Dissipative chaos, Shilnikov chaos and bursting oscillations in a three-dimensional autonomous system: theory and electr
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O R I G I N A L PA P E R
Dissipative chaos, Shilnikov chaos and bursting oscillations in a three-dimensional autonomous system: theory and electronic implementation Sifeu Takougang Kingni · Lars Keuninckx · Paul Woafo · Guy Van der Sande · Jan Danckaert
Received: 5 January 2013 / Accepted: 6 March 2013 / Published online: 30 March 2013 © Springer Science+Business Media Dordrecht 2013
Abstract A three-dimensional autonomous chaotic system is presented and physically implemented. Some basic dynamical properties and behaviors of this system are described in terms of symmetry, dissipative system, equilibria, eigenvalue structures, bifurcations, and phase portraits. By tuning the parameters, the system displays chaotic attractors of different shapes. For specific parameters, the system exhibits periodic and chaotic bursting oscillations which resemble the conventional heart sound signals. The existence of Shilnikov type of heteroclinic orbit in the three-dimensional system is proven using the undetermined coefficients method. As a result, Shilnikov criterion guarantees that the three-dimensional system has the horseshoe chaos. The corresponding electronic circuit is designed and implemented, exhibiting experimental chaotic attractors in accord with numerical simulations.
S.T. Kingni () · L. Keuninckx · G. Van der Sande · J. Danckaert Applied Physics Research Group (APHY), Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium e-mail: [email protected] S.T. Kingni · P. Woafo Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes, Department of Physics, Faculty of Science, University of Yaoundé I, Po. Box 812, Yaoundé, Cameroon
Keywords Chaos · Bursting oscillations · Three-dimensional autonomous chaotic system · Electronic implementation · Bifurcation · Shilnikov criterion
1 Introduction Since Lorenz discovered chaos in a simple system of three autonomous ordinary differential equations in order to describe the simplified Rayleigh–Benard problem in 1963 [1], chaos has been found in many other systems and has great potential use in many technological disciplines, such as in information and communication technologies, flow dynamics, power system protection, biomedical system analysis, and so on [2]. Systems such as that of Lorenz are described by a set of three differential equations quadratic flow of three dynamical variables x, y, and z whose time derivatives, equivariant under the action of the rotation through π radian around the z-axis are functions having linear and nonlinear terms depending on x, y, and z. It is notable that the Lorenz system has seven terms on the right-hand side, two of which are nonlinear xz and xy. The nonlinear characteristics and basic dynamic properties of the Lorenz system are well studied by many papers and monographs [3, 4]. In 1976, Rössler constructed three-dimensional quadratic autonomous chaotic systems [5], which also have seven terms on the right-hand side, but with only one quadratic nonlinearity xz. Obviously, the Rössler sys-
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