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Volumn 32, Issue 8, 2012, Pages 2825-2851

How to find a codimension-one heteroclinic cycle between two periodic orbits

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EID: 84859772535     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2012.32.2825     Document Type: Article
Times cited : (32)

References (55)
  • 1
    • 79954562301 scopus 로고    scopus 로고
    • Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields
    • P. Aguirre, E. J. Doodol, B. Krauskopf and H. M. Osinga, Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields, Discr. Contin. Dynam. Syst., 29 (2011), 1309-1344.
    • (2011) Discr. Contin. Dynam. Syst , vol.29 , pp. 1309-1344
    • Aguirre, P.1    Doodol, E.J.2    Krauskopf, B.3    Osinga, H.M.4
  • 2
    • 33745292753 scopus 로고    scopus 로고
    • Crossing bifurcations and unstable dimension variability
    • K. T. Alligood, E. Sander and J. A. Yorko, Crossing bifurcations and unstable dimension variability, Phys. Rev. Lett., 96 (2006), 244103.
    • (2006) Phys. Rev. Lett , vol.96 , pp. 244103
    • Alligood, K.T.1    Sander, E.2    Yorko, J.A.3
  • 3
    • 0027421180 scopus 로고
    • A single-pool model for intracellular calcium oscillations and waves in the xenopus laevis oocyte
    • A. Atri, J. Amundsen, D. Clapham and J. Sneyd, A single-pool model for intracellular calcium oscillations and waves in the Xenopus laevis oocyte, Biophysical Journal, 65 (1993), 17271739.
    • (1993) Biophysical Journal , vol.65 , pp. 17271739
    • Atri, A.1    Amundsen, J.2    Clapham, D.3    Sneyd, J.4
  • 4
    • 0001340513 scopus 로고
    • The numerical computation of connecting orbits in dynamical systems
    • W.-J. Boyn, The numerical computation of connecting orbits in dynamical systems, IMA J. Numer. Anal., 10 (1990), 379-405.
    • (1990) IMA J. Numer. Anal , vol.10 , pp. 379-405
    • Boyn, W.-J.1
  • 5
    • 0012912040 scopus 로고
    • On well-posed problems for connecting orbits in dynamical systems
    • in "Chaotic Numerics" (Goolong, 1993), Amor. Math. Soc, Providence, RI
    • W.-J. Boyn, On well-posed problems for connecting orbits in dynamical systems, in "Chaotic Numerics" (Goolong, 1993), Cont. Math., 172, Amor. Math. Soc, Providence, RI, (1994), 131-168.
    • (1994) Cont. Math , vol.172 , pp. 131-168
    • Boyn, W.-J.1
  • 6
    • 0001842302 scopus 로고    scopus 로고
    • Homoclinic and heteroclinic orbits to a cycle in a tri-trophic food chain
    • M. P. Boor, B. W. Kooi and S. A. L. M. Kooijman, Homoclinic and heteroclinic orbits to a cycle in a tri-trophic food chain, J. Math. Biology, 39 (1999), 19-38.
    • (1999) J. Math. Biology , vol.39 , pp. 19-38
    • Boor, M.P.1    Kooi, B.W.2    Kooijman, S.A.L.M.3
  • 7
    • 0035109332 scopus 로고    scopus 로고
    • Multiple attractors and boundary crises in a tri-trophic food chain
    • M. P. Boor, B. W. Kooi and S. A. L. M. Kooijman, Multiple attractors and boundary crises in a tri-trophic food chain, Math. Bioscioncos, 169 (2001), 109-128.
    • (2001) Math. Bioscioncos , vol.169 , pp. 109-128
    • Boor, M.P.1    Kooi, B.W.2    Kooijman, S.A.L.M.3
  • 13
    • 0034456294 scopus 로고    scopus 로고
    • Computing connecting orbits via an improved algorithm for continuing invariant subspaces
    • J. W. Dommel, L. Dioci and M. J. Friedman, Computing connecting orbits via an improved algorithm for continuing invariant subspaces, SIAM J. Sci. Comput., 22 (2000), 81-94.
    • (2000) SIAM J. Sci. Comput , vol.22 , pp. 81-94
    • Dommel, J.W.1    Dioci, L.2    Friedman, M.J.3
  • 14
    • 58149362431 scopus 로고
    • Šil'nikov-hopf bifurcations
    • B. Dong and K. Sakamoto, Šil'nikov-Hopf bifurcations, J. Diff. Eqns., 119 (1995), 1-23.
