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:
Buchtitelbild

2012 | OriginalPaper | Buchkapitel

1. Introduction

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

Chapter 1 is an introduction, presenting the background for nonlinear dynamics, bifurcation and stability, normal form method, Melnikov function and Hilbert’s 16th problem.

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
Nächstes Kapitel Hopf Bifurcation and Normal Form Computation
Literatur
4.
Doedel, E.J.: AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations. Concordia University, Montreal, Canada (August 2007)
5.
Arnold, V.I.: Lectures on bifurcations in versal families. Russ. Math. Surv. 27, 54–123 (1972) CrossRef
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
7.
Arnold, V.I.: Geometric Methods in the Theory of Ordinary Differential Equations. Springer, New York (1983) CrossRef
8.
Ashkenazi, M., Chow, S.N.: Normal forms near critical points for differential equations and maps. IEEE Trans. Circuits Syst. 35, 850–862 (1988) MathSciNetMATHCrossRef
9.
Atherton, D.P.: Stability of Nonlinear Systems. Research Studies Press, New York (1981) MATH
10.
11.
12.
13.
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
15.
16.
Belitskii, G.: Normal forms relative to a filtering action of a group. Trans. Mosc. Math. Soc. 2, 1–39 (1981) MathSciNet
19.
Bhatia, N.P., Szegö, G.P.: Stability Theory of Dynamical Systems. Springer, Berlin (1970) MATH
25.
Birkhoff, G.D.: Dynamical Systems. AMS, Providence (1927) MATH
26.
Campbell, S.A., Stone, E.: Analysis of the chatter instability in a nonlinear model for drilling. ASME J. Comput. Nonlinear Dyn. 1, 294–306 (2006) CrossRef
34.
Chen, G.R., Dong, X.: From Chaos to Order. World Scientific, Singapore (1998) MATH
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
45.
Chow, S.N., Drachman, B., Wang, D.: Computation of normal forms. J. Comput. Appl. Math. 29, 1290–1430 (1990) MathSciNetCrossRef
47.
Chow, S.-N., Li, C.-C., Wang, D.: Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press, Cambridge (1994) MATHCrossRef
48.
Chow, S.-N., Li, C., Yi, Y.: The cyclicity of period annulus of degenerate quadratic Hamiltonian system with elliptic segment. Ergod. Theory Dyn. Syst. 22(4), 1233–1261 (2002) MathSciNet
50.
Christopher, C.J., Lloyd, N.G.: Polynomial systems: a lower bound for the Hilbert numbers. Proc. R. Soc. Lond. A 450, 219–224 (1995) MathSciNetMATHCrossRef
51.
Christopher, C.J., Lloyd, N.G.: Small-amplitude limit cycles in polynomial Liénard systems. Nonlinear Differ. Equ. Appl. 3(2), 183–190 (1996) MathSciNetMATHCrossRef
52.
Christopher, C.J., Lynch, S.: Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces. Nonlinearity 12(4), 1099–1112 (1999) MathSciNetMATHCrossRef
54.
Chua, L.O., Kokubu, H.: Normal forms for nonlinear vector fields—Part I: Theory and algorithm. IEEE Trans. Circuits Syst. 35, 863–880 (1988) MathSciNetMATHCrossRef
55.
Chua, L.O., Kokubu, H.: Normal forms for nonlinear vector fields—Part II: Applications. IEEE Trans. Circuits Syst. 36, 51–70 (1988) MathSciNetCrossRef
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
66.
Gasull, A., Torregrosa, J.: Small-amplitude limit cycles in Liénard systems via multiplicity. J. Differ. Equ. 159(1), 186–211 (1999) MathSciNetMATHCrossRef
68.
Gazor, M., Yu, P.: Infinite order parametric normal form of Hopf singularity. Int. J. Bifurc. Chaos 18(11), 3393–3408 (2008) MathSciNetMATHCrossRef
69.
Golubitsky, M.S., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory. Springer, New York (1985) MATH
72.
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 4th edn. Springer, New York (1993)
74.
Han, M.: On the number of limit cycles bifurcating from a homoclinic or heteroclinic loop. Sci. China Ser. A 36(2), 113–132 (1993)
75.
Han, M.: Bifurcations of invariant tori and subharmonic solutions for periodic perturbed systems. Sci. China Ser. A 37(11), 1325–1336 (1994) MathSciNetMATH
76.
Han, M.: Cyclicity of planar homoclinic loops and quadratic integrable systems. Sci. China Ser. A 40(12), 1247–1258 (1997) MathSciNetMATHCrossRef
77.
Han, M.: Bifurcations of limit cycles from a heteroclinic cycle of Hamiltonian systems. Chin. Ann. Math., Ser. B 19(2), 189–196 (1998) MATH
78.
