Homoclinic and heteroclinic bifurcations close to a twisted heteroclinic cycle

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 8, Issue 2, 1998, Pages 359-375

  Zimmermann, Martín G ^a,c   Natiello, Mario A ^b  

^a UPPSALA UNIVERSITY   (Sweden)

^b ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

^c INSTITUTO MEDITERRÁNEO DE ESTUDIOS AVANZADOS   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032008443     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127498000218     Document Type: Article

References (24)

1. Bykov, V. V. [1993] "The bifurcations of separatrix contours and chaos," Physica D62, 290-299.
2. Champneys, A. R. & Kuznetsov, Yu. A. [1994] "Numerical detection and continuation of codimension-two homoclinic bifurcations," Int. J. Bifurcation and Chaos 4(4), 785-822.
3. Chow, S.-N., Deng, B. & Terman, D. [1990] "The bifurcations of homoclinic and periodic orbits from two heteroclinic orbits," SIAM J. Math. Anal. 21, 179-204.
4. 1-linearization, and homoclinic bifurcation," J. Diff. Eqs. 79, 189-231.
5. Deng, B. [1990] "Homoclinic bifurcations with nonhyperbolic equilibria," SIAM J. Math. Anal. 21(3), 693-720.
6. Deng, B. [1991a] "The bifurcations of countable connections from a twisted heteroclinic loop," SIAM J. Math. Anal. 22(3), 653-679.
7. Deng, B. [1991b] "The existence of infinitely many traveling front and back waves in the FitzHugh-Nagumo equations," SIAM J. Math. Anal. 22(6), 1631-1650.
8. Deng, B. [1993] "Homoclinic twisting bifurcations and cusp horseshoe maps," J. Dyn. Diff. Eqs. 5(3), 417-467.
9. Doedel, E. & Kernévez, J. [1986] "AUTO: Software for continuation problems in ordinary differential equations with applications," Technical Report, Applied Mathematics, California Institute of Technology.
10. Doelman, A. [1993] "Traveling waves in the complex Ginzburg-Landau equation," J. Nonlin. Sci. 3, 225-266.
11. Doelman, A. & Holmes, P. [1996] "Homoclinic explosions and implosions," Phil. Trans. R. Soc. Lond. A354, 845-893.
12. Glendinning, P. & Sparrow, C. [1984] "Local and global behavior near homoclinic orbits," J. Stat. Phys. 35, 645.
13. Glendinning, P. & Sparrow, C. [1986] "T-points: A codimension-two heteroclinic bifurcation," J. Stat. Phys. 43(3/4), 479-488.
14. Homburg, A. J., Kokubu, H. & Krupta, M. [1994] "The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit," Ergod. Theor. Dyn. Sys. 14, 667-693.
15. Ilyashenko, Yu. S. & Yakovenko, S. Yu. [1991] "Finitely-smooth normal forms of local families of diffeomorphisms and vector fields," Russian Math. Surv. 46(1), 1-43.
16. Kokubu, H. [1988] "Homoclinic and heteroclinic bifurcations of vector fields," Japan J. Appl. Math. 5, 455-501.
17. Rinzel, J. & Terman, D. [1982] "Propagation phenomena in a bistable reaction-diffusion system," SIAM J. Appl. Math. 42, 1111-1137.
18. Sandstede, B. [1993] "Verzweigungstheorie homok-liner Verdopplungen," PhD thesis, Berlin University, Berlin.
19. Sparrow, C. [1982] The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors (Springer-Verlag, New York).
20. Šil'nikov, L. P. [1965] "A case for the existance of a denumerable set of periodic motions," Sov. Math. Dokl. 6, 163-166.
21. Šošitaǐšvili, A. N. [1973] "Bifurcations of topological type of a vector field near a singular point;" Functional Anal. Appl. 6, 97.
22. Wiggins, S. [1988] Global Bifurcations and Chaos, Applied Mathematical Science (Springer-Verlag).
23. Wiggins, S. [1991] Introduction to Applied Nonlinear Dynamical Systems and Chaos, Applied Mathematics 2, (Springer-Verlag).
24. Zimmermann, M. G., Firle, S. O., Natiello, M. A., Hildebrand, M., Eiswirth, M., Bär, M., Bangia, A. K. & Kevrekidis, I. G. [1997] "Pulse bifurcation and transition to spatiotemporal chaos in an excitable reaction-diffusion model," Physica D110, 92-104.