For citation:
Kuznetsov A. P., Kuznetsov S. P., Stankevich N. V. Four-dimensional system with torus attractor birth via saddle-node bifurcation of limit cycles in context of family of blue sky catastrophes. Izvestiya VUZ. Applied Nonlinear Dynamics, 2015, vol. 23, iss. 4, pp. 32-39. DOI: 10.18500/0869-6632-2015-23-4-32-39
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517.9
Four-dimensional system with torus attractor birth via saddle-node bifurcation of limit cycles in context of family of blue sky catastrophes
Autors:
Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Kuznetsov Sergey Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Stankevich Nataliya Vladimirovna, National Research University "Higher School of Economics"
Abstract:
A new four-dimensional model with quasi-periodic dynamics is suggested. The torus attractor originates via the saddle-node bifurcation, which may be regarded as a member of a bifurcation family embracing different types of blue sky catastrophes. Also the torus birth through the Neimark-Sacker bifurcation occurs in some other region of the parameter space.
Reference:
- Glazier J.A., Libchaber A. Quasi-periodicity and dynamical systems: an experimentalist’s view // IEEE Transactions on Circuits and Systems. 1988. Vol.35, No 7. P. 790.
- Broer H. W., Huitema G. B., Sevryuk M. B. Quasi-periodic motions in families of dynamical systems: order amidst chaos. Lecture Notes in Mathematics. Vol. 1645. Springer, 1996. 200 p.
- Palis J., Pugh C.C. Fifty problems in dynamical systems //Dynamical Systems-Warwick 1974. - Springer Berlin Heidelberg, 1975. P. 345–353.
- Turaev D.V., Shilnikov L.P. Blue sky catastrophes // Dokl. Math. 1995. Vol. 51. P. 404–407.
- Shilnikov L.P., Turaev D.V. Simple bifurcations leading to hyperbolic attractors // Computers & Mathematics with Applications. 1995. Vol. 34, No 2. P. 173.
- Shilnikov L. P., Turaev D. V. A new simple bifurcation of a periodic orbit of blue sky catastrophe type //Translations of the American Mathematical Society–Series 2. 2000. Vol. 200. P. 165.
- Gavrilov N., Shilnikov A. Example of a blue sky catastrophe //Translations of the American Mathematical Society-Series 2. 2000. Vol. 200. P.99.
- Shilnikov A. L., Shilnikov L. P., Turaev D. V. Blue sky catastrophe in singularly perturbed systems // Moscow Math. J. 2005. Vol. 5. P. 205.
- Shilnikov L. P., Shilnikov A. L., Turaev D. V. Showcase of Blue Sky Catastrophes // International Journal of Bifurcation and Chaos. 2014. Vol. 24, No 08. P. 1440003.
- Shilnikov A. and Turaev D. Blue-sky catastrophe // Scholarpedia. 2007. Vol.2, No 8. P. 1889.
- Shilnikov A., Cymbalyuk G. Transition between tonic spiking and bursting in a neuron model via the blue-sky catastrophe // Physical Review Letters. 2005. Vol. 94, No 4. P. 048101.
- Meca E., Mercader I., Batiste O. and Ramirez-Piscina L. Blue sky catastrophe in double-diffusive convection // Physical Review Letters. 2004. Vol.92, No. 23. P. 234501.
- Kuznetsov S.P. Example of blue sky catastrophe accompanied by a birth of Smale– Williams attractor // Regular and Chaotic Dynamics. 2010. Vol. 15, No 2–1. P. 348.
- Katayama K., Horiguchi T. Synchronous phenomena of neural network models using Hindmarsh–Rose equation // Interdisciplinary Information Sciences. 2005. Vol. 11, No. 1. P. 11.
- Sherman A. Anti-phase, asymmetric and aperiodic oscillations in excitable cells-I. Coupled bursters //Bulletin of mathematical biology. 1994. Vol. 56, No 5. P. 811–835.
Received:
25.05.2015
Accepted:
25.05.2015
Published:
31.03.2016
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