Bulletin of Mathematical Biology

, Volume 50, Issue 5, pp 465–491 | Cite as

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

  • Nikola Samardzija
  • Larry D. Greller


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.


Steady State Solution Chaotic Attractor Strange Attractor Interarrival Time Stability Diagram 
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Copyright information

© Society for Mathematical Biology 1988

Authors and Affiliations

  • Nikola Samardzija
    • 1
  • Larry D. Greller
    • 1
  1. 1.Engineering Department, and Central Research & Development DepartmentE. I. du Pont de Nemours & Co.WilmingtonU.S.A.

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