# UNIFORMLY HYPERBOLIC ATTRACTOR IN A SYSTEM BASED ON COUPLED OSCILLATORS WITH «FIGURE-EIGHT» SEPARATRIX

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Kruglov V. P., Khadzhieva L. M. UNIFORMLY HYPERBOLIC ATTRACTOR IN A SYSTEM BASED ON COUPLED OSCILLATORS WITH «FIGURE-EIGHT» SEPARATRIX. Izvestiya VUZ, Applied Nonlinear Dynamics, 2016, vol. 24, iss. 6, pp. 54-64 DOI: 10.18500/0869-6632-2016-24-6-54-64

A new autonomous system with chaotic dynamics corresponding to Smale–Williams attractor in Poincare map is introduced. The system is constructed on the basis of the model with «figure-eight» separatrix on the phase plane discussed in former times by Y.I. Neimark. Our system is composed of two Neimark subsystems with generalized coordinates x and y. It is described by the equations with additional terms due to which the system becomes self-oscillating. Furthermore, a special coupling between subsystems provides the tripling of the angle of vector (x, y) rotation when returning to the neighborhood of the origin in successive rounds on separatrix. Study is based on the numerical solution of the dynamical equations with the construction of the Poincare map. Results of numerical simulation (iteration diagram for the angular variable, Lyapunov exponents) demonstrate that the angular variable undergoes expanding circle map, while in the other directions there is a strong compression of the phase volume element. Distribution of angles between stable and unstable manifolds of the attractor is obtained and it confirms the property of transversal manifolds of the attractor. Structural stability of the attractor is confirmed by the smooth dependence of the highest Lyapunov exponent on the parameters. With this we conclude that the attractor of the Smale–Williams type is observed in the phase space of the proposed system in a certain range of parameters.

DOI: 10.18500/0869-6632-2016-24-6-54-64

*Paper reference:* Kruglov V.P., Khadzhieva L.M.-B. Uniformly hyperbolic attractor in a system based on coupled oscillators with «figure-eight» separatrix.* Izvestiya VUZ. Applied Nonlinear Dynamics.* 2016. Vol. 24. Issue 6. P.54-64 .

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## BibTeX

author = {V. P. Kruglov and L. M.-B. Khadzhieva},

title = {UNIFORMLY HYPERBOLIC ATTRACTOR IN A SYSTEM BASED ON COUPLED OSCILLATORS WITH «FIGURE-EIGHT» SEPARATRIX},

year = {2016},

journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},

volume = {24},number = {6},

url = {http://andjournal.sgu.ru/en/articles/uniformly-hyperbolic-attractor-in-system-based-on-coupled-oscillators-with-figure-eight},

address = {Саратов},

language = {russian},

doi = {10.18500/0869-6632-2016-24-6-54-64},pages = {54--64},issn = {0869-6632},

keywords = {chaos,Attractor,Smale–Williams solenoid,Lyapunov exponent},

abstract = {A new autonomous system with chaotic dynamics corresponding to Smale–Williams attractor in Poincare map is introduced. The system is constructed on the basis of the model with «figure-eight» separatrix on the phase plane discussed in former times by Y.I. Neimark. Our system is composed of two Neimark subsystems with generalized coordinates x and y. It is described by the equations with additional terms due to which the system becomes self-oscillating. Furthermore, a special coupling between subsystems provides the tripling of the angle of vector (x, y) rotation when returning to the neighborhood of the origin in successive rounds on separatrix. Study is based on the numerical solution of the dynamical equations with the construction of the Poincare map. Results of numerical simulation (iteration diagram for the angular variable, Lyapunov exponents) demonstrate that the angular variable undergoes expanding circle map, while in the other directions there is a strong compression of the phase volume element. Distribution of angles between stable and unstable manifolds of the attractor is obtained and it confirms the property of transversal manifolds of the attractor. Structural stability of the attractor is confirmed by the smooth dependence of the highest Lyapunov exponent on the parameters. With this we conclude that the attractor of the Smale–Williams type is observed in the phase space of the proposed system in a certain range of parameters. DOI: 10.18500/0869-6632-2016-24-6-54-64 Paper reference: Kruglov V.P., Khadzhieva L.M.-B. Uniformly hyperbolic attractor in a system based on coupled oscillators with «figure-eight» separatrix. Izvestiya VUZ. Applied Nonlinear Dynamics. 2016. Vol. 24. Issue 6. P.54-64 . Download full version }}