Han, M.: Liapunov constants and Hopf cyclicity of Liénard systems. Ann. Differ. Equ. 15(2), 113–126 (1999) MATH
79.
81.
Han, M.: Periodic Solution and Bifurcation Theory of Dynamical Systems. Science Publication, Beijing (2002) (in Chinese)
82.
Han, M.: Bifurcation theory of limit cycles of planar systems. In: Canada, A., Drabek, P., Fonda, A. (eds.) Handbook of Differential Equations, Ordinary Differential Equations, vol. 3. Elsevier, Amsterdam (2006). Chapter 4 CrossRef
83.
Han, M., Hu, S., Liu, X.: On the stability of double homoclinic and heteroclinic cycles. Nonlinear Anal. 53, 701–713 (2003) MathSciNetMATHCrossRef
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
85.
Han, M., Luo, D., Zhu, D.: Uniqueness of limit cycles bifurcating from a singular closed orbit. II. Acta Math. Sin. 35(4), 541–548 (1992) (in Chinese) MathSciNetMATH
86.
Han, M., Shang, D., Wang, Z., Yu, P.: Bifurcation of limit cycles in a 4th-order near-Hamiltonian systems. Int. J. Bifurc. Chaos 17(11) (2007)
87.
Han, M., Wang, Z., Zang, H.: Limit cycles by Hopf and homoclinic bifurcations for near-Hamiltonian systems. Chin. J. Contemp. Math. 28(4), 423–434 (2007) MathSciNet
88.
Han, M., Wu, Y., Bi, P.: Bifurcation of limit cycles near polycycles with n vertices. Chaos Solitons Fractals 22(2), 383–394 (2004) MathSciNetMATHCrossRef
89.
90.
Han, M., Yang, C.: On the cyclicity of a 2-polycycle for quadratic systems. Chaos Solitons Fractals 23, 1787–1794 (2005) MathSciNetMATH
91.
Han, M., Yang, J., Tarta, A., Yang, G.: Limit cycles near homoclinic and heteroclinic loops. J. Dyn. Differ. Equ. 20, 923–944 (2008) MathSciNetMATHCrossRef
92.
93.
Han, M., Ye, Y.: On the coefficients appearing in the expansion of Melnikov function in homoclinic bifurcations. Ann. Differ. Equ. 14(2), 156–162 (1998) MathSciNetMATH
94.
Han, M., Ye, Y., Zhu, D.: Cyclicity of homoclinic loops and degenerate cubic Hamiltonians. Sci. China Ser. A 42(6), 605–617 (1999) MathSciNetMATHCrossRef
95.
Han, M., Zhang, T., Zang, H.: On the number and distribution of limit cycles in a cubic system. Int. J. Bifurc. Chaos 14(12), 4285–4292 (2004) MathSciNetMATHCrossRef
96.
Han, M., Zhang, T., Zang, H.: Bifurcation of limit cycles near equivariant compound cycles. Sci. China Ser. A 50(4), 503–514 (2007) MathSciNetMATHCrossRef
97.
98.
Hassard, B.D., Kazarinoff, N.D., Wan, Y.-H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981) MATH
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
100.
Hopf, E.: Abzweigung einer periodischen Losung von stationaren Losung einers differential-systems. Ber. Math. Phys. Kl. Sachs Acad. Wiss. Leipzig 94, 1–22 (1942); and Ber. Math. Phys. Kl. Sachs Acad. Wiss. Leipzig Math.-Nat. Kl. 95, 3–22 (1942)
101.
Hou, Y., Han, M.: Melnikov functions for planar near-Hamiltonian systems and Hopf bifurcations. J. Shanghai Norm. Univ. (Nat. Sci.) 35(1), 1–10 (2006)
105.
Iooss, G.: Bifurcation of Maps and Applications. North-Holland, New York (1979) MATH
106.
109.
111.
Jiang, Q., Han, M.: Melnikov functions and perturbation of a planar Hamiltonian system. Chin. Ann. Math., Ser. B 20(2), 233–246 (1999) MathSciNetMATHCrossRef
113.
Jiang, X., Yuan, Z., Yu, P., Zou, X.: Dynamics of an HIV-1 therapy model of fighting a virus with another virus. J. Biol. Dyn. 3(4), 387–409 (2009) MathSciNetCrossRef
114.
124.
Langford, W.F., Zhan, K.: Strongly resonant Hopf bifurcation and vortex induced vibrations. In: IUTAM Symposium, Nonlinearity and Chaos in Engineering Dynamics. Springer, New York (1994)
125.
