Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Introduction Kapitel lesen Erstes Kapitel lesen

2012 | OriginalPaper | Buchkapitel

4. Application (I)—Hilbert’s 16th Problem

verfasst von : Maoan Han, Pei Yu

Erschienen in: Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles

Verlag: Springer London

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In Chap. 4, Hopf bifurcation and computation of normal forms are applied to consider planar vector fields and focus on the well-known Hilbert’s 16th problem. Attention is given to general cubic order and higher order systems are considered to find the maximal number of limit cycles possible for such systems i.e., to find the lower bound of the Hilbert number for certain vector fields. The Liénard system is also investigated and critical periods of bifurcating periodic solutions from two special type of planar systems are studied.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel Comparison of Methods for Computing Focus Values
Nächstes Kapitel Application (II)—Practical Problems
Fußnoten
1
The command realroot, which uses dyadic rationals and binary splitting after a method of Collins, provides reliable intervals guaranteed to contain roots. However, fsolve has evolved to be equally reliable.
 
Literatur
6.
Arnold, V.I.: Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields. Funct. Anal. Appl. 11, 85–92 (1977) CrossRef
14.
Bautin, N.N.: On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type. Mat. Sb. (N. S.) 30(72), 181–196 (1952) MathSciNet
18.
Beyn, W.J., Champneys, A., Doedel, E., Govaerts, W., Kuznetsov, Yu.A., Sandstede, B.: Numerical continuation, and computation of normal forms. In: Fiedler, B. (ed.) Handbook of Dynamical Systems, vol. 2, pp. 149–219. North-Holland, Amsterdam (2002) CrossRef
29.
Caubergh, M., Dumortier, F.: Hilbert’s 16th problem for classical Liénard equations of even degree. J. Differ. Equ. 244(6), 1359–1394 (2008) MathSciNetMATHCrossRef
30.
Caubergh, M., Dumortier, F., Luca, S.: Cyclicity of unbounded semihyperbolic 2-saddle cycles in polynomial Liénard systems. Discrete Contin. Dyn. Syst. 27(3), 963–980 (2010) MathSciNetMATHCrossRef
31.
Caubergh, M., Françoise, J.P.: Generalized Liénard equations, cyclicity and Hopf–Takens bifurcations. Qual. Theory Dyn. Syst. 5(2), 195–222 (2004) MathSciNetMATHCrossRef
38.
Chen, L.S., Wang, M.S.: The relative position, and the number, of limit cycles of a quadratic differential system. Acta Math. Sin. 22, 751–758 (1979) MATHCrossRef
40.
Chen, G., Wu, Y., Yang, X.: The number of limit cycles for a class of quintic Hamiltonian systems under quintic perturbations. J. Aust. Math. Soc. 73, 37–53 (2002) MathSciNetMATHCrossRef
44.
Chicone, C., Jacobs, M.: Bifurcation of critical periods for plane vector fields. Trans. Am. Math. Soc. 312, 433–486 (1989) MathSciNetMATHCrossRef
46.
56.
Dhooge, A., Govaerts, W., Kuznetsov, Yu.A.: MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw. 29, 141–164 (2003) MathSciNetMATHCrossRef
57.
Dhooge, A., Govaerts, W., Kuznetsov, Yu.A., Meijer, H.G.E., Sautois, B.: New features of the software MatCont for bifurcation analysis of dynamical systems. Math. Comput. Model. Dyn. Syst. 14, 145–175 (2008) MathSciNetCrossRef
58.
Dumortier, F.: Compactification and desingularization of spaces of polynomial Liénard equations. J. Differ. Equ. 224(2), 296–313 (2006) MathSciNetMATHCrossRef
59.
61.
Dumortier, F., Panazzolo, D., Roussarie, R.: More limit cycles than expected in Liénard equations. Proc. Am. Math. Soc. 135, 1895–1904 (2007) MathSciNetMATHCrossRef
70.
Gonzalez-Vega, L., Rouillier, F., Roy, M.-F., Trujillo, G.: Symbolic Recipes for Real Solutions, in Some Tapas of Computer Algebra. Springer, Heidelberg (1999)
84.
