zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Codimension 2 bifurcation of twisted double homoclinic loops. (English) Zbl 1186.34054
Summary: A local active coordinates approach is employed to obtain bifurcation equations of twisted double homoclinic loops. Under the condition of one twisted orbit, we obtain the existence and uniqueness and of the 1-1 double homoclinic loop, 2-1 double homoclinic loop, 2-1 right homoclinic loop, 1-1 large homoclinic loop, 2-1 large homoclinic loop and 2-1 large period orbit. For the case of double twisted orbits, we obtain the existence or non-existence of 1-1 double homoclinic loop, 1-2 double homoclinic loop, 2-1 double homoclinic loop, 2-2 double homoclinic loop, 2-1 large homoclinic loop, 1-2 large homoclinic loop, 2-2 large homoclinic loop, 2-2 right homoclinic loop, 2-2 large homoclinic loop, 2-2 left homoclinic loop and 2-2 large period orbit. Moreover, the bifurcation surfaces and their existence regions are given. Besides, bifurcation sets are presented on the 2 dimensional subspace spanned by the first two Melnikov vectors.
MSC:
34C23Bifurcation (ODE)
37G99Local and nonlocal bifurcation theory
References:
[1]Dumortier, F.; Roussarie, R.: On the saddle loop bifurcation, Lnm 1455 (1990) · Zbl 0731.34010
[2]Dumortier, F.; Roussarie, R.; Sotomayor, J.: Generic 3-parameter families of planar vector fields, unfoldings of saddle, focus and elliptic singularities with nilpotent linear parts, Lnm 1480 (1991)
[3]Du, Z.; Zhang, W.: Melnikov method for homoclinic bifurcation in nonlinear impact oscillators, Comput. math. Appl. 50, 445-458 (2005) · Zbl 1097.37043 · doi:10.1016/j.camwa.2005.03.007
[4]Feng, B.: The stability of heteroclinic loop under the critical condition, Sci. China ser. A 34, 673-684 (1991)
[5]Feng, B.; Xiao, D.: Homolinic and heteroclinic bifurcations of heteroclinic loops, Acta math. Sin. 35, 815-830 (1992)
[6]Homburg, A. J.: Global aspects of homoclinic bifurcations of vector fields, Global aspects of homoclinic bifurcations of vector fields 121 (1996)
[7]Luo, D.; Wang, X.; Zhu, D.: Bifurcation theory and methods of dynamical systems, (1997)
[8]Mourtada, A.: Degenerate and nontrivial hyperbolic polycycles with vertices, J. differential equations 113, 68-83 (1994) · Zbl 0810.58027 · doi:10.1006/jdeq.1994.1114
[9]Naudot, V.: Strange attractor in the unfolding of an inclination-flip homoclinic orbit, Ergod. theory dynam. Systems 16, 1071-1086 (1996) · Zbl 0866.58045 · doi:10.1017/S014338570001018X
[10]Reyn, J. W.: Generation of limit cycles from separatrix polygons in the phase plane, Lect. notes math. 810, 264-289 (1980) · Zbl 0437.34025
[11]Roussarie, R.: On the number of limit cycles which appear by perturbation of separatrix loop of planar fields, Bol. soc. Brasil mat. 17, 67-101 (1986) · Zbl 0628.34032 · doi:10.1007/BF02584827
[12]B. Sandstede, Verzweigungstheorie homokliner Verdopplungen, Ph.D. Thesis, University of Stuttgart, 1993 · Zbl 0850.58012
[13]Shil’nikov, L. P.: A case of the existence of a denumerable set of periodic motions, Sov. math. Dockl. 6, 163-166 (1965) · Zbl 0136.08202
[14]Wu., Y.; Han, M.: New configurations of 24 limit cycles in a quintic system, Comput. math. Appl. 55, 2064-2075 (2008) · Zbl 1153.37027 · doi:10.1016/j.camwa.2007.08.034
