Non-existence of Shilnikov Chaos in Continuous-time Systems

Zeraoulia Elhadj1 and J. C. Sprott2

1Department of Mathematics, University of Tébessa, (12002), Algeria;
2Department of Physics, University of Wisconsin, Madison, WI 53706, USA


In this paper, a non-existence condition for homoclinic and heteroclinic orbits in n-dimensional, continuous-time, and smooth systems is obtained. Based on this result and an elementary example, it can be conjectured that there is a fourth kind of chaos in polynomial ordinary differential equation (ODE) systems characterized by the non-existence of homoclinic and heteroclinic orbits.

Ref: E. Zeraoulia and  J. C. Sprott, Applied Mathematics and Mechanics (English Edition) 33, 371-374 (2012)

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