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

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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

Wenjun Zhang ^1, , Bernd Krauskopf ^1, and Vivien Kirk ^1,

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

Abstract
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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.

Keywords: boundary value problem formulation, Periodic orbit, Lin's method., heteroclinic cycle.

Mathematics Subject Classification: 34C37, 34C23, 37G1.

Citation: Wenjun Zhang, Bernd Krauskopf, Vivien Kirk. How to find a codimension-one heteroclinic cycle between two periodic orbits. Discrete and Continuous Dynamical Systems, 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), 1309-1344. 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), 244103. 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), 1727-1739. 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), 379-405. doi: 10.1093/imanum/10.3.379.
[5]	W.-J. Beyn, On well-posed problems for connecting orbits in dynamical systems, in "Chaotic Numerics" (Geelong, 1993), Cont. Math., 172, Amer. Math. Soc., Providence, RI, (1994), 131-168.
[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), 19-38. 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), 109-128. 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), 469-525. 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, Mathematical Physics, III, Springer-Verlag, Berlin, 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), 663-693. 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), 1261-1304. 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), 867-887. 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), 81-94. doi: 10.1137/S1064827598344868.
[14]	B. Deng and K. Sakamoto, Šil'nikov-Hopf bifurcations, J. Diff. Eqns., 119 (1995), 1-23. 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}., ().
[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), 141-164. Available at: http://www.matcont.ugent.be/. 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), 25-76.
[18]	L. Dieci and J. Rebaza, Point-to-periodic and periodic-to-periodic connections, BIT, 44 (2004), 41-62. 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), 617-618. doi: 10.1023/B:BITN.0000046846.33609.da.
[20]	E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations, in "Numerical Continuation Methods for Dynamical Systems'' (eds. B. Krauskopf, H. M. Osinga and J. Galán-Vioque), Underst. Complex Syst., Springer, Dordrecht, (2007), 1-49.
[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/}., ().
[22]	E. J. Doedel and M. J. Friedman, Numerical computation of heteroclinic orbits, J. Comput. Appl. Math., 26 (1989), 155-170. 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), 1889-1903. 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), 159-169. doi: 10.1142/S0218127409022804.
[25]	E. J. Doedel, B. Krauskopf and H. M. Osinga, Global bifurcations of the Lorenz manifold, Nonlinearity, 19 (2006), 2947-2972. 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), 1008-1041. doi: 10.1137/05062408X.
[27]	M. Falcke, Reading the patterns in living cells: The physics of $Ca^{2+}$ signaling, Adv. Phys., 53 (2004), 255-440. 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), 230-253. 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), 789-808. 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), 37-57. doi: 10.1007/BF01063734.
[31]	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'' 2^nd edition, Applied Mathematical Sciences, 42, Springer-Verlag, New York, 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), 639-683. doi: 10.1007/s00332-011-9096-z.
[33]	P. Hirschberg and E. Knobloch, Šil'nikov-Hopf bifurcation, Phys. D, 62 (1993), 202-216. doi: 10.1016/0167-2789(93)90282-6.
[34]	A. J. Homburg and B, Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, in "Handbook of Dynamical Systems III'' (eds. H. Broer, F. Takens and B. Hasselblatt), Elsevier, (2010), 379-524.
[35]	J. Knobloch, Lin's method for discrete dynamical systems, J. Difference Equations and Applications, 6 (2000), 577-623. doi: 10.1080/10236190008808247.
[36]	J. Knobloch, "Lin's Method for Discrete and Continuous Dynamical Systems and Applications,'' Habilitationsschrift, TU Ilmenau, 2004.
[37]	J. Knobloch and T. Rieß, Lin's method for heteroclinic chains involving periodic orbits, Nonlinearity, 23 (2010), 23-54. doi: 10.1088/0951-7715/23/1/002.
[38]	J. Knobloch, T, Rieß and M. Vielitz, Nonreversible homoclinic snaking, Dynamical Systems, 26 (2011), 335-365.
[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), 81-90. 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), 2149-2167. 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, Springer, Dordrecht, 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), 1655-1690. doi: 10.1088/0951-7715/21/8/001.
[43]	Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,'' 3^rd edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 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), 462-487. 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), 295-325. 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), 2977-2999. 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, Springer-Verlag, New York-Berlin, 1982. doi: 10.1007/978-1-4612-5703-5.
[48]	T. Pampel, Numerical approximation of connecting orbits with asymptotic rate, Numerische Mathematik, 90 (2001), 309-348. 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), 390-443. 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), 305-348. doi: 10.1016/j.jde.2010.04.007.
[51]	T. Rieß, "Using Lin's Method for an Almost Shilnikov Problem,'' Diploma Thesis, TU Ilmenau, 2003.
[52]	B. Sandstede, "Verzweigungstheorie Homokliner Verdopplungen,'' Ph.D thesis, University of Stuttgart, 1993.
[53]	S. M. Wieczorek and B. Krauskopf, Bifurcations of $n$-homoclinic orbits in optically injected lasers, Nonlinearity, 18 (2005), 1095-1120. doi: 10.1088/0951-7715/18/3/010.
[54]	A. C. Yew, Multipulses of nonlinearly-coupled Schrödinger equations, J. Diff. Eqns., 173 (2001), 92-137. 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).