    • (1995) J. Diff. Eqns , vol.119 , pp. 1-23
    • Dong, B.1    Sakamoto, K.2
  • 16
    • 8744230577 scopus 로고    scopus 로고
    • Matcont: A matlab package for numerical bifurcation analysis of ODEs
    • A. Dhooge, W. Govaorts and Yu. A. Kuznotsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Software, 29 (2003), 141-164. Available at: http://www.matcont.ugent.be/.
    • (2003) ACM Trans. Math. Software , vol.29 , pp. 141-164
    • Dhooge, A.1    Govaorts, W.2    Kuznotsov Yu., A.3
  • 17
    • 0035587772 scopus 로고    scopus 로고
    • Partially hyperbolic and transitive dynamics generated by heteroclinic cycles
    • L. Diaz and J. Rocha, Partially hyperbolic and transitive dynamics generated by heteroclinic cycles, Ergod. Th. Dynam. Sys., 21 (2001), 25-76.
    • (2001) Ergod. Th. Dynam. Sys , vol.21 , pp. 25-76
    • Diaz, L.1    Rocha, J.2
  • 20
    • 84892356236 scopus 로고    scopus 로고
    • Lecture notes on numerical analysis of nonlinear equations, in
    • (ods. B. Krauskopf, H. M. Osinga and J. Galán-Vioque), Underst. Complex Syst., Springer, Dordrecht
    • E. J. Doodol, Lecture notes on numerical analysis of nonlinear equations, in "Numerical Continuation Methods for Dynamical Systems" (ods. B. Krauskopf, H. M. Osinga and J. Galán-Vioque), Underst. Complex Syst., Springer, Dordrecht, (2007), 1-49.
    • (2007) Numerical Continuation Methods for Dynamical Systems , pp. 1-49
    • Doodol, E.J.1
  • 21
    • 84962866297 scopus 로고    scopus 로고
    • with major contributions from A. R. Champnoys, T. F. Fairgriovo, Yu. A. Kuznetsov, B. E. Oldeman, R. C. Paffenroth, B. Sandstede, X. J. Wang and C. Zhang
    • E. J. Doodol, with major contributions from A. R. Champnoys, T. F. Fairgriovo, Yu. A. Kuznetsov, B. E. Oldeman, R. C. Paffenroth, B. Sandstede, X. J. Wang and C. Zhang, AUTO-07P: Continuation and bifurcation software for ordinary differential equations. Available at: http://cmvl.cs.concordia.ca/.
    • AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations
    • Doodol, E.J.1
  • 22
    • 0001058067 scopus 로고
    • Numerical computation of heteroclinic orbits
    • E. J. Doodol and M. J. Friedman, Numerical computation of heteroclinic orbits, J. Comput. Appl. Math., 26 (1989), 155-170.
    • (1989) J. Comput. Appl. Math , vol.26 , pp. 155-170
    • Doodol, E.J.1    Friedman, M.J.2
  • 25
    • 33846058378 scopus 로고    scopus 로고
    • Global bifurcations of the lorenz manifold
    • E. J. Doodol, B. Krauskopf and H. M. Osinga, Global bifurcations of the Lorenz manifold, Nonlinoarity, 19 (2006), 2947-2972.
    • (2006) Nonlinoarity , vol.19 , pp. 2947-2972
    • Doodol, E.J.1    Krauskopf, B.2    Osinga, H.M.3
  • 26
    • 33644887198 scopus 로고    scopus 로고
    • Computing one-dimensional global manifolds of poincaré maps by continuation
    • DOI 10.1137/05062408X
    • J. P. England, B. Krauskopf and H. M. Osinga, Computing one-dimensional global manifolds of Poincaré maps by continuation, SIAM J. Appl. Dynam. Syst., 4 (2005), 1008-1041. (Pubitemid 43385157)
    • (2005) SIAM Journal on Applied Dynamical Systems , vol.4 , Issue.4 , pp. 1008-1041
    • England, J.P.1    Krauskopf, B.2    Osinga, H.M.3
  • 27
    • 3142773575 scopus 로고    scopus 로고
    • 2+ signaling
    • 2+ signaling, Adv. Phys., 53 (2004), 255-440.
    • (2004) Adv. Phys , vol.53 , pp. 255-440
    • Falcko, M.1
  • 29
    • 0026170304 scopus 로고
    • Numerical computation and continuation of invariant manifolds connecting fixed points
    • M. Friedman and E. J. Doodol, Numerical computation and continuation of invariant manifolds connecting fixed points, SIAM J. Numer. Anal., 28 (1991), 789-808. (Pubitemid 21650677)
    • (1991) SIAM Journal on Numerical Analysis , vol.28 , Issue.3 , pp. 789-808
    • Friedman, M.J.1    Doedel, E.J.2
  • 30
    • 2042442839 scopus 로고
    • Computational methods for global analysis of homoclinic and heteroclinic orbits: A case study
    • M. Friedman and E. J. Doodol, Computational methods for global analysis of homoclinic and heteroclinic orbits: A case study, J. Dyn. Diff. Eq., 5 (1993), 37-57.
