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The singular limit of a Hopf bifurcation
How to find a codimension-one heteroclinic cycle between two periodic orbits
1. | Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand, New Zealand |
  We are concerned here with the case of heteroclinic connections between two saddle periodic orbits, called PtoP orbits for short. A homoclinic orbit from a periodic orbit to itself is an example of a PtoP connection, but is generically structurally stable in a phase space of any dimension. The issue that we address here is that, until now, no example of a concrete vector field with a non-structurally stable PtoP connection was known. We present an example of a PtoP heteroclinic cycle of codimension one between two different saddle periodic orbits in a four-dimensional vector field model of intracellular calcium dynamics. We first show that this model is a good candidate system for the existence of such a PtoP cycle and then demonstrate how a PtoP cycle can be detected and continued in system parameters using a numerical setup that is based on Lin's method.
References:
[1] |
P. Aguirre, E. J. Doedel, 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.
doi: 10.3934/dcds.2011.29.1309. |
[2] |
K. T. Alligood, E. Sander and J. A. Yorke, Crossing bifurcations and unstable dimension variability,, Phys. Rev. Lett., 96 (2006).
doi: 10.1103/PhysRevLett.96.244103. |
[3] |
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), 1727.
doi: 10.1016/S0006-3495(93)81191-3. |
[4] |
W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems,, IMA J. Numer. Anal., 10 (1990), 379.
doi: 10.1093/imanum/10.3.379. |
[5] |
W.-J. Beyn, On well-posed problems for connecting orbits in dynamical systems,, in, 172 (1994), 131.
|
[6] |
M. P. Boer, 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.
doi: 10.1007/s002850050161. |
[7] |
M. P. Boer, B. W. Kooi and S. A. L. M. Kooijman, Multiple attractors and boundary crises in a tri-trophic food chain,, Math. Biosciences, 169 (2001), 109.
doi: 10.1016/S0025-5564(00)00058-4. |
[8] |
C. Bonatti and L. Díaz, Robust heteroclinic cycles and $C^1$-generic dynamics,, J. Inst. Math. Jussieu, 7 (2008), 469.
doi: 10.1017/S1474748008000030. |
[9] |
C. Bonatti, L. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,'', Encyclopaedia of Mathematical Sciences, 102 (2005).
|
[10] |
A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman and J. Sneyd, When Shil'nikov Meets Hopf in excitable systems,, SIAM J. Appl. Dynam. Syst., 6 (2007), 663.
doi: 10.1137/070682654. |
[11] |
A. R. Champneys, E. Knobloch, V. Kirk, B. E. Oldeman and J. D. M. Rademacher, Unfolding a tangent equilibrium-to-periodic heteroclinic cycle,, SIAM J. App. Dyn. Sys., 8 (2009), 1261.
doi: 10.1137/080734923. |
[12] |
A. R. Champneys, Yu. A. Kuznetsov and B. Sandstede, A numerical toolbox for homoclinic bifurcation analysis,, Int. J. Bif. Chaos Appl. Sci. Engrg., 6 (1996), 867.
doi: 10.1142/S0218127496000485. |
[13] |
J. W. Demmel, L. Dieci and M. J. Friedman, Computing connecting orbits via an improved algorithm for continuing invariant subspaces,, SIAM J. Sci. Comput., 22 (2000), 81.
doi: 10.1137/S1064827598344868. |
[14] |
B. Deng and K. Sakamoto, Šil'nikov-Hopf bifurcations,, J. Diff. Eqns., 119 (1995), 1.
doi: 10.1006/jdeq.1995.1082. |
[15] |
F. Dercole, User guide to BPCONT, Dipartimento di Elettronica e Informazione, Politecnico di Milano, 2007., Available at: \url{http://ftp.elet.polimi.it/outgoing/Fabio.Dercole/bpcont/bpcont.tar.gz}., (). Google Scholar |
[16] |
A. Dhooge, W. Govaerts and Yu. A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs,, ACM Trans. Math. Software, 29 (2003), 141.
doi: 10.1145/779359.779362. |
[17] |
L. Díaz and J. Rocha, Partially hyperbolic and transitive dynamics generated by heteroclinic cycles,, Ergod. Th. Dynam. Sys., 21 (2001), 25.
|
[18] |
L. Dieci and J. Rebaza, Point-to-periodic and periodic-to-periodic connections,, BIT, 44 (2004), 41.
doi: 10.1023/B:BITN.0000025093.38710.f6. |
[19] |
L. Dieci and J. Rebaza, Erratum: "Point-to-periodic and periodic-to-periodic connections",, BIT, 44 (2004), 617.
doi: 10.1023/B:BITN.0000046846.33609.da. |
[20] |
E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations,, in, (2007), 1.