Han, M., Lin, Y., Yu, P.: A study on the existence of limit cycles of a planar system with 3rd-degree polynomials. Int. J. Bifurc. Chaos 14(1), 41–60 (2004) MathSciNetMATHCrossRef
99.
Hilbert, D.: Mathematical problems (M. Newton, transl.). Bull. Am. Math. Soc. 8, 437–479 (1902); reprinted in Bull. Am. Math. Soc. (N. S.), 37, 407–436 (2000) MathSciNetMATHCrossRef
107.
Ilyashenko, Yu., Panov, A.: Some upper estimations of the number of limit cycles of planar vector fields with application to the Liénard equation. Mosc. Math. J. 1(4), 583–599 (2001) MathSciNetMATH
109.
112.
Jiang, J., Han, M., Yu, P., Lynch, S.: Limit cycles in two types of symmetric Liénard systems. Int. J. Bifurc. Chaos 17(6), 2169–2174 (2007) MathSciNetMATHCrossRef
118.
Kolutsky, G.: One upper estimate on the number of limit cycles in even degree Liénard equations in the focus case. arXiv:​0911.​3516v1 [math.DS], November 18 (2009)
120.
Kuznetsov, Yu.A.: Numerical normalization techniques for all codimension 2 bifurcations of equilibria in ODEs. SIAM J. Numer. Anal. 36, 1104–1124 (1999) MathSciNetMATHCrossRef
121.
Kuznetsov, Yu.A.: Elements of Applied Bifurcation Theory, 3rd edn. Springer, New York (2004) MATH
122.
Kuznetsov, Yu.A.: Practical computation of normal forms on center manifolds at degenerate Bogdanov–Takens bifurcations. Int. J. Bifurc. Chaos 15(11), 3535–3546 (2005) MATHCrossRef
127.
Li, J.: Chaos and Melnikov Method. Chongqing University Press, Chongqing (1989)
128.
Li, J.: Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. Int. J. Bifurc. Chaos 13, 47–106 (2003) MATHCrossRef
129.
Li, J., Bai, J.X.: The cyclicity of multiple Hopf bifurcation in planar cubic differential systems: M(3)≥7. Preprint, Kunming Institute of Technology (1989)
130.
Li, J., Chan, H.S.Y., Chung, K.W.: Bifurcations of limit cycles in a Z 3-equivariant planar vector field of degree 5. Int. J. Bifurc. Chaos 11, 2287–2298 (2001) MathSciNetMATHCrossRef
131.
Li, J., Chan, H.S.Y., Chung, K.W.: Investigations of bifurcations of limit cycles in Z 2-equivariant planar vector fields of degree 5. Int. J. Bifurc. Chaos 12(10), 2137–2157 (2002) MathSciNetMATHCrossRef
133.
Li, J., Li, C.F.: Planar cubic Hamiltonian systems and distribution of limit cycles of (E 3). Acta Math. Sin. 28, 509–521 (1985) MATH
137.
138.
Li, J., Zhang, M.Q.: Bifurcations of limit cycles in a Z 8-equivariant planar vector field of degree 7. J. Differ. Equ. Dyn. Syst. 16(4), 1123–1139 (2004) MATHCrossRef
141.
Li, J., Zhou, H.: On the control of parameters of distributions of limit cycles for a Z 2-equivariant perturbed planar Hamiltonian polynomial vector field. Int. J. Bifurc. Chaos 15, 137–155 (2005) MATHCrossRef
144.
Liénard, A.: Etude des oscillations entretenues. Rev. Gén. électr. 23, 901–912 (1928)
145.
Lins, A., De Melo, W., Pugh, C.C.: On Liénard equation. Lect. Notes Math. 597, 335–357 (1977) CrossRef
147.
Liu, Y., Li, J.: New results on the study of Z q -equivariant planar polynomial vector fields. Qual. Theory Dyn. Syst. 9, 167–219 (2010) MathSciNetMATHCrossRef
148.
Liu, Y., Li, J., Huang, W.: Singular Point Values, Center Problem and Bifurcations of Limit Cycles of Two Dimensional Differential Autonomous Systems. Science Press, Beijing (2008)
152.
153.
157.
158.
Lynch, S., Christopher, C.J.: Limit cycles in highly non-linear differential equations. J. Sound Vib. 224(3), 505–517 (1999) MathSciNetMATHCrossRef
161.
Mañosas, F., Villadelprat, J.: A note on the critical periods of potential systems. Int. J. Bifurc. Chaos 16, 765–774 (2006) MATHCrossRef
180.
Rayleigh, J.: The Theory of Sound. Dover, New York (1945) MATH
184.
185.
Rousseau, C., Toni, B.: Local bifurcation of critical periods in vector fields with homogeneous nonlinearities of the third degree. Can. Math. Bull. 36, 473–484 (1993) MathSciNetMATHCrossRef
189.
