Continuation of connecting orbits in 3D-ODEs (II): (II) cycle-to-cycle connections

International Journal of Bifurcation and Chaos

Volumn 19, Issue 1, 2009, Pages 159-169

  Doedel, E J ^a   Kooi, B W ^b   Van Voorn, G A K ^b   Kuznetsov, Yu A ^c  

^a Gina Cody School of Engineering and Computer Science   (Canada)

^b VU UNIVERSITY AMSTERDAM   (Netherlands)

^c UTRECHT UNIVERSITY   (Netherlands)

Author keywords

Boundary value problems; Cycle to cycle connections; Global bifurcations; Projection boundary conditions

Indexed keywords

BOUNDARY CONDITIONS; BOUNDARY VALUE PROBLEMS; EIGENVALUES AND EIGENFUNCTIONS; ORDINARY DIFFERENTIAL EQUATIONS;

ADJOINTS; GLOBAL BIFURCATIONS; HOMOTOPY METHOD; MONODROMY; NUMERICAL CONTINUATION; PROJECTION BOUNDARY CONDITIONS; VARIATIONAL EQUATIONS;

NUMERICAL METHODS;

EID: 65249145224     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127409022804     Document Type: Article

References (31)

1. Antonovsky, M. Ya., Fleming, R. A., Kuznetsov, Yu. A. & Clark, W. C. [1990] "Forest-pest interaction dynamics: The simplest mathematical models," Theor. Popul. Biol. 37, 343-367.
2. Bazykin, A. D. [1998] Nonlinear Dynamics of Interacting Populations (World Scientific, Singapore).
3. Beyn, W.-J. [1990] "The numerical computation of connecting orbits in dynamical systems," IMA J. Numer. Anal. 9, 379-405.
4. Beyn, W.-J. [1994] "On well-posed problems for connecting orbits in dynamical systems," in Chaotic Numerics (Geelong, 1993 ), Contemp. Math., Vol. 172, (Amer. Math. Soc., Providence, RI), pp. 131-168.
5. Birkhoff, G. [1935] "Nouvelles recherches sur les systèmes dynamiques," Memoriae Pont. Acad. Sci. Novi. Lincaei, Ser. 3, 1, 85-216.
6. Boer, M. P., Kooi, B. W. & Kooijman, S. [1999] "Homoclinic and heteroclinic orbits in a tri-trophic food chain," J. Math. Biol. 39, 19-38.
7. Boer, M. P., Kooi, B. W. & Kooijman, S. [2001] "Multiple attractors and boundary crises in a tri-trophic food chain," Math. Biosci. 169, 109-128.
8. Champneys, A. R. & Kuznetsov, Yu. A. [1994] "Numerical detection and continuation of codimension-two homoclinic bifurcations," Int. J. Bifurcation and Chaos 4, 785-822.
9. Champneys, A. R., Kuznetsov, Yu. A. & Sandstede, B. [1996] "A numerical toolbox for homoclinic bifurcation analysis," Int. J. Bifurcation and Chaos 6, 867-887.
10. Demmel, J. W., Diece, L. & Friedman, M. J. [2000] "Computing connecting orbits via an improved algorithm for continuing invariant subspaces," SIAM J. Sci. Comput. 22, 81-94.
11. Dieci, L. & Rebaza, J. [2004a] "Point-to-periodic and periodic-to-periodic connections," BIT Numer. Math. 44, 41-62.
12. Dieci, L. & Rebaza, J. [2004b] "Erratum: 'Point-toperiodic and periodic-to-periodic connections'," BIT Numer. Math. 44, 617-618.
13. Doedel, E. J., Friedman, M. J. & Monteiro, A. C. [1994] "On locating connecting orbits," Appl. Math. Comput. 65, 231-239.
14. Doedel, E. J., Champneys, A. R., Fairgrieve, T. F., Kuznetsov, Yu. A., Sandstede, B. & Wang, X. [1997] "auto97: Continuation and bifurcation software for ordinary differential equations," Technical Report, Concordia University, Montreal, Quebec, Canada.
15. Doedel, E. J., Kooi, B. W., Kuznetsov, Yu. A. & Van Voorn, G. A. K. [2008] "Continuation of connecting orbits in 3D-ODEs: (I) Point-to-cycle connections," Int. J. Bifurcation and Chaos 18, 1889-1903.
16. Gavrilov, N. K. & Shilnikov, L. P. [1972] "On three-dimensional systems close to systems with a structurally unstable homoclinic curve: I," Math. USSR-Sb. 17, 467-485.
17. Gavrilov, N. K. & Shilnikov, L. P. [1973] "On three-dimensional systems close to systems with a structurally unstable homoclinic curve: II," Math. USSR-Sb. 19, 139-156.
18. Hastings, A. & Powell, T. [1991] "Chaos in a threespecies food chain," Ecol. 72, 896-903.
19. Klebanoff, A. & Hastings, A. [1994] "Chaos in three species food chains," J. Math. Biol. 32, 427-451.
20. Kuznetsov, Yu. A. & Rinaldi, S. [1996] "Remarks on food chain dynamics," Math. Biosci. 134, 1-33.
21. Kuznetsov, Yu. A., De Feo, O. & Rinaldi, S. [2001] "Belyakov homoclinic bifurcations in a tritrophic food chain model," SIAM J. Appl. Math. 62, 462-487.
22. Kuznetsov, Yu. A. [2004] Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, 3rd edition, Vol. 112 (Springer).
23. McCann, K. & Yodzis, P. [1995] "Bifurcation structure of a three-species food chain model," Theor. Popul. Biol. 48, 93-125.
24. Neimark, Ju. I. [1967] "Motions close to doublyasymptotic motion," Sov. Math. Dokl. 8, 228-231.
25. Palis, J. & Takens, F. [1993] Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors, Cambridge Studies in Advanced Mathematics, Vol. 35 (Cambridge University Press, Cambridge).
26. Pampel, T. [2001] "Numerical approximation of connecting orbits with asymptotic rate," Numerische Mathematik 90, 309-348.
27. Poincaré, H. [1879] Sur les Propriétés des Fonctions Définies par les Équations aux Différences Partielles. Thèse (Gauthier-Villars, Paris).
28. Rosenzweig, M. L. & MacArthur, R. H. [1963] "Graphical representation and stability conditions of predator-prey interactions," Am. Nat. 97, 209-223.
29. Shil'nikov, L. P. [1967] "On a Poincaré-Birkhoff problem," Math. USSR-Sb. 3, 353-371.
30. Smale, S. [1963] "Diffeomorphisms with many periodic points," Differential and Combinatorial Topology, ed. Carins, S. (Princeton University Press, Princeton, NJ), pp. 63-80.
31. Van Voorn, G. A. K., Hemerik, L., Boer, M. P. & Kooi, B. W. [2007] "Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect," Math. Biosci. 209, 451-469.