Globally Asymptotically Stable Equilibrium Points in Kukles Systems
The problem of determining the basin of attraction of equilibrium points is of great importance for applications of stability theory. In this article, we address the global asymptotic stability problem of an equilibrium point of an ordinary differential equation on the plane. More precisely, we stud...
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| Veröffentlicht in: | Qualitative theory of dynamical systems Jg. 19; H. 3 |
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| Abstract | The problem of determining the basin of attraction of equilibrium points is of great importance for applications of stability theory. In this article, we address the global asymptotic stability problem of an equilibrium point of an ordinary differential equation on the plane. More precisely, we study equilibrium points of Kukles systems from the global asymptotic stability point of view. First of all, we classify the Kukles systems satisfying the assumptions: the origin is the unique equilibrium point which is locally asymptotically stable, and the divergence is negative except possibly at the origin. Then, for each of such Kukles system, we prove that the origin is globally asymptotically stable. Poincaré compactification is used to study the systems on the complements of compact sets. |
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| AbstractList | The problem of determining the basin of attraction of equilibrium points is of great importance for applications of stability theory. In this article, we address the global asymptotic stability problem of an equilibrium point of an ordinary differential equation on the plane. More precisely, we study equilibrium points of Kukles systems from the global asymptotic stability point of view. First of all, we classify the Kukles systems satisfying the assumptions: the origin is the unique equilibrium point which is locally asymptotically stable, and the divergence is negative except possibly at the origin. Then, for each of such Kukles system, we prove that the origin is globally asymptotically stable. Poincaré compactification is used to study the systems on the complements of compact sets. |
| ArticleNumber | 94 |
| Author | Mello, Luis Fernando Dias, Fabio Scalco |
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| Keywords | Global asymptotic stability Poincaré compactification 34D23 34A26 Kukles system 34D05 |
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| References | ChiconeCOrdinary Differential Equations with Applications1999New YorkSpringer0937.34001 Ahmadi, A.A., Krstic, M., Parrilo, P.A.: A globally asymptotically stable polynomial vector field with no polynomial Lyapunov function. In: Proceedings of the IEEE Conference on Decision and Control, pp. 7579–7580 (2011) ChamberlandMLlibreJŚwirszczGWeakened Markus-Yamabe conditions for 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2$$\end{document}-dimensional global asymptotic stabilityNonlinear Anal.200459951958209637010.1016/j.na.2004.01.010 DumortierFLlibreJArtésJCQualitative Theory of Planar Differential Systems2006BerlinSpringer1110.34002 HubbardJHWestBHDifferential Equations: A Dynamical Systems Approach. Higher-Dimensional Systems1995New YorkSpringer10.1007/978-1-4612-4192-8 ChristopherCJInvariant algebraic curves and conditions for a centreProc. R. Soc. Edinburgh Sect. A199412412091229131319910.1017/S0308210500030213 FeßlerRA proof of the two-dimensional Markus-Yamabe stability conjecture and a generalizationAnn. Polon. Math.1995624574134821710.4064/ap-62-1-45-74 GinéJLlibreJVallsCCenters for the Kukles homogeneous systems with even degreeJ. Appl. Anal. Comput.2017715341548372393507247212 Glutsyuk, A.A.: The asymptotic stability of the linearization of a vector field on the plane with a singular point implies global stability, Funktsional. Anal. i Prilozhen. 29, 17–30 (1995); translation in Funct. Anal. Appl. 29, 238–247 (1995) GutiérrezCA solution to the bidimensional global asymptotic stability conjectureAnn. Inst. H. Poincaré Anal. Non Linéaire199512627671136054010.1016/S0294-1449(16)30147-0 GinéJConditions for the existence of a center for the Kukles homogeneous systemsComput. Math. Appl.20024312611269190635310.1016/S0898-1221(02)00098-6 MarkusLYamabeHGlobal stability criteria for differential systemsOsaka Math. J.1960123053171260190096.28802 PearsonJMLloydNGKukles revisited: Advances in computing techniquesComput. Math. Appl.20106027972805273432110.1016/j.camwa.2010.09.034 KuklesISSur quelques cas de distinction entre un foyer et un centreDokl. Akad. Nauk. SSSR194442208211113560063.03332 GaikoVGlobal bifurcation analysis of the Kukles cubic systemInt. J. Dyn. Syst. Differ. Equ.2018832633638913491442.34069 JinXWangDOn the conditions of Kukles for the existence of a centreBull. London Math. Soc.19902214102676410.1112/blms/22.1.1 432_CR1 cr-split#-432_CR10.2 M Chamberland (432_CR2) 2004; 59 IS Kukles (432_CR14) 1944; 42 cr-split#-432_CR10.1 JH Hubbard (432_CR12) 1995 C Chicone (432_CR3) 1999 F Dumortier (432_CR5) 2006 L Markus (432_CR15) 1960; 12 J Giné (432_CR8) 2002; 43 CJ Christopher (432_CR4) 1994; 124 J Giné (432_CR9) 2017; 7 V Gaiko (432_CR7) 2018; 8 C Gutiérrez (432_CR11) 1995; 12 R Feßler (432_CR6) 1995; 62 JM Pearson (432_CR16) 2010; 60 X Jin (432_CR13) 1990; 22 |