|
[21] |
E. J. Doedel, with major contributions from A. R. Champneys, T. F. Fairgrieve, 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: \url{http://cmvl.cs.concordia.ca/}., (). Google Scholar |
[22] |
E. J. Doedel and M. J. Friedman, Numerical computation of heteroclinic orbits,, J. Comput. Appl. Math., 26 (1989), 155.
doi: 10.1016/0377-0427(89)90153-2. |
[23] |
E. J. Doedel, B. W. Kooi, Yu. A. Kuznetsov and G. A. K. van Voorn, Continuation of connecting orbits in 3D-ODES: I. Point-to-cycle connections,, Int. J. Bifurc. Chaos Appl. Sci. Engrg., 18 (2008), 1889.
doi: 10.1142/S0218127408021439. |
[24] |
E. J. Doedel, B. W. Kooi, Yu. A. Kuznetsov and G. A. K. van Voorn, Continuation of connecting orbits in 3D-ODES: II. Cycle-to-cycle connections,, Int. J. Bifurc. Chaos Appl. Sci. Engrg., 19 (2009), 159.
doi: 10.1142/S0218127409022804. |
[25] |
E. J. Doedel, B. Krauskopf and H. M. Osinga, Global bifurcations of the Lorenz manifold,, Nonlinearity, 19 (2006), 2947.
doi: 10.1088/0951-7715/19/12/013. |
[26] |
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.
doi: 10.1137/05062408X. |
[27] |
M. Falcke, Reading the patterns in living cells: The physics of $Ca^{2+}$ signaling,, Adv. Phys., 53 (2004), 255.
doi: 10.1080/00018730410001703159. |
[28] |
E. Freire, A. J. Rodríguez-Luis, E. Gamero and E. Ponce, A case study for homoclinic chaos in an autonomous electronic circuit: A trip from Takens-Bogdanov to Hopf-Šil'nikov,, Physica D, 62 (1993), 230.
doi: 10.1016/0167-2789(93)90284-8. |
[29] |
M. Friedman and E. J. Doedel, Numerical computation and continuation of invariant manifolds connecting fixed points,, SIAM J. Numer. Anal., 28 (1991), 789.
doi: 10.1137/0728042. |
[30] |
M. Friedman and E. J. Doedel, Computational methods for global analysis of homoclinic and heteroclinic orbits: A case study,, J. Dyn. Diff. Eq., 5 (1993), 37.
doi: 10.1007/BF01063734. |
[31] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'', 2nd edition, 42 (1986).
|
[32] |
E. Harvey, V. Kirk, J. Sneyd and M. Wechselberger, Multiple time-scales, mixed mode oscillations and canards in intracellular calcium models,, J. Nonlinear Science, 21 (2011), 639.
doi: 10.1007/s00332-011-9096-z. |
[33] |
P. Hirschberg and E. Knobloch, Šil'nikov-Hopf bifurcation,, Phys. D, 62 (1993), 202.
doi: 10.1016/0167-2789(93)90282-6. |
[34] |
A. J. Homburg and B, Sandstede, Homoclinic and heteroclinic bifurcations in vector fields,, in, (2010), 379. Google Scholar |
[35] |
J. Knobloch, Lin's method for discrete dynamical systems,, J. Difference Equations and Applications, 6 (2000), 577.
doi: 10.1080/10236190008808247. |
[36] |
J. Knobloch, "Lin's Method for Discrete and Continuous Dynamical Systems and Applications,'', Habilitationsschrift, (2004). Google Scholar |
[37] |
J. Knobloch and T. Rieß, Lin's method for heteroclinic chains involving periodic orbits,, Nonlinearity, 23 (2010), 23.
doi: 10.1088/0951-7715/23/1/002. |
[38] |
J. Knobloch, T, Rieß and M. Vielitz, Nonreversible homoclinic snaking,, Dynamical Systems, 26 (2011), 335.
|
[39] |
E. J. Kostelich, I. Kan, C. Grebogi, E. Ott and J. A. Yorke, Unstable dimension variability: A source of nonhyperbolicity in chaotic systems,, Physica D, 109 (1997), 81.
doi: 10.1016/S0167-2789(97)00161-9. |
[40] |
B. Krauskopf and B. E. Oldeman, Bifurcations of global reinjection orbits near a saddle-node Hopf bifurcation,, Nonlinearity, 19 (2006), 2149.
doi: 10.1088/0951-7715/19/9/010. |
[41] |
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, (2007).