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), 1309-1344. 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), 244103. 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), 1727-1739. 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), 379-405. doi: 10.1093/imanum/10.3.379.
[5]	W.-J. Beyn, On well-posed problems for connecting orbits in dynamical systems, in "Chaotic Numerics" (Geelong, 1993), Cont. Math., 172, Amer. Math. Soc., Providence, RI, (1994), 131-168.
[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), 19-38. 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), 109-128. 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), 469-525. 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, Mathematical Physics, III, Springer-Verlag, Berlin, 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), 663-693. 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), 1261-1304. 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), 867-887. 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), 81-94. doi: 10.1137/S1064827598344868.
[14]	B. Deng and K. Sakamoto, Šil'nikov-Hopf bifurcations, J. Diff. Eqns., 119 (1995), 1-23. 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}., ().
[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), 141-164. Available at: http://www.matcont.ugent.be/. 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), 25-76.
[18]	L. Dieci and J. Rebaza, Point-to-periodic and periodic-to-periodic connections, BIT, 44 (2004), 41-62. 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), 617-618. doi: 10.1023/B:BITN.0000046846.33609.da.
[20]	E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations, in "Numerical Continuation Methods for Dynamical Systems'' (eds. B. Krauskopf, H. M. Osinga and J. Galán-Vioque), Underst. Complex Syst., Springer, Dordrecht, (2007), 1-49.
[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/}., ().
[22]	E. J. Doedel and M. J. Friedman, Numerical computation of heteroclinic orbits, J. Comput. Appl. Math., 26 (1989), 155-170. 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), 1889-1903. 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), 159-169. doi: 10.1142/S0218127409022804.
[25]	E. J. Doedel, B. Krauskopf and H. M. Osinga, Global bifurcations of the Lorenz manifold, Nonlinearity, 19 (2006), 2947-2972. 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), 1008-1041. doi: 10.1137/05062408X.
[27]	M. Falcke, Reading the patterns in living cells: The physics of $Ca^{2+}$ signaling, Adv. Phys., 53 (2004), 255-440. 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), 230-253. 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), 789-808. 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), 37-57. doi: 10.1007/BF01063734.
[31]	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'' 2^nd edition, Applied Mathematical Sciences, 42, Springer-Verlag, New York, 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), 639-683. doi: 10.1007/s00332-011-9096-z.
[33]	P. Hirschberg and E. Knobloch, Šil'nikov-Hopf bifurcation, Phys. D, 62 (1993), 202-216. doi: 10.1016/0167-2789(93)90282-6.
[34]	A. J. Homburg and B, Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, in "Handbook of Dynamical Systems III'' (eds. H. Broer, F. Takens and B. Hasselblatt), Elsevier, (2010), 379-524.
[35]	J. Knobloch, Lin's method for discrete dynamical systems, J. Difference Equations and Applications, 6 (2000), 577-623. doi: 10.1080/10236190008808247.
[36]	J. Knobloch, "Lin's Method for Discrete and Continuous Dynamical Systems and Applications,'' Habilitationsschrift, TU Ilmenau, 2004.
[37]	J. Knobloch and T. Rieß, Lin's method for heteroclinic chains involving periodic orbits, Nonlinearity, 23 (2010), 23-54. doi: 10.1088/0951-7715/23/1/002.
[38]	J. Knobloch, T, Rieß and M. Vielitz, Nonreversible homoclinic snaking, Dynamical Systems, 26 (2011), 335-365.
[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), 81-90. 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), 2149-2167. 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, Springer, Dordrecht, 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), 1655-1690. doi: 10.1088/0951-7715/21/8/001.
[43]	Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,'' 3^rd edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 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), 462-487. 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), 295-325. 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), 2977-2999. 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, Springer-Verlag, New York-Berlin, 1982. doi: 10.1007/978-1-4612-5703-5.
[48]	T. Pampel, Numerical approximation of connecting orbits with asymptotic rate, Numerische Mathematik, 90 (2001), 309-348. 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), 390-443. 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), 305-348. doi: 10.1016/j.jde.2010.04.007.
[51]	T. Rieß, "Using Lin's Method for an Almost Shilnikov Problem,'' Diploma Thesis, TU Ilmenau, 2003.
[52]	B. Sandstede, "Verzweigungstheorie Homokliner Verdopplungen,'' Ph.D thesis, University of Stuttgart, 1993.
[53]	S. M. Wieczorek and B. Krauskopf, Bifurcations of $n$-homoclinic orbits in optically injected lasers, Nonlinearity, 18 (2005), 1095-1120. doi: 10.1088/0951-7715/18/3/010.
[54]	A. C. Yew, Multipulses of nonlinearly-coupled Schrödinger equations, J. Diff. Eqns., 173 (2001), 92-137. doi: 10.1006/jdeq.2000.3922.
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