    • (1993) J. Dyn. Diff. Eq , vol.5 , pp. 37-57
    • Friedman, M.1    Doodol, E.J.2
  • 32
    • 80053346248 scopus 로고    scopus 로고
    • Multiple time-scales, mixed mode oscillations and canards in intracellular calcium models
    • E. Harvey, V. Kirk, J. Snoyd and M. Wochsolborgor, Multiple time-scales, mixed mode oscillations and canards in intracellular calcium models, J. Nonlinear Science, 21 (2011), 639-683.
    • (2011) J. Nonlinear Science , vol.21 , pp. 639-683
    • Harvey, E.1    Kirk, V.2    Snoyd, J.3    Wochsolborgor, M.4
  • 33
    • 0000036850 scopus 로고
    • Šil'nikov-Hopf bifurcation
    • P. Hirschborg and E. Knobloch, Šil'nikov-Hopf bifurcation, Phys. D, 62 (1993), 202-216.
    • (1993) Phys. D , vol.62 , pp. 202-216
    • Hirschborg, P.1    Knobloch, E.2
  • 34
    • 77956987116 scopus 로고    scopus 로고
    • Homoclinic and heteroclinic bifurcations in vector fields
    • (ods. H. Broor, F. Takons and B. Hassolblatt), Elsevier
    • A. J. Homburg and B, Sandstodo, Homoclinic and heteroclinic bifurcations in vector fields, in "Handbook of Dynamical Systems III" (ods. H. Broor, F. Takons and B. Hassolblatt), Elsevier, (2010), 379-524.
    • (2010) Handbook of Dynamical Systems III , pp. 379-524
    • Homburg, A.J.1    Sandstodo, B.2
  • 35
    • 0242648399 scopus 로고    scopus 로고
    • Lin's method for discrete dynamical systems, J
    • J. Knobloch, Lin's method for discrete dynamical systems, J. Difference Equations and Applications, 6 (2000), 577-623.
    • (2000) Difference Equations and Applications , vol.6 , pp. 577-623
    • Knobloch, J.1
  • 36
    • 19944418492 scopus 로고    scopus 로고
    • Lin's method for discrete and continuous dynamical systems and applications
    • TU Ilmenau
    • J. Knobloch, "Lin's Method for Discrete and Continuous Dynamical Systems and Applications," Habilitationsschrift, TU Ilmenau, 2004.
    • (2004) Habilitationsschrift
    • Knobloch, J.1
  • 37
    • 76149123202 scopus 로고    scopus 로고
    • Lin's method for heteroclinic chains involving periodic orbits
    • J. Knobloch and T. Rieß, Lin's method for heteroclinic chains involving periodic orbits, Nonlinoarity, 23 (2010), 23-54.
    • (2010) Nonlinoarity , vol.23 , pp. 23-54
    • Knobloch, J.1    Rieß, T.2
  • 39
    • 0002623916 scopus 로고    scopus 로고
    • Unstable dimension variability: A source of nonhyperbolicity in chaotic systems
    • PII S0167278997001619
    • E. J. Kostolich, I. Kan, C. Grobogi, E. Ott and J. A. Yorko, Unstable dimension variability: A source of nonhyperbolicity in chaotic systems, Physica D, 109 (1997), 81-90. (Pubitemid 127701567)
    • (1997) Physica D: Nonlinear Phenomena , vol.109 , Issue.1-2 , pp. 81-90
    • Kostelich, E.J.1    Kan, I.2    Grebogi, C.3    Ott, E.4    Yorke, J.A.5
  • 40
    • 33747418994 scopus 로고    scopus 로고
    • Bifurcations of global reinjection orbits near a saddle-node Hopf bifurcation
    • DOI 10.1088/0951-7715/19/9/010, PII S0951771506240027, 010
    • B. Krauskopf and B. E. Oldoman, Bifurcations of global reinjection orbits near a saddle-node Hopf bifurcation, Nonlinoarity, 19 (2006), 2149-2167. (Pubitemid 44249948)
    • (2006) Nonlinearity , vol.19 , Issue.9 , pp. 2149-2167
    • Krauskopf, B.1    Oldeman, B.E.2
  • 41
    • 57549119107 scopus 로고    scopus 로고
    • Numerical continuation methods for dynamical systems. Path following and boundary value problems
    • Eds., Springer, Dordrecht
    • B. Krauskopf, H. M. Osinga and J. Galán-Vioque, eds., "Numerical Continuation Methods for Dynamical Systems. Path Following and Boundary Value Problems," Understanding Complex Systems, Springer, Dordrecht, 2007.