|
[42] |
B. Krauskopf and T. Rieß, A Lin's method approach to finding and continuing heteroclinic orbits connections involving periodic orbits,, Nonlinearity, 21 (2008), 1655.
doi: 10.1088/0951-7715/21/8/001. |
[43] |
Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,'', 3rd edition, 112 (2004).
|
[44] |
Yu. A. Kuznetsov, O. De Feo and S. Rinaldi, Belyakov homoclinic bifurcations in a tritrophic food-chain model,, SIAM J. Appl. Math., 62 (2001), 462.
doi: 10.1137/S0036139900378542. |
[45] |
X.-B. Lin, Using Mel'nikov's method to solve Šil'nikov's problems,, Proc. R. Soc. Edinb. Sect. A, 116 (1990), 295.
doi: 10.1017/S0308210500031528. |
[46] |
B. E. Oldeman, A. R. Champneys and B. Krauskopf, Homoclinic branch switching: A numerical implementation of Lin's method,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2977.
doi: 10.1142/S0218127403008326. |
[47] |
J. Palis, Jr., and W. de Melo, "Geometric Theory of Dynamical Systems. An Introduction,'', Translated from the Portuguese by A. K. Manning, (1982).
doi: 10.1007/978-1-4612-5703-5. |
[48] |
T. Pampel, Numerical approximation of connecting orbits with asymptotic rate,, Numerische Mathematik, 90 (2001), 309.
doi: 10.1007/s002110100302. |
[49] |
J. D. M. Rademacher, Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit,, J. Diff. Eqns., 218 (2005), 390.
doi: 10.1016/j.jde.2005.03.016. |
[50] |
J. D. M. Rademacher, Lyapunov-Schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies,, J. Diff. Eqns., 249 (2010), 305.
doi: 10.1016/j.jde.2010.04.007. |
[51] |
T. Rieß, "Using Lin's Method for an Almost Shilnikov Problem,'', Diploma Thesis, (2003). Google Scholar |
[52] |
B. Sandstede, "Verzweigungstheorie Homokliner Verdopplungen,'', Ph.D thesis, (1993). Google Scholar |
[53] |
S. M. Wieczorek and B. Krauskopf, Bifurcations of $n$-homoclinic orbits in optically injected lasers,, Nonlinearity, 18 (2005), 1095.
doi: 10.1088/0951-7715/18/3/010. |
[54] |
A. C. Yew, Multipulses of nonlinearly-coupled Schrödinger equations,, J. Diff. Eqns., 173 (2001), 92.
doi: 10.1006/jdeq.2000.3922. |
[55] |
W. Zhang, V. Kirk, J. Sneyd and M. Wechselberger, Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales,, J. Math. Neuroscience, 1 (2011). Google Scholar |
show all references
References:
[1] |
P. Aguirre, E. J. Doedel, 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.
doi: 10.3934/dcds.2011.29.1309. |
[2] |
K. T. Alligood, E. Sander and J. A. Yorke, Crossing bifurcations and unstable dimension variability,, Phys. Rev. Lett., 96 (2006).
doi: 10.1103/PhysRevLett.96.244103. |
[3] |
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), 1727.
doi: 10.1016/S0006-3495(93)81191-3. |
[4] |
W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems,, IMA J. Numer. Anal., 10 (1990), 379.
doi: 10.1093/imanum/10.3.379. |
[5] |
W.-J. Beyn, On well-posed problems for connecting orbits in dynamical systems,, in, 172 (1994), 131.
|
[6] |
M. P. Boer, 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.
doi: 10.1007/s002850050161. |
[7] |
M. P. Boer, B. W. Kooi and S. A. L. M. Kooijman, Multiple attractors and boundary crises in a tri-trophic food chain,, Math. Biosciences, 169 (2001), 109.
doi: 10.1016/S0025-5564(00)00058-4. |
[8] |
C. Bonatti and L. Díaz, Robust heteroclinic cycles and $C^1$-generic dynamics,, J. Inst. Math. Jussieu, 7 (2008), 469.
doi: 10.1017/S1474748008000030. |
[9] |
C. Bonatti, L. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,'', Encyclopaedia of Mathematical Sciences, 102 (2005).
|
[10] |
A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman and J. Sneyd, When Shil'nikov Meets Hopf in excitable systems,, SIAM J. Appl. Dynam. Syst., 6 (2007), 663.
doi: 10.1137/070682654. |
[11] |
A. R. Champneys, E. Knobloch, V. Kirk, B. E. Oldeman and J. D. M. Rademacher, Unfolding a tangent equilibrium-to-periodic heteroclinic cycle,, SIAM J. App. Dyn. Sys., 8 (2009), 1261.