| References_xml | – reference: Ahmadi, A.A., Krstic, M., Parrilo, P.A.: A globally asymptotically stable polynomial vector field with no polynomial Lyapunov function. In: Proceedings of the IEEE Conference on Decision and Control, pp. 7579–7580 (2011) – reference: Glutsyuk, A.A.: The asymptotic stability of the linearization of a vector field on the plane with a singular point implies global stability, Funktsional. Anal. i Prilozhen. 29, 17–30 (1995); translation in Funct. Anal. Appl. 29, 238–247 (1995) – reference: FeßlerRA proof of the two-dimensional Markus-Yamabe stability conjecture and a generalizationAnn. Polon. Math.1995624574134821710.4064/ap-62-1-45-74 – reference: GutiérrezCA solution to the bidimensional global asymptotic stability conjectureAnn. Inst. H. Poincaré Anal. Non Linéaire199512627671136054010.1016/S0294-1449(16)30147-0 – reference: ChamberlandMLlibreJŚwirszczGWeakened Markus-Yamabe conditions for 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2$$\end{document}-dimensional global asymptotic stabilityNonlinear Anal.200459951958209637010.1016/j.na.2004.01.010 – reference: GinéJLlibreJVallsCCenters for the Kukles homogeneous systems with even degreeJ. Appl. Anal. Comput.2017715341548372393507247212 – reference: MarkusLYamabeHGlobal stability criteria for differential systemsOsaka Math. J.1960123053171260190096.28802 – reference: GaikoVGlobal bifurcation analysis of the Kukles cubic systemInt. J. Dyn. Syst. Differ. Equ.2018832633638913491442.34069 – reference: PearsonJMLloydNGKukles revisited: Advances in computing techniquesComput. Math. Appl.20106027972805273432110.1016/j.camwa.2010.09.034 – reference: KuklesISSur quelques cas de distinction entre un foyer et un centreDokl. Akad. Nauk. SSSR194442208211113560063.03332 – reference: ChristopherCJInvariant algebraic curves and conditions for a centreProc. R. Soc. Edinburgh Sect. A199412412091229131319910.1017/S0308210500030213 – reference: HubbardJHWestBHDifferential Equations: A Dynamical Systems Approach. Higher-Dimensional Systems1995New YorkSpringer10.1007/978-1-4612-4192-8 – reference: JinXWangDOn the conditions of Kukles for the existence of a centreBull. London Math. Soc.19902214102676410.1112/blms/22.1.1 – reference: GinéJConditions for the existence of a center for the Kukles homogeneous systemsComput. Math. Appl.20024312611269190635310.1016/S0898-1221(02)00098-6 – reference: ChiconeCOrdinary Differential Equations with Applications1999New YorkSpringer0937.34001 – reference: DumortierFLlibreJArtésJCQualitative Theory of Planar Differential Systems2006BerlinSpringer1110.34002 – volume: 7 start-page: 1534 year: 2017 ident: 432_CR9 publication-title: J. Appl. Anal. Comput. – volume: 42 start-page: 208 year: 1944 ident: 432_CR14 publication-title: Dokl. Akad. Nauk. SSSR – volume: 60 start-page: 2797 year: 2010 ident: 432_CR16 publication-title: Comput. Math. Appl. doi: 10.1016/j.camwa.2010.09.034 – volume-title: Differential Equations: A Dynamical Systems Approach. Higher-Dimensional Systems year: 1995 ident: 432_CR12 doi: 10.1007/978-1-4612-4192-8 – volume: 59 start-page: 951 year: 2004 ident: 432_CR2 publication-title: Nonlinear Anal. doi: 10.1016/j.na.2004.01.010 – volume-title: Ordinary Differential Equations with Applications year: 1999 ident: 432_CR3 – volume-title: Qualitative Theory of Planar Differential Systems year: 2006 ident: 432_CR5 – volume: 124 start-page: 1209 year: 1994 ident: 432_CR4 publication-title: Proc. R. Soc. Edinburgh Sect. A doi: 10.1017/S0308210500030213 – ident: #cr-split#-432_CR10.2 doi: 10.1007/BF01077471 – volume: 12 start-page: 305 year: 1960 ident: 432_CR15 publication-title: Osaka Math. J. – volume: 8 start-page: 326 year: 2018 ident: 432_CR7 publication-title: Int. J. Dyn. Syst. Differ. Equ. – volume: 43 start-page: 1261 year: 2002 ident: 432_CR8 publication-title: Comput. Math. Appl. doi: 10.1016/S0898-1221(02)00098-6 – volume: 12 start-page: 627 year: 1995 ident: 432_CR11 publication-title: Ann. Inst. H. Poincaré Anal. Non Linéaire doi: 10.1016/S0294-1449(16)30147-0 – volume: 22 start-page: 1 year: 1990 ident: 432_CR13 publication-title: Bull. London Math. Soc. doi: 10.1112/blms/22.1.1 – ident: 432_CR1 doi: 10.1109/CDC.2011.6161499 – volume: 62 start-page: 45 year: 1995 ident: 432_CR6 publication-title: Ann. Polon. Math. doi: 10.4064/ap-62-1-45-74 – ident: #cr-split#-432_CR10.1 doi: 10.1007/BF01077471 |
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| SubjectTerms | Asymptotic properties Difference and Functional Equations Differential equations Dynamical Systems and Ergodic Theory Mathematics Mathematics and Statistics Ordinary differential equations Stability |
| Title | Globally Asymptotically Stable Equilibrium Points in Kukles Systems |
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