Spiral periodic structure inside chaotic region in parameter‐space of a Chua circuit

Abstract

SUMMARYIn this letter we investigate, via numerical simulations, the parameter-space of the set of autonomous firstorder differential equations of a Chua circuit. We show that this parameter-space presents self-organized periodic structures immersed in a chaotic region, forming a single spiral structure that coils up around a focal point. Additionally, bifurcation diagrams are used to show that those periodic structures also organize themselves in period-adding cascades, along specific directions that point towards this same focal point.

High Resolution Parameter-Space from a Two-Level Model on Semi-Insulating GaAs

“…6. Similar structures are already present in the literature for other systems [Hoff et al, 2014;Bonatto & Gallas, 2008;Albuquerque & Rech, 2010;Stoop et al, 2010].…”

Bifurcation structures and transient chaos in a four-dimensional Chua model

“…3. The periodicity spiral was reported in some recent works about three-dimensional Chua's model [20,29], in other systems [30,31], and references therein. Here, we show the periodicity spiral for a four-dimensional Chua's model, and the dimensional parameter d, that controls the additional dimension of system (1), has a fundamental role in this organization.…”

Spiral periodic structures in a parameter plane of an ecological model

“…1. As we said before in Section 1, spiral organization of periodic structures have been observed in parameter planes of different systems [1,[29][30][31]43,49,50], modeled by different sets of nonlinear differential equations, that may involve polynomial functions and exponential functions among other mathematical functions. This may be an indicator of the importance of this type of organization of periodic structures, which is present in various fields of the knowledge.…”

“…One such type of organization consists of a set of shrimp-shaped periodic structures forming a spiral that coils up around a focal point while period-adding bifurcations take place. To our knowledge these spiral bifurcations have been observed in parameter planes of electronic circuits [1,29,49], a Rössler model [50], a chemical oscillator [1], a Hopfield neural network [30], modified optical injection semiconductor lasers [31], and a tumor growth mathematical model [43]. The spiral periodic structures were experimentally detected in electronic circuits [51], and the global mechanism responsible for its origin and organization was reported simultaneously by Vitolo et al [14] and Barrio et al [15].…”