doi: 10.1137/080734923. |
[12] |
A. R. Champneys, Yu. A. Kuznetsov and B. Sandstede, A numerical toolbox for homoclinic bifurcation analysis,, Int. J. Bif. Chaos Appl. Sci. Engrg., 6 (1996), 867.
doi: 10.1142/S0218127496000485. |
[13] |
J. W. Demmel, L. Dieci and M. J. Friedman, Computing connecting orbits via an improved algorithm for continuing invariant subspaces,, SIAM J. Sci. Comput., 22 (2000), 81.
doi: 10.1137/S1064827598344868. |
[14] |
B. Deng and K. Sakamoto, Šil'nikov-Hopf bifurcations,, J. Diff. Eqns., 119 (1995), 1.
doi: 10.1006/jdeq.1995.1082. |
[15] |
F. Dercole, User guide to BPCONT, Dipartimento di Elettronica e Informazione, Politecnico di Milano, 2007., Available at: \url{http://ftp.elet.polimi.it/outgoing/Fabio.Dercole/bpcont/bpcont.tar.gz}., (). Google Scholar |
[16] |
A. Dhooge, W. Govaerts and Yu. A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs,, ACM Trans. Math. Software, 29 (2003), 141.
doi: 10.1145/779359.779362. |
[17] |
L. Díaz and J. Rocha, Partially hyperbolic and transitive dynamics generated by heteroclinic cycles,, Ergod. Th. Dynam. Sys., 21 (2001), 25.
|
[18] |
L. Dieci and J. Rebaza, Point-to-periodic and periodic-to-periodic connections,, BIT, 44 (2004), 41.
doi: 10.1023/B:BITN.0000025093.38710.f6. |
[19] |
L. Dieci and J. Rebaza, Erratum: "Point-to-periodic and periodic-to-periodic connections",, BIT, 44 (2004), 617.
doi: 10.1023/B:BITN.0000046846.33609.da. |
[20] |
E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations,, in, (2007), 1.
|
[21] |
E. J. Doedel, with major contributions from A. R. Champneys, T. F. Fairgrieve, 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: \url{http://cmvl.cs.concordia.ca/}., (). Google Scholar |
[22] |
E. J. Doedel and M. J. Friedman, Numerical computation of heteroclinic orbits,, J. Comput. Appl. Math., 26 (1989), 155.
doi: 10.1016/0377-0427(89)90153-2. |
[23] |
E. J. Doedel, B. W. Kooi, Yu. A. Kuznetsov and G. A. K. van Voorn, Continuation of connecting orbits in 3D-ODES: I. Point-to-cycle connections,, Int. J. Bifurc. Chaos Appl. Sci. Engrg., 18 (2008), 1889.
doi: 10.1142/S0218127408021439. |
[24] |
E. J. Doedel, B. W. Kooi, Yu. A. Kuznetsov and G. A. K. van Voorn, Continuation of connecting orbits in 3D-ODES: II. Cycle-to-cycle connections,, Int. J. Bifurc. Chaos Appl. Sci. Engrg., 19 (2009), 159.
doi: 10.1142/S0218127409022804. |
[25] |
E. J. Doedel, B. Krauskopf and H. M. Osinga, Global bifurcations of the Lorenz manifold,, Nonlinearity, 19 (2006), 2947.
doi: 10.1088/0951-7715/19/12/013. |
[26] |
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.
doi: 10.1137/05062408X. |
[27] |
M. Falcke, Reading the patterns in living cells: The physics of $Ca^{2+}$ signaling,, Adv. Phys., 53 (2004), 255.
doi: 10.1080/00018730410001703159. |
[28] |
E. Freire, A. J. Rodríguez-Luis, E. Gamero and E. Ponce, A case study for homoclinic chaos in an autonomous electronic circuit: A trip from Takens-Bogdanov to Hopf-Šil'nikov,, Physica D, 62 (1993), 230.
doi: 10.1016/0167-2789(93)90284-8. |
[29] |
M. Friedman and E. J. Doedel, Numerical computation and continuation of invariant manifolds connecting fixed points,, SIAM J. Numer. Anal., 28 (1991), 789.
doi: 10.1137/0728042. |
[30] |
M. Friedman and E. J. Doedel, Computational methods for global analysis of homoclinic and heteroclinic orbits: A case study,, J. Dyn. Diff. Eq., 5 (1993), 37.
doi: 10.1007/BF01063734. |
[31] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'', 2nd edition, 42 (1986).