    • (2007) Understanding Complex Systems
    • Krauskopf, B.1    Osinga, H.M.2    Galán-Vioque, J.3
  • 42
    • 47849115552 scopus 로고    scopus 로고
    • A Lin's method approach to finding and continuing heteroclinic orbits connections involving periodic orbits
    • B. Krauskopf and T. Rieß, A. Lin's method approach to finding and continuing heteroclinic orbits connections involving periodic orbits, Nonlinoarity, 21 (2008), 1655-1690.
    • (2008) Nonlinoarity , vol.21 , pp. 1655-1690
    • Krauskopf, B.1    Rieß, T.2
  • 44
    • 0036252851 scopus 로고    scopus 로고
    • Belyakov homoclinic bifurcations in a tritrophic food chain model
    • PII S0036139900378542
    • Yu. A. Kuznotsov, O. Do Feo and S. Rinaldi, Belyakov homoclinic bifurcations in a tritrophic food-chain model, SIAM J. Appl. Math., 62 (2001), 462-487. (Pubitemid 33779885)
    • (2001) SIAM Journal on Applied Mathematics , vol.62 , Issue.2 , pp. 462-487
    • Kuznetsov, Y.A.1    De Feo, O.2    Rinaldi, S.3
  • 45
    • 84972438215 scopus 로고
    • Using mel'nikov's method to solve šil'nikov's problems
    • X.-B. Lin, Using Mel'nikov's method to solve Šil'nikov's problems, Proc. R. Soc. Edinb. Sect. A, 116 (1990), 295-325.
    • (1990) Proc. R. Soc. Edinb. Sect. A , vol.116 , pp. 295-325
    • Lin, X.-B.1
  • 47
    • 0003633463 scopus 로고
    • An Introduction, Translated from the Portuguese by A. K. Manning, Springer-Vorlag, New York-Berlin
    • J. Palis, Jr., and W. do Melo, "Geometric Theory of Dynamical Systems. An Introduction," Translated from the Portuguese by A. K. Manning, Springer-Vorlag, New York-Berlin, 1982.
    • (1982) Geometric Theory of Dynamical Systems
    • Palis Jr., J.1    Do Melo, W.2
  • 48
    • 0035545015 scopus 로고    scopus 로고
    • Numerical approximation of connecting orbits with asymptotic rate
    • T. Pampol, Numerical approximation of connecting orbits with asymptotic rate, Numerische Mathematik, 90 (2001), 309-348.
    • (2001) Numerische Mathematik , vol.90 , pp. 309-348
    • Pampol, T.1
  • 49
    • 27744536022 scopus 로고    scopus 로고
    • Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit
    • J. D. M. Radomachor, Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit, J. Diff. Eqns., 218 (2005), 390-443.
    • (2005) J. Diff. Eqns , vol.218 , pp. 390-443
    • Radomachor, J.D.M.1
  • 50
    • 77952642583 scopus 로고    scopus 로고
    • Lyapunov-Schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies
    • J. D. M. Radomachor, Lyapunov-Schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies, J. Diff. Eqns., 249 (2010), 305-348.
    • (2010) J. Diff. Eqns , vol.249 , pp. 305-348
    • Radomachor, J.D.M.1
  • 53
    • 24144495753 scopus 로고    scopus 로고
    • Bifurcations of n-homoclinic orbits in optically injected lasers
    • DOI 10.1088/0951-7715/18/3/010
    • S. M. Wioczorok and B. Krauskopf, Bifurcations of n-homoclinic orbits in optically injected lasers, Nonlinoarity, 18 (2005), 1095-1120. (Pubitemid 43033007)
    • (2005) Nonlinearity , vol.18 , Issue.3 , pp. 1095-1120
    • Wieczorek, S.1    Krauskopf, B.2
  • 54
    • 0010765318 scopus 로고    scopus 로고
    • Multipulses of nonlinearly-coupled Schrödinger equations
    • A. C. Yew, Multipulses of nonlinearly-coupled Schrödinger equations, J. Diff. Eqns., 173 (2001), 92-137.
    • (2001) J. Diff. Eqns , vol.173 , pp. 92-137
    • Yew, A.C.1
  • 55
    • 84859739166 scopus 로고    scopus 로고
    • Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales
    • W. Zhang, V. Kirk, J. Snoyd and M. Wochsolborgor, Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales, J. Math. Nouroscionco, 1 (2011).
    • (2011) J. Math. Nouroscionco , vol.1
    • Zhang, W.1    Kirk, V.2    Snoyd, J.3    Wochsolborgor, M.4


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