Dynamic elements of chaos in the Willamowski–Rössler network

Abstract

Two dynamic elements of the Willamowski–Rössler network are identified: one of which is a Lotka–Volterra oscillator involving two autocatalytic species X and Y, while the other is a switch between X and a third autocatalytic species Z. These two dynamic elements are coupled via X. Nonperiodic oscillations arise only if X autocatalyzes faster than Z. The chaotic nature of the oscillations is confirmed using the Shil’nikov theorem which requires the existence of a homoclinic orbit doubly asymptotic to a steady state of the saddle-focus type. Under chaotic conditions, two steady states coexist inside the positive orthant of concentration space and both satisfy the conditions of the Shil’nikov theorem. Chaos is further shown by the existence of several unstable period-3 fixed points of first-return Poincaré maps. A chaotic attractor is found and its changing structure under various sets of parameters is established.