Shi, S.: A concrete example of the existence of four limit cycles for plane quadratic systems. Sci. Sin. 11, 1051–1056 (1979) (in Chinese); 23, 153–158 (1980) (in English)
190.
191.
196.
Van der Pol, B.: On relaxation-oscillations. Philos. Mag. 2(7), 978–992 (1926)
199.
Wang, D.M.: A class of cubic differential systems with 6-tuple focus. J. Differ. Equ. 87, 305–315 (1990) MATHCrossRef
200.
Wang, S.: Hilbert’s 16th problem and computation of limit cycles. PhD Thesis, The University of Western Ontario, Canada (2004)
203.
Wang, S., Yu, P.: Bifurcation of limit cycles in a quintic Hamiltonian system under sixth-order perturbation. Chaos Solitons Fractals 26(5), 1317–1335 (2005) MathSciNetMATHCrossRef
204.
Wang, S., Yu, P.: Existence of 121 limit cycles in a perturbed planar polynomial Hamiltonian vector field of degree 11. Chaos Solitons Fractals 30(3), 606–621 (2006) MathSciNetMATHCrossRef
205.
Wang, S., Yu, P., Li, J.: Bifurcation of limit cycles in Z 10-equivariant vector fields of degree 9. Int. J. Bifurc. Chaos 16(8), 2309–2324 (2006) MathSciNetMATHCrossRef
215.
Yang, J., Han, M., Li, J., Yu, P.: Existence conditions of thirteen limit cycles in a cubic system. Int. J. Bifurc. Chaos 20(8), 2569–2577 (2010) MathSciNetMATHCrossRef
216.
Yao, W., Yu, P.: Bifurcation of small limit cycles in Z 5-equivariant planar vector fields of order 5. J. Math. Anal. Appl. 328(1), 400–413 (2007) MathSciNetMATHCrossRef
218.
Yu, P.: Computation of normal forms via a perturbation technique. J. Sound Vib. 211(1), 19–38 (1998) MATHCrossRef
222.
Yu, P., Han, M.: Limit cycles in 3rd-order planar system. In: International Congress of Mathematicians, Beijing, China, August 20–28, 2002
229.
Yu, P.: Computation of limit cycles—the second part of Hilbert’s 16th problem. Fields Inst. Commun. 49, 151–177 (2006)
235.
Yu, P., Chen, R.: Bifurcation of limit cycles in a 5th-order Z 6-equivariant planar vector field. Preprint
236.
Yu, P., Corless, R.M.: Symbolic computation of limit cycles associated with Hilbert’s 16th problem. Commun. Nonlinear Sci. Numer. Simul. 14(12), 4041–4056 (2009) MathSciNetMATHCrossRef
237.
Yu, P., Han, M.: Twelve limit cycles in a 3rd-order planar system with Z 2 symmetry. Commun. Pure Appl. Anal. 3(3), 515–526 (2004) MathSciNetMATHCrossRef
238.
Yu, P., Han, M.: Twelve limit cycles in a cubic case of the 16th Hilbert problem. Int. J. Bifurc. Chaos 15(7), 2191–2205 (2005) MathSciNetMATHCrossRef
239.
Yu, P., Han, M.: Small limit cycles from fine focus points in cubic order Z 2-equivariant vector fields. Chaos Solitons Fractals 24(1), 329–348 (2005) MathSciNetMATH
240.
241.
Yu, P., Han, M.: On limit cycles of the Liénard equations with Z 2 symmetry. Chaos Solitons Fractals 31(3), 617–630 (2007) MathSciNetMATHCrossRef
242.
Yu, P., Han, M.: Critical periods of planar reversible vector field with 3rd-degree polynomial functions. Int. J. Bifurc. Chaos 19(1), 419–433 (2009) MathSciNetMATHCrossRef
243.
Yu, P., Han, M., Yuan, Y.: Analysis on limit cycles of Z q -equivariant polynomial vector fields with degree 3 or 4. J. Math. Anal. Appl. 322(1), 51–65 (2006) MathSciNetMATHCrossRef
257.
Zhang, T.H., Han, M., Zang, H., Meng, X.Z.: Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations. Chaos Solitons Fractals 22, 1127–1138 (2004) MathSciNetMATHCrossRef
258.
Zhang, W.N., Hou, X.R., Zeng, Z.B.: Weak centers and bifurcation of critical periods in reversible cubic systems. Comput. Math. Appl. 40, 771–782 (2000) MathSciNetMATHCrossRef
261.
Metadaten
Titel
Application (I)—Hilbert’s 16th Problem
verfasst von
Maoan Han
Pei Yu
Copyright-Jahr
2012
Verlag
Springer London
Buch
Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles

Print ISBN: 978-1-4471-2917-2

Electronic ISBN: 978-1-4471-2918-9

Copyright-Jahr: 2012

DOI
https://doi.org/10.1007/978-1-4471-2918-9_4