Homoclinic chaos in the Rössler model

Abstract

We study the origin of homoclinic chaos in the classical 3D model proposed by O. Rössler in 1976. Of our particular interest are the convoluted bifurcations of the Shilnikov saddle-foci and how their synergy determines the global unfolding of the model, along with transformations of its chaotic attractors. We apply two computational methods proposed, 1D return maps and a symbolic approach specifically tailored to this model, to scrutinize homoclinic bifurcations, as well as to detect the regions of structurally stable and chaotic dynamics in the parameter space of the Rössler model.

On the Organization of Homoclinic Bifurcation Curves in Systems with Shilnikov Spiral Attractors

“…., because of the splitting homoclinic loop "outside", which prevents the formation of secondary loops. The graphs of the distance between the saddle-focus and the attractor confirm it, i.e., the distance does not vanish in the regions between the teeth (see [10] for system (1.1)), which means that the saddle-focus does not belong to the attractor and there is no homoclinic loop in the system for these values of parameters.…”

“…There is a well-known fact, repeatedly proven for both the Arneodo -Coullet -Tresser model and the Rössler model (see, e.g., [10,11]), that chaos in these systems is developed via the Shilnikov scenario [12] from the stable equilibrium. Let us demonstrate it using system (1.2) as an example.…”

“…In order to illustrate it, let us provide the following numerical experiment based on the fact that, when a homoclinic loop exists, the saddle-focus equilibrium becomes part of an attractor, i.e., orbits in the attractor pass arbitrarily close to this equilibrium. Following [10,14], we compute a distance between a sufficiently long orbit in the attractor and the saddle-focus equilibrium. If this distance is less than some small threshold, we assume that a homoclinic orbit exists.…”

Self-pulsing and chaos in the asymmetrically-driven dissipative photonic Bose-Hubbard dimer: A bifurcation analysis

“…7 for the homoclinic orbits. The type of chaos associated with these attractors is commonly known as spiral chaos [54,55]. However, due to the exponentially shrinking of the chaotic intervals when approaching Hom b 1 we have not been able to confirm these two hypotheses.…”

“…δ < 1), chaotic dynamics is expected in the neighborhood of the homoclinic orbit [44]. The interplay be- tween chaotic dynamics and homoclinic orbits has been studied by different authors, in particular in the context of the Rossler model [46,[53][54][55].…”