|
[32] |
E. Harvey, V. Kirk, J. Sneyd and M. Wechselberger, Multiple time-scales, mixed mode oscillations and canards in intracellular calcium models,, J. Nonlinear Science, 21 (2011), 639.
doi: 10.1007/s00332-011-9096-z. |
[33] |
P. Hirschberg and E. Knobloch, Šil'nikov-Hopf bifurcation,, Phys. D, 62 (1993), 202.
doi: 10.1016/0167-2789(93)90282-6. |
[34] |
A. J. Homburg and B, Sandstede, Homoclinic and heteroclinic bifurcations in vector fields,, in, (2010), 379. Google Scholar |
[35] |
J. Knobloch, Lin's method for discrete dynamical systems,, J. Difference Equations and Applications, 6 (2000), 577.
doi: 10.1080/10236190008808247. |
[36] |
J. Knobloch, "Lin's Method for Discrete and Continuous Dynamical Systems and Applications,'', Habilitationsschrift, (2004). Google Scholar |
[37] |
J. Knobloch and T. Rieß, Lin's method for heteroclinic chains involving periodic orbits,, Nonlinearity, 23 (2010), 23.
doi: 10.1088/0951-7715/23/1/002. |
[38] |
J. Knobloch, T, Rieß and M. Vielitz, Nonreversible homoclinic snaking,, Dynamical Systems, 26 (2011), 335.
|
[39] |
E. J. Kostelich, I. Kan, C. Grebogi, E. Ott and J. A. Yorke, Unstable dimension variability: A source of nonhyperbolicity in chaotic systems,, Physica D, 109 (1997), 81.
doi: 10.1016/S0167-2789(97)00161-9. |
[40] |
B. Krauskopf and B. E. Oldeman, Bifurcations of global reinjection orbits near a saddle-node Hopf bifurcation,, Nonlinearity, 19 (2006), 2149.
doi: 10.1088/0951-7715/19/9/010. |
[41] |
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, (2007).
|
[42] |
B. Krauskopf and T. Rieß, A Lin's method approach to finding and continuing heteroclinic orbits connections involving periodic orbits,, Nonlinearity, 21 (2008), 1655.
doi: 10.1088/0951-7715/21/8/001. |
[43] |
Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,'', 3rd edition, 112 (2004).
|
[44] |
Yu. A. Kuznetsov, O. De Feo and S. Rinaldi, Belyakov homoclinic bifurcations in a tritrophic food-chain model,, SIAM J. Appl. Math., 62 (2001), 462.
doi: 10.1137/S0036139900378542. |
[45] |
X.-B. Lin, Using Mel'nikov's method to solve Šil'nikov's problems,, Proc. R. Soc. Edinb. Sect. A, 116 (1990), 295.
doi: 10.1017/S0308210500031528. |
[46] |
B. E. Oldeman, A. R. Champneys and B. Krauskopf, Homoclinic branch switching: A numerical implementation of Lin's method,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2977.
doi: 10.1142/S0218127403008326. |
[47] |
J. Palis, Jr., and W. de Melo, "Geometric Theory of Dynamical Systems. An Introduction,'', Translated from the Portuguese by A. K. Manning, (1982).
doi: 10.1007/978-1-4612-5703-5. |
[48] |
T. Pampel, Numerical approximation of connecting orbits with asymptotic rate,, Numerische Mathematik, 90 (2001), 309.
doi: 10.1007/s002110100302. |
[49] |
J. D. M. Rademacher, Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit,, J. Diff. Eqns., 218 (2005), 390.
doi: 10.1016/j.jde.2005.03.016. |
[50] |
J. D. M. Rademacher, Lyapunov-Schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies,, J. Diff. Eqns., 249 (2010), 305.
doi: 10.1016/j.jde.2010.04.007. |
[51] |
T. Rieß, "Using Lin's Method for an Almost Shilnikov Problem,'', Diploma Thesis, (2003). Google Scholar |
[52] |
B. Sandstede, "Verzweigungstheorie Homokliner Verdopplungen,'', Ph.D thesis, (1993). Google Scholar |
[53] |
S. M. Wieczorek and B. Krauskopf, Bifurcations of $n$-homoclinic orbits in optically injected lasers,, Nonlinearity, 18 (2005), 1095.
doi: 10.1088/0951-7715/18/3/010. |
[54] |
A. C. Yew, Multipulses of nonlinearly-coupled Schrödinger equations,, J. Diff. Eqns., 173 (2001), 92.
doi: 10.1006/jdeq.2000.3922. |
[55] |
W. Zhang, V. Kirk, J. Sneyd and M. Wechselberger, Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales,, J. Math. Neuroscience, 1 (2011). Google Scholar |
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