While virtually everyone recognizes the famous Lorenz “Butterfly”, the strange attractor that is one of the central icons of chaos theory, in my opinion Hamiltonian chaos generates far more interesting patterns. This is because Hamiltonians conserve phase-space volume, stretching and folding small volumes of initial conditions as they evolve in time, until they span large sections of phase space. Hamiltonian chaos is usually displayed as multi-color Poincaré sections (also known as first-return maps) that are created when a set of single trajectories, each represented by a single color, pierce the Poincaré plane over and over again.
The archetype of all Hamiltonian systems is the harmonic oscillator.
The classic Hamiltonian system, perhaps the archetype of all Hamiltonian systems, is the harmonic oscillator. The physics of the harmonic oscillator are taught in the most elementary courses, because every stable system in the world is approximated, to lowest order, as a harmonic oscillator. As the simplest dynamical system, one would think that it held no surprises. But surprisingly, it can create the most beautiful tapestries of color when pulsed periodically and mapped onto the Poincaré plane.
The Hamiltonian of the periodically kicked harmonic oscillator is converted into the Web Map, represented as an iterative mapping as
There can be resonance between the sequence of kicks and the natural oscillator frequency such that α = 2π/q. At these resonances, intricate web patterns emerge. The Web Map produces a web of stochastic layers when plotted on an extended phase plane. The symmetry of the web is controlled by the integer q, and the stochastic layer width is controlled by the perturbation strength K.
Web Map Python Program
Iterated maps are easy to implement in code. Here is a simple Python code to generate maps of different types. You can play with the coupling constant K and the periodicity q. For small K, the tapestries are mostly regular. But as the coupling K increases, stochastic layers emerge. When q is a small even number, tapestries of regular symmetric are generated. However, when q is an odd small integer, the tapestries turn into quasi-crystals.
# -*- coding: utf-8 -*-
import numpy as np
from scipy import integrate
from matplotlib import pyplot as plt
phi = (1+np.sqrt(5))/2
K = 1-phi # (0.618, 4) (0.618,5) (0.618,7) (1.2, 4)
q = 4 # 4, 5, 6, 7
alpha = 2*np.pi/q
for eloop in range(0,1000):
xlast = 50*np.random.random()
ylast = 50*np.random.random()
xnew = np.zeros(shape=(300,))
ynew = np.zeros(shape=(300,))
for loop in range(0,300):
xnew[loop] = (xlast + K*np.sin(ylast))*np.cos(alpha) + ylast*np.sin(alpha)
ynew[loop] = -(xlast + K*np.sin(ylast))*np.sin(alpha) + ylast*np.cos(alpha)
xlast = xnew[loop]
ylast = ynew[loop]
References and Further Reading
D. D. Nolte, Introduction to Modern Dynamics: Chaos, Networks, Space and Time (Oxford, 2015)
G. M. Zaslavsky, Hamiltonian chaos and fractional dynamics. (Oxford, 2005)