journal of Physics A Mathematical General
- Pub Date:
- November 2010
- Nonlinear Sciences - Chaotic Dynamics;
- High Energy Physics - Theory;
- Mathematical Physics
- This paper, and its precedent arXiv:0909.2424 [math-ph], are dedicated to Murray Gell-Mann on the occasion of his 80th birthday