Leblanc, V.G., Langford, W.F.: Classification and unfoldings of 1:2 resonant Hopf bifurcation. Arch. Ration. Mech. Anal. 136, 305–357 (1996) MathSciNetMATHCrossRef
128.
Li, J.: Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. Int. J. Bifurc. Chaos 13, 47–106 (2003) MATHCrossRef
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
132.
Li, J., Huang, Q.: Bifurcations of limit cycles forming compound eyes in the cubic system. Chin. Ann. Math., Ser. B 8, 391–403 (1987) MATH
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
134.
Li, J., Lin, Y.: Global bifurcations in a perturbed cubic system with Z 2-symmetry. Acta Math. Appl. Sin. 8(2), 131–143 (1992) MathSciNetMATHCrossRef
135.
Li, J., Liu, Z.: Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system. Publ. Math. 35, 487–506 (1991) MATH
136.
Li, J., Liu, Z.: Bifurcation set and compound eyes in a perturbed cubic Hamiltonian system, in ordinary and delay differential equations. In: Pitman Research Notes in Mathematics Ser., vol. 272, pp. 116–128. Longman, Harlow (1991)
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
139.
Li, J., Zhang, M., Li, S.: Bifurcations of limit cycles in a Z 2-equivariant planar polynomial vector field of degree 7. Int. J. Bifurc. Chaos 16(4), 925–943 (2006) MATHCrossRef
140.
Li, J., Zhao, X.: Rotation symmetry groups of planar Hamiltonian systems. Ann. Differ. Equ. 5, 25–33 (1989) MATH
142.
Liao, X.X., Wang, L., Yu, P.: Stability of Dynamical Systems. Elsevier, Amsterdam (2007) MATH
143.
Liao, X.X., Yu, P.: Absolute Stability of Nonlinear Control Systems. Springer, New York (2008) MATHCrossRef
144.
Liénard, A.: Etude des oscillations entretenues. Rev. Gén. électr. 23, 901–912 (1928)
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
149.
Llibre, J.: Integrability of polynomial differential systems. In: Canada, A., Drabek, P., Fonda, A. (eds.) Handbook of Differential Equations, Ordinary Differential Equations, vol. 1. Elsevier, Amsterdam (2004). Chapter 5
150.
Llibre, J., Rodríguez, G.: Configurations of limit cycles and planar polynomial vector fields. J. Differ. Equ. 198(2), 374–380 (2004) MATHCrossRef
151.
Lloyd, N.G.: Limit cycles of polynomial systems. In: Bedford, T., Swift, J. (eds.) New Directions in Dynamical Systems. London Mathematical Society Lecture Notes, vol. 40, pp. 192–234 (1988) CrossRef
156.
Luo, D., Wang, X., Zhu, D., Han, M.: Bifurcation Theory and Methods of Dynamical Systems. Advanced Series in Dynamical Systems, vol. 15. World Scientific, Singapore (1997) MATH
159.
163.
Mebatsion, T., Finke, S., Weiland, F., Conzelmann, K.: A CXCR4/CD4 pseudotype rhabdovirus that selectively infects HIV-1 envelope protein-expressing cells. Cell 90, 841–847 (1997) CrossRef
164.
Melnikov, V.K.: On the stability of the center for time periodic perturbations. Trans. Mosc. Math. Soc. 12, 1–57 (1963)
178.
Poincaré, H.: Les Méthodes Nouvelles de la Mécanique Céleste (1892–1899)
183.
Roussarie, R.: On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields. Bol. Soc. Bras. Mat. 17(2), 57–101 (1986) MathSciNetCrossRef
186.
Sanders, J.A.: Normal form theory and spectral sequences. J. Differ. Equ. 192, 536–552 (2003) MATHCrossRef
187.
Schlomiuk, D.: Algebraic and geometric aspects of the theory of polynomial vector fields. In: Schlomiuk, D. (ed.) Bifurcations and Periodic Orbits of Vector Fields. NATO ASI Series C, vol. 408, pp. 429–467. Kluwer Academic, London (1993)
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.
193.
Takens, F.: Unfoldings of certain singularities of vector fields: generalized Hopf bifurcations. J. Differ. Equ. 14, 476–493 (1973) MathSciNetMATHCrossRef
195.
196.
Van der Pol, B.: On relaxation-oscillations. Philos. Mag. 2(7), 978–992 (1926)
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
206.
Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990) MATH
208.
Wu, J., Faria, T., Huang, Y.S.: Synchronization and stable phase-locking in a network of neurons with memory. Math. Comput. Model. 30, 117–138 (1999) MathSciNetMATHCrossRef
209.
Wu, Y., Han, M.: On the study of limit cycles of the generalized Rayleigh–Liénard oscillator. Int. J. Bifurc. Chaos 14(8), 2905–2914 (2004) MathSciNetMATHCrossRef
210.
