New class of chaotic systems with circular equilibrium
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This paper brings a new mathematical model of the third-order autonomous deterministic dynamical system with associated chaotic motion. Its unique property lies in the existence of circular equilibrium which was not, by referring to the best knowledge of the authors, so far reported. Both mathematical analysis and circuitry implementation of the corresponding differential equations are presented. It is shown that discovered system provides a structurally stable strange attractor which fulfills fractal dimensionality and geometrical density and is bounded into a finite state space volume.
KeywordsAutonomous system Attracting set Circular equilibrium Chaos Nonlinear dynamics Vector field
It is well known that chaotic dynamics is not restricted only to complicated and strongly nonlinear vector fields  but can be observed also in the case of algebraically simple systems with six terms including nonlinearity . Recent progress in overall performance of the personal computers and possibility of multi-grid calculation allows to implement fast-to-be-calculated quantifier of the dynamical motion inside a procedure for chaos or hyper-chaos localization . Doing this we can start searching for irregular behavior of arbitrary-order nonlinear dynamical system. Such process begins with analytical definition of dimensionless mathematical models and continues with specification of the internal system parameters which are so far unknown. Since coexistence of multiple different attractors is possible in such systems, the initial conditions are randomly and, more importantly, repeatedly chosen. Each time a routine comes across vector field which provides the so-called folding and stretching mechanism, the dynamical system is remembered for consequent numerical analysis.
This work has been primarily motivated by two recently published research papers where a group of dynamical systems with very specific properties have been presented. In paper  a class of the dynamical systems without equilibrium has been presented. Similarly paper  introduces several dynamical systems with a line equilibrium. Both works can be considered as a breakthrough idea since chaos is often put into the context of the singular saddle-type fixed points; the most common configuration of the vector field contains two  or three of them. From this point of view a system with circular equilibrium (CES) represents somehow future logical progress.
2 Mathematical models under inspection
The initial conditions can be taken as \(\mathbf x _0=(0, 0, 0)^T\). Typical property of this dynamical system is long spiral-type transient behavior and dissipative dynamical flow given by parameter \(d\).
3 Experimental verification
In this short paper a novel dynamical system with circular equilibrium is uncovered and numerically confirmed as well as experimentally measured. Brief nature of this paper leaves the place for upcoming deeper investigation of the class of dynamical system with circular equilibrium. It is believed that brute-force method that combines stochastic search routine with objective function in the form of precise motion quantifier is powerful tool which can be utilized for discovering interesting dynamical systems with prescribed features. As indicated by new publications [6, 12] research in this particular area will proceed in the near future.
Research described in this paper was financed by Czech Ministry of Education in frame of National Sustainability Program under grant LO1401. For research, infrastructure of the SIX Center was used.
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