Математика и математическое моделирование. 2015; : 17-27
Исследование динамической системы взаимосвязанных осцилляторов Рёсслера
Аннотация
Список литературы
1. Nandan M., Hens C. R., Pal P., Dana S. K. Transition from amplitude to oscillation death in a network of oscillators // Chaos. 2014. Vol. 24. 043103. DOI: 10.1063/1.4897446
2. Kondor D., Vattay G. Dynamics and Structure in Cell Signaling Networks: Off-State Stability and Dynamically Positive Cycles // PLoS One. 2013. Vol. 8. № 3. e57653. DOI: 10.1371/journal.pone.0057653
3. Motter A. E., Myers S. A., Anghel M., Nishikawa T. Spontaneous synchrony in power-grid networks // Nat. Phys. 2013. Vol. 9. Pp. 191-197. DOI: 10.1038/nphys2535
4. Wu Q., Fu X., Small M., Xu X.-J. The impact of awareness on epidemic spreading in networks // Chaos. 2012. Vol. 22. 013101. DOI: 10.1063/1.3673573
5. Крищенко А.П. Локализация инвариантных компактов динамических систем // Дифференциальные уравнения. 2005. Вып. 41. № 12. С. 1597-1604.
6. Канатников А.Н., Крищенко А.П. Инвариантные компакты динамических систем. М.: МГТУ им. Н.Э. Баумана, 2011. 231 с.
7. Канатников А.Н., Крищенко А.П. Реализация итерационной процедуры в задачах локализации автономных систем // Наука и образование: научное издание МГТУ им. Н.Э. Баумана. 2014. № 11. С. 307-319. DOI: 10.7463/1114.0734649
8. Канатников А.Н., Михайлова О.В. Локализация инвариантных компактов дискретной системы Лози // Наука и образование: научное издание МГТУ им. Н.Э. Баумана. 2013. № 8. С. 121-134.
9. Канатников А.Н., Фёдорова Ю.П. Локализация инвариантных компактов двумерных непрерывных динамических систем // Наука и образование: научное издание МГТУ им. Н.Э. Баумана. 2013. № 7. С. 159-174.
10. Канатников А.Н. Локализирующие множества для инвариантных компактов непрерывных динамических систем с возмущением // Дифференциальные уравнения. 2012. Вып. 48. № 11. С. 1483.
11. Krishchenko A.P., Starkov K.E. Dynamical Analysis of Raychaudhuri Equations Based on the Localization Method of Compact Invariant Sets // International Journal of Bifurcation and Chaos. 2014. Vol. 24. № 11. 1450136. DOI: 10.1142/S0218127414501363
Mathematics and Mathematical Modeling. 2015; : 17-27
The Research of the Dynamical System of Globally Coupled R ossler Oscillators
Abstract
The paper studies a dynamical system of globally coupled Rossler oscillators previously con- sidered in terms of oscillation quenching. There are two types of quenching, namely homogeneous steady state and inhomogeneous steady state. A transition from the former to the latter can cause a disease in biological structure, a defect in synchronized power grid, as well as it can be used for epidemic delimitation.
The investigation of the system of Rossler oscillators is conducted in two directions: finding the equilibrium points and localization of invariant compact sets.
The equilibrium points are found for the systems of two oscillators, at most. In particular, a system consisting of one oscillator has, depending on the values of parameters, one, two, or infinitely many equilibrium points, and in the latter case the equilibrium points form a line in 3D space. The set of equilibrium points of the system of two oscillators is explicitly described as well; its structure to an even greater degree depends on the values of parameters.
Localization of invariant compact sets involves the method concerned with constructing a localizing function. One knows that its values on each invariant compact set are bounded above and below, respectively, by the supremum and the infimum of its values on the set of zeros of its gradient. This method revealed, as a localizing set, the intersection of a family of closed subsets each of them being bounded by some parabolic surface and located on its outer side. The paper presents illustration of this localizing set for one certain set of values of parameters.
References
1. Nandan M., Hens C. R., Pal P., Dana S. K. Transition from amplitude to oscillation death in a network of oscillators // Chaos. 2014. Vol. 24. 043103. DOI: 10.1063/1.4897446
2. Kondor D., Vattay G. Dynamics and Structure in Cell Signaling Networks: Off-State Stability and Dynamically Positive Cycles // PLoS One. 2013. Vol. 8. № 3. e57653. DOI: 10.1371/journal.pone.0057653
3. Motter A. E., Myers S. A., Anghel M., Nishikawa T. Spontaneous synchrony in power-grid networks // Nat. Phys. 2013. Vol. 9. Pp. 191-197. DOI: 10.1038/nphys2535
4. Wu Q., Fu X., Small M., Xu X.-J. The impact of awareness on epidemic spreading in networks // Chaos. 2012. Vol. 22. 013101. DOI: 10.1063/1.3673573
5. Krishchenko A.P. Lokalizatsiya invariantnykh kompaktov dinamicheskikh sistem // Differentsial'nye uravneniya. 2005. Vyp. 41. № 12. S. 1597-1604.
6. Kanatnikov A.N., Krishchenko A.P. Invariantnye kompakty dinamicheskikh sistem. M.: MGTU im. N.E. Baumana, 2011. 231 s.
7. Kanatnikov A.N., Krishchenko A.P. Realizatsiya iteratsionnoi protsedury v zadachakh lokalizatsii avtonomnykh sistem // Nauka i obrazovanie: nauchnoe izdanie MGTU im. N.E. Baumana. 2014. № 11. S. 307-319. DOI: 10.7463/1114.0734649
8. Kanatnikov A.N., Mikhailova O.V. Lokalizatsiya invariantnykh kompaktov diskretnoi sistemy Lozi // Nauka i obrazovanie: nauchnoe izdanie MGTU im. N.E. Baumana. 2013. № 8. S. 121-134.
9. Kanatnikov A.N., Fedorova Yu.P. Lokalizatsiya invariantnykh kompaktov dvumernykh nepreryvnykh dinamicheskikh sistem // Nauka i obrazovanie: nauchnoe izdanie MGTU im. N.E. Baumana. 2013. № 7. S. 159-174.
10. Kanatnikov A.N. Lokaliziruyushchie mnozhestva dlya invariantnykh kompaktov nepreryvnykh dinamicheskikh sistem s vozmushcheniem // Differentsial'nye uravneniya. 2012. Vyp. 48. № 11. S. 1483.
11. Krishchenko A.P., Starkov K.E. Dynamical Analysis of Raychaudhuri Equations Based on the Localization Method of Compact Invariant Sets // International Journal of Bifurcation and Chaos. 2014. Vol. 24. № 11. 1450136. DOI: 10.1142/S0218127414501363