Wu, Y., Han, M., Liu, X.: On the study of limit cycles of a cubic polynomials system under Z 4-equivariant quintic perturbation. Chaos Solitons Fractals 24(4), 999–1012 (2005) MathSciNetMATHCrossRef
211.
Xia, H., Wolkowicz, G.S.K., Wang, L.: Transient oscillations induced by delayed growth response in the chemostat. J. Math. Biol. 50, 489–530 (2005) MathSciNetMATHCrossRef
212.
Xu, Z., Hangan, H., Yu, P.: Analytical solutions for invisicid Gaussian impinging jets. ASME J. Appl. Mech. 75, 021019 (2008) CrossRef
213.
Xu, F., Yu, P., Liao, X.: Global analysis on n-scroll-chaotic attractors of modified Chua’s circuit. Int. J. Bifurc. Chaos 19(1), 135–157 (2009) MathSciNetMATHCrossRef
214.
Yang, J., Han, M.: Limit cycles near a double homoclinic loop. Ann. Differ. Equ. 23(4), 536–545 (2007) MathSciNetMATH
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
217.
Ye, Y.Q.: Theory of Limit Cycles. Transl. Math. Monographs, vol. 66. AMS, Providence (1986) MATH
223.
Yu, P.: Analysis on double Hopf bifurcation using computer algebra with the aid of multiple scales. Ann. Differ. Equ. 27, 19–53 (2002) MATH
224.
Yu, P.: Computation of the simplest normal forms with perturbation parameters based on Lie transform and rescaling. J. Comput. Appl. Math. 144(1–2), 359–373 (2002) MathSciNetMATHCrossRef
225.
Yu, P.: Bifurcation dynamics in control systems. In: Chen, R., Hill, D.J., Yu, X. (eds.) Chaos and Bifurcation Control: Theory and Applications, vol. 293, pp. 99–126. Springer, New York (2003)
226.
Yu, P.: A simple and efficient method for computing center manifold and normal forms associated with semi-simple cases. Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 10(1–3), 273–286 (2003) MathSciNetMATH
227.
Yu, P.: Closed-form conditions of bifurcation points for general differential equations. Int. J. Bifurc. Chaos 15(4), 1467–1483 (2005) MATHCrossRef
230.
Yu, P.: Local and global bifurcations to limit cycles in a class of Liénard equation. Int. J. Bifurc. Chaos 17(1), 183–198 (2007) MATHCrossRef
232.
Yu, P., Chen, G.: Hopf bifurcation control using nonlinear feedback with polynomial functions. Int. J. Bifurc. Chaos 14(5), 1683–1704 (2004) MathSciNetMATHCrossRef
233.
Yu, P., Chen, R.: The simplest parametrized normal forms of Hopf and generalized Hopf bifurcations. Ann. Differ. Equ. 50(1–2), 297–313 (2007) MathSciNetMATH
234.
Yu, P., Chen, R.: Computation of focus values with applications. Ann. Differ. Equ. 51(3), 409–427 (2008) MathSciNetMATH
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
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
247.
248.
Yu, P., Leung, A.Y.T.: A perturbation method for computing the simplest normal forms of dynamical systems. J. Sound Vib. 261(1), 123–151 (2003) MathSciNetCrossRefMATH
249.
Yu, P., Leung, A.: Normal forms of vector fields with perturbation parameters. Chaos Solitons Fractals 34(2), 564–579 (2007) MathSciNetMATHCrossRef
250.
Yu, P., Leung, A.: The simplest normal form and its application to bifurcation control. Chaos Solitons Fractals 33(3), 845–863 (2007) MathSciNetMATHCrossRef
251.
Yu, P., Yuan, Y., Xu, J.: Study of double Hopf bifurcation and chaos for an oscillator with time delayed feedback. Commun. Nonlinear Sci. Numer. Simul. 7(1–2), 69–91 (2002) MathSciNetCrossRef
252.
Yu, P., Yuan, Y.: A matching pursuit technique for computing the simplest normal forms of vector fields. J. Symb. Comput. 35(5), 591–615 (2003) MathSciNetMATHCrossRef
253.
Yu, P., Yuan, Y.: An efficient method for computing the simplest normal forms of vector fields. Int. J. Bifurc. Chaos 13(1), 19–46 (2003) MathSciNetMATHCrossRef
254.
Yu, P., Zhu, S.: Computation of the normal forms for general M-DOF systems using multiple time scales. Part I: Autonomous systems. Commun. Nonlinear Sci. Numer. Simul. 10(8), 869–905 (2005) 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
259.
Metadaten
Titel
Introduction
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_1