Analytical Conditions for the Existence of a Homoclinic Loop in Chua Circuits

The presence of a saddle-focus separatrix — a homoclinic curve in Poincaré’s terminology — plays a fundamental role in the onset and existence of chaos in dynamical systems. To observe chaotic behavior of trajectories, for instance, it is sufficient to have two conditions: the existence of a separatrix loop and Shil’nikov’s saddle-focus condition [1]. Consider a three-dimensional dynamical system with a saddle-focus equilibrium. This system may contain a trajectory that leads from the saddle-focus to itself, i.e., a homoclinic loop. Assume that the roots of the equilibrium characteristic equation are such that Im p1 = 0, Re p1 > 0, Re pk < 0, k = 2, 3. Then by Shil’nikov’s theorem, the inequality Re p2,3 + p1 > 0 in the neighborhood of the homoclinic loop (both when it exists and when it is breaking down) ensures the existence of a countable set of periodic saddle trajectories. If in addition to this condition we also have 2 Re p2,3 +p1 < 0, then systems with infinitely many stable periodic trajectories are dense on the set of systems with a saddle-focus homoclinic loop. If on the other hand 2 Re p2,3 + p1 > 0, then systems with infinitely many completely unstable periodic trajectories are dense on this set. Such homoclinic loops have been discovered and investigated for Chua equations [2], for phase synchronization systems [3], etc. However, there are difficulties with exact quantitative bounds and analytical expressions for the bifurcation parameters corresponding to the separatrix loop. Such bounds and expressions can be obtained for piecewise-linear systems. In this article, we present analytical relationships for the bifurcation curves of Chua equations. The relationships are derived using the same technique as previously applied for the n th order piecewise-linear equation [4].