[15]Zhang, Z.; Li, C.; Zheng, Z.: The foundation of bifurcation theory for vector field, (1997)
[16]Chow, S. N.; Deng, B.; Fiedler, B.: Homoclinic bifurcation at resonant eigenvalues, J. dyn. Syst. diff. Eqs. 2, 177-244 (1990) · Zbl 0703.34050 · doi:10.1007/BF01057418
[17]Jin, Y.; Zhu, D.: Degenerated homoclinic bifurcations with higher dimensions, Chin. ann. Math. ser. B 21, 201-210 (2000) · Zbl 0974.37014 · doi:10.1142/S0252959900000224
[18]Jin, Y.; Zhu: Bifurcations of rough heteroclinic loops with three saddle points, Acta math. Sin. (Engl. Ser.) 18, 199-208 (2002) · Zbl 1010.34037 · doi:10.1007/s101140100139
[19]Sun, J.: Bifurcations of heteroclinic loop with nonhyperbolic critical points in rn, Sci. China, ser. A 24, 1145-1151 (1994)
[20]Tian, Q.; Zhu, D.: Bifurcations of nontwisted heteroclinic loops, Sci. China ser. A 30, 193-202 (2000)
[21]Zhu, D.: Problems in homoclinic bifurcation with higher dimensions, Acta math. Sin. new series 14, 341-352 (1998) · Zbl 0932.37032 · doi:10.1007/BF02580437
[22]Zhu, D.; Xia, Z.: Bifurcations of heteroclinic loops, Sci. China ser. A 41, 837-848 (1998)
[23]Han, M.; Chen, J.: The number of limit cycles in double homoclinic bifurcations, Chin. ann. Math. 43, 914-928 (2000) · Zbl 1013.34026 · doi:10.1007/BF02879797
[24]Han, M.; Wu, Y.: The stability of double homoclinic loops, Appl. math. Lett. 17, 1291-1298 (2004) · Zbl 1122.37312 · doi:10.1016/j.aml.2003.10.012
[25]Homburg, A. J.; Knobloch, J.: Multiple homoclinic orbits in conservative and reversible systems, Trans. amer. Math. soc. 358, 1715-1740 (2006) · Zbl 1162.34034 · doi:10.1090/S0002-9947-05-03793-1
[26]Morales, C. A.; Pacifico, M. J.; Martin, B. San: Contracting Lorenz attractors through resonant double homoclinic loops, SIAM J. Math. anal. 38, 309-332 (2006) · Zbl 1107.37041 · doi:10.1137/S0036141004443907
[27]Rychlik, M. R.: Lorenz attractors through shil’nikov-type bifurcation. I., Ergod. theory dynam. Systems 10, 793-821 (1990) · Zbl 0715.58027 · doi:10.1017/S0143385700005915
[28]Robinson, C.: Homoclinic bifurcation to a transitive attractor of Lorenz type, Nonlinearity 2, 495-518 (1989) · Zbl 0704.58031 · doi:10.1088/0951-7715/2/4/001
[29]Turaev, D. V.: Bifurcations of a homoclinic ”figure eight” of a multidimensional saddle, Uspekhi mat. Nauk 43, 223-224 (1988) · Zbl 0679.58032 · doi:10.1070/RM1988v043n05ABEH001952
[30]Turaev, D. V.; Shil’nikov, L. P.: Bifurcation of a homoclinic ”figure eight” saddle with a negative saddle value, Dokl. akad. Nauk SSSR 290, 1301-1304 (1986) · Zbl 0638.34037
[31]Zou, Y.; She, Y.: Homoclinic bifurcation properties near eight-figure homoclinic orbit, Northeast. math. J. 18, 79-88 (2002) · Zbl 1028.37031
[32]Zang, H.; Zhang, T.; Han, M.: Bifurcations of limit cycles from quintic Hamiltonian systems with a double figure eight loop, Bull. sci. Math. 130, 71-86 (2006) · Zbl 1101.34027 · doi:10.1016/j.bulsci.2005.07.001
[33]Sell; George, R.: Smooth linearization near a fixed point, Amer. J. Math. 107, 1035-1091 (1985) · Zbl 0574.34025 · doi:10.2307/2374346
[34]Deng, B.: Homoclinic twisting bifurcations and cusp horseshoe, J. dynam. Differential equations 5, 417-467 (1993) · Zbl 0782.34042 · doi:10.1007/BF01053531