August  2012, 32(8): 2825-2851. doi: 10.3934/dcds.2012.32.2825

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

Received  August 2011 Revised  November 2011 Published  March 2012

Global bifurcations involving saddle periodic orbits have recently been recognized as being involved in various new types of organizing centers for complicated dynamics. The main emphasis has been on heteroclinic connections between saddle equilibria and saddle periodic orbits, called EtoP orbits for short, which can be found in vector fields in $\mathbb{R}^3$. Thanks to the development of dedicated numerical techniques, EtoP orbits have been found in a number of three-dimensional model vector fields arising in applications.
    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.
Citation: Wenjun Zhang, Bernd Krauskopf, Vivien Kirk. How to find a codimension-one heteroclinic cycle between two periodic orbits. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2825-2851. doi: 10.3934/dcds.2012.32.2825
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.  Google Scholar

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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.  Google Scholar

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W.-J. Beyn, On well-posed problems for connecting orbits in dynamical systems,, in, 172 (1994), 131.   Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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C. Bonatti, L. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,'', Encyclopaedia of Mathematical Sciences, 102 (2005).   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[27]

M. Falcke, Reading the patterns in living cells: The physics of $Ca^{2+}$ signaling,, Adv. Phys., 53 (2004), 255.  doi: 10.1080/00018730410001703159.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'', 2nd edition, 42 (1986).   Google Scholar

[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.  Google Scholar

[33]

P. Hirschberg and E. Knobloch, Šil'nikov-Hopf bifurcation,, Phys. D, 62 (1993), 202.  doi: 10.1016/0167-2789(93)90282-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

J. Knobloch, T, Rieß and M. Vielitz, Nonreversible homoclinic snaking,, Dynamical Systems, 26 (2011), 335.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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).   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[48]

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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.  Google Scholar

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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.  Google Scholar

[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

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

W.-J. Beyn, On well-posed problems for connecting orbits in dynamical systems,, in, 172 (1994), 131.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Bonatti, L. Díaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,'', Encyclopaedia of Mathematical Sciences, 102 (2005).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

B. Deng and K. Sakamoto, Šil'nikov-Hopf bifurcations,, J. Diff. Eqns., 119 (1995), 1.  doi: 10.1006/jdeq.1995.1082.  Google Scholar

[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.  Google Scholar

[17]

L. Díaz and J. Rocha, Partially hyperbolic and transitive dynamics generated by heteroclinic cycles,, Ergod. Th. Dynam. Sys., 21 (2001), 25.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations,, in, (2007), 1.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. Falcke, Reading the patterns in living cells: The physics of $Ca^{2+}$ signaling,, Adv. Phys., 53 (2004), 255.  doi: 10.1080/00018730410001703159.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'', 2nd edition, 42 (1986).   Google Scholar

[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.  Google Scholar

[33]

P. Hirschberg and E. Knobloch, Šil'nikov-Hopf bifurcation,, Phys. D, 62 (1993), 202.  doi: 10.1016/0167-2789(93)90282-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

J. Knobloch, T, Rieß and M. Vielitz, Nonreversible homoclinic snaking,, Dynamical Systems, 26 (2011), 335.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[43]

Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,'', 3rd edition, 112 (2004).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

T. Pampel, Numerical approximation of connecting orbits with asymptotic rate,, Numerische Mathematik, 90 (2001), 309.  doi: 10.1007/s002110100302.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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]

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