Explosive route to chaos through a fractal torus in a generalized Lotka-Volterra model

Abstract

The behavior of a model that generalizes the Lotka-Volterra problem into three dimensions is presented. The results show the analytic derivation of stability diagrams that describe the system's qualitative features. In particular, we show that for a certain value of the bifurcation parameter the system instantly jumps out of a steady state solution into a chaotic solution that portrays a fractal torus in the three-dimensional phase space. This scenario, is referred to as the explosive route to chaos and is attributed to the non-transversal saddle connection type bifurcation. The stability diagrams also present a region in which the Hopf type bifurcation leads to periodic and chaotic solutions. In addition, the bifurcation diagrams reveal a qualitative similarity to the data obtained in the Texas and Bordeaux experiments on the Belousov-Zhabotinskii chemical reaction. The paper is concluded by showing that the model can be useful for representing dynamics associated with biological and chemical phenomena.

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Samardzija, N., Greller, L.D. Explosive route to chaos through a fractal torus in a generalized Lotka-Volterra model. Bltn Mathcal Biology 50, 465–491 (1988). https://doi.org/10.1007/BF02458847

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Keywords

  • Steady State Solution
  • Chaotic Attractor
  • Strange Attractor
  • Interarrival Time
  • Stability Diagram