The simplest normal form of Hopf bifurcation

Nonlinearity

Volumn 16, Issue 1, 2003, Pages 277-300

  Yu, P ^a   Leung, A Y T ^b  

^a UNIVERSITY OF WESTERN ONTARIO   (Canada)

^b CITY UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0242371109     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/16/1/317     Document Type: Article

References (37)

1. Algaba A, Freire E and Gamero E 1998 Hypernormal form for the Hopf-zero bifurcation Int. J. Bifurcation Chaos 8 1857-87
2. Algaba A, Freire E and Gamero E 1999 Hypernormal forms for equilibria of vector fields: Codimension one linear degeneracies Rocky Mountain J. Math. 29 13-45
3. Baider A and Churchill R 1988 Unique normal forms for planar vector fields Math. Z. 199 303-10
4. Baider A and Sanders J A 1992 Further reduction of the Takens-Bogdanov normal forms J. Diff. Eqn. 99 205-44
5. Belitskii G 1979 Invariant normal forms of formal series Func. Anal Appl. 13 46-7
6. Belitskii G 1981 Normal forms relative to a filtering action of a group Trans. Mask. Math. Soc. 2 1-39
7. Bi Q and Yu P 1998 Computation of normal forms of differential equations associated with non-semisimple zero eigenvalues Int. J. Bifurcation Chaos 8 2279-319
8. Chen Y S and Leung A Y T 1998 Bifurcation and Chaos in Engineering (London: Springer)
9. Chow S-N, Li C-Z and Wang D 1994 Normal Forms and Bifurcation of Planar Vector Fields (Cambridge: Cambridge University Press)
10. Cushman R and Sanders J A 1986 Nilpotent normal forms and representation theory of s/(2,R) Contemp. Math. 56 31-51
11. Cushman R and Sanders J A 1988 Normal form for (2;n-nilpotent vector field, using invariant theory Physica D 30 399-412
12. Chua L O and Kokubu H 1988 Normal forms for nonlinear vector fields - Part I: Theory and algorithm IEEE Trans. Circ. Syst. 35 863-80
13. Chua L O and Kokubu H 1988 Normal forms for nonlinear vector fields - Part II: Applications IEEE Trans. Circ. Syst. 3651-70
14. Dulac M H 1912 Solutions d'un systém d'éuations diffééilles dans le voisinage de valeure singuliées Bull. Soc. Math. France 40 324-83
15. Elphick C, Tirapegui E, Brachet M E, Coullet P and Iooss G 1987 A simple global characterization for normal forms of singular vector fields Physica D 29 95-127
16. Golubisky M S and Schaeffer D G 1985 Singularities and Croups in Bifurcation Theory (New York: Springer)
17. Guckenheimer J and Holmes P 1992 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (New York: Springer)
18. Huseyin K 1986 Multiple-Parameter Stability Theory and its Applications (Oxford: Oxford University Press)
19. Kokubu H 1984 Normal forms for parametrized vector fields and its applications to bifurcations of some reaction diffusion equations Japan. J. Appl. Math. 1273-97
20. Kokubu H, Oka H and Wang D 1996 Linear grading function and further reduction of normal forms J. Diff. Eqn. 132 293-318
21. Leung A Y T and Ge T 1995 An algorithm for higher order Hopf normal form J. Shock and Vibration 2 307-19
22. Leung A Y T and Zhang Q C 1998 Higher-order normal form and period averaging J. Sound Vibration 217 795-806
23. Leung A Y T, Zhang Q C and Chen Y S 1994 Normal form analysis of Hopf bifurcation exemplified by Duffing's equation J. Shock Vibration 1 233-40
24. Nayfeh A H 1993 Methods of Normal Forms (New York: Wiley)
25. Poincaré H 1879 Sur les Propriétés des Fonctions Définies par des Equations aux Différences Partielles Thèse Inaugural (Paris: Gauthier-Villars)
26. Takens F 1973 Normal forms for certain singularities of vector fields Ann. Inst. Fourier 23 163-5
27. Ushiki S 1984 Normal forms for singularities of vector fields Japan. J. Appl. Math. 1 1-37
28. Wang D., Li J., Huang M. and Jiang Y. 2000 Unique normal forms of Bogdanov-Tiakens singularity J. Diff. Eqn. 163 223-38
29. Yu P and Huseyin K 1988 A perturbation analysis of interactive static and dynamic bifurcations IEEE Trans. Automat. Control. 33 28-41
30. Yu P 1998 Computation of normal forms via a perturbation technique J. Sound Vibration 211 19-38
31. Yu P 1999 Simplest normal forms of Hopf and generalized Hopf bifurcations Int. J. Bifurcation Chaos 9 1917-39
32. Yu P 2002 Computation of the simplest normal forms with perturbation parameters based on Lie transform and rescaling J. Comput. Appl. Math. 144 359-73
33. Yu P 2002 A simple and efficient method for computing centre manifold and normal forms associated with semi-simple cases. Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis at press
34. Yu P and Yuan Y 2001 The simplest normal form for the singularity of a pure imaginary pair and a zero eigenvalue Continuous, Discrete and Impulsive Systems (DCDIS) Series B: Applications and Algorithms 8 219-49
35. Yu P and Yuan Y 2003 An efficient method for computing the simplest normal forms of vector fields Int. J. Bifurcation Chaos 13
36. Yu P and Yuan Y 2002 An matching pursuit technique for computing the simplest normal forms of vector fields J. Symbolic Computation to appear
37. Yuan Y and Yu P 2001 Computation of simplest normal forms of differential equations associated with a double-zero eigenvalues Int. J. Bifurcation Chaos 11 1307-30