A new kind of parametric singularities and their use for the study of the bifurcation structure of an n-dimensional map

Nonlinear Analysis, Theory, Methods and Applications

Volumn 28, Issue 5, 1997, Pages 917-946

  Carcasses, Jean Pierre ^a  

^a DGE   (France)

Author keywords

Contour line ; Crossroad area; Flip bifurcation curve; Fold bifurcation curve; Reduced multiplier; Saddle area; Spring area

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; CURVE FITTING; DIFFERENTIAL EQUATIONS; DIFFERENTIATION (CALCULUS); EIGENVALUES AND EIGENFUNCTIONS; FUNCTIONS; MATRIX ALGEBRA; TOPOLOGY;

CONTOUR LINE; CROSSROAD AREA; FLIP BIFURCATION CURVE; FOLD BIFURCATION CURVE; PARAMETRIC SINGULARITIES; REDUCED MULTIPLIER; SADDLE AREA; SPRING AREA;

BIFURCATION (MATHEMATICS);

EID: 0031104061     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/0362-546x(95)00185-x     Document Type: Article

References (20)

1. SHIRAIWA K., A generalisation of the Levinson-Massera's equalities, Nagoya Math. J. 67, 121-138 (1977).
2. KAWAKAMI H., Bifurcations of periodic responses in forced dynamic non-linear circuits. Computation of bifurcation values of the systems parameters, IEEE Trans. Circuits and Systems 31, 248-260 (1984).
3. YOSHINAGA T. & KAWAKAMI H., Codimension two bifurcation problems in forced non-linear circuits, Trans. IEICE E73(6), 000-000 (1990).
4. MIRA C., Chaotic Dynamics from the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism. World Scientific, Singapore (1987).
5. MIRA C., CARCASSES J-P., BOSCH M., SIMO C. & TATJER J. C., Crossroad area-spring area transition, (II) foliated parametric representation, Int. J. Bifurcation and Chaos 1(2), 339-348 (1991).
6. CARCASSES J-P. & KAWAKAMI H., Necessary and sufficient condition for the existence of a cusp point on a fold bifurcation curve in the parameter plane of a n-dimensional map. Preprint (1993).
7. EL HAMOULY H. & MIRA C., Singularitiés dues au feuilletage du plan des bifurcations d'un difféomorphisme bi-dimensionnel, C. r. Acad. Sci. Paris 294, 387-390 (1982).
8. FOURNIER-PRUNARET D., KAWAKAMI H. & MIRA C., Feuilletage du plan des bifurcations d'un difféopmorphisme bi-dimensionnel. Doublement de l'ordre des zones source et des zones échangeur, C. r. Acad. Sci. Paris 301, 223-226 (1985).
9. CARCASSES J-P., MIRA C., BOSCH M., SIMO C. & TATJER J. C., Crossroad area-Spring area transition, (I) parameter plane representation, Int. J. Bifurcation and Chaos 1(1), 183-196 (1991).
10. CARCASSES J-P., Sur quelques structures complexes de bifurcation de systèmes dynamique, Thèse de Doctorat de l'Univ. Paul Sabatier, Toulouse, France (1990).
11. MIRA C., On some codimension - three bifurcations occurring in maps. Spring area-crossroad area transitions, Proceedings ECIT 91, pp. 168-177. World Scientific (1991).
12. TAHA A. K., Organisation of bifurcations generated from a Matthieu-Duffing equation with a double periodic excitation, Proceedings ECIT 91, pp. 285-294. World Scientific (1991).
13. YAMAMOTO N., The computation of nodal points generated by doubling bifurcation points on a locus of turning points, Proceedings of The Second Symposium on Non-linear Theory and its Applications, pp. 115-118 (1991).
14. YAMAMOTO N., Singular points of plane curves generated by period doubling bifurcation points and a method for computing them, Int. J. Bifurcation and Chaos 1(4), 795-802 (1991).
15. CARCASSES J-P., A necessary and sufficient condition to have r pairs of complex multipliers with modulus equal to unity in the case of an n-dimensional map, Int. J. Bifurcation and Chaos (to appear) (1995).
16. CARCASSES J-P., Determination of different configurations of fold and flip bifurcation curves of a one or two-dimensional map, Int. J. Bifurcation and Chaos 3(4), 869-902 (1993).
17. 2 = 1 in the bifurcation curves of maps, Int. J. Bifurcation and Chaos 2(4), 1001-1004 (1992).
18. MIRA C. & CARCASSES J-P., On the "crossroad area-saddle area" and "crossroad area-spring area" transitions, Int. J. Bifurcation and Chaos 1(3), 641-655 (1991).
19. ANDRONOV A. A., WITT A. A. & KHAIKIN S., Theory of Oscillators. Pergamon Press, Oxford (1966).
20. ANDRONOV A. A., LEONTOVITCH E. A., GORDON I. I. & MAIER A. G., Theory of Bifurcations of Dynamic Systems in the Phase Plane. Nauka, Moscow (1967).