October  2013, 33(10): 4435-4471. doi: 10.3934/dcds.2013.33.4435

About the unfolding of a Hopf-zero singularity

1. 

Hasselt University, Campus Diepenbeek, Agoralaan gebouw D, B-3590 Diepenbeek

2. 

Departamento de Matemáticas, Universidad de Oviedo, Avda. Calvo Sotelo s/n, 33007 Oviedo, Spain

3. 

Department of Mathematics/JST-CREST, Kyoto University, Kyoto 606-8502, Japan

4. 

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via, 585, 08071 Barcelona, Spain

Received  September 2012 Revised  January 2013 Published  April 2013

We study arbitrary generic unfoldings of a Hopf-zero singularity of codimension two. They can be written in the following normal form: \begin{eqnarray*} \left\{ \begin{array}{l} x'=-y+\mu x-axz+A(x,y,z,\lambda,\mu) \\ y'=x+\mu y-ayz+B(x,y,z,\lambda,\mu) \\ z'=z^2+\lambda+b(x^2+y^2)+C(x,y,z,\lambda,\mu), \end{array} \right. \end{eqnarray*} with $a>0$, $b>0$ and where $A$, $B$, $C$ are $C^\infty$ or $C^\omega$ functions of order $O(\|(x,y,z,\lambda,\mu)\|^3)$.
    Despite that the existence of Shilnikov homoclinic orbits in unfoldings of Hopf-zero singularities has been discussed previously in the literature, no result valid for arbitrary generic unfoldings is available. In this paper we present new techniques to study global bifurcations from Hopf-zero singularities. They allow us to obtain a general criterion for the existence of Shilnikov homoclinic bifurcations and also provide a detailed description of the bifurcation set. Criteria for the existence of Bykov cycles are also provided. Main tools are a blow-up method, including a related invariant theory, and a novel approach to the splitting functions of the invariant manifolds. Theoretical results are applied to the Michelson system and also to the so called extended Michelson system. Paper includes thorough numerical explorations of dynamics for both systems.
Citation: Freddy Dumortier, Santiago Ibáñez, Hiroshi Kokubu, Carles Simó. About the unfolding of a Hopf-zero singularity. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4435-4471. doi: 10.3934/dcds.2013.33.4435
References:
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K. L. Adams, J. R. King and R. H. Tew, Beyond-all-orders effects in multiple-scales asymptotics: Travelling-wave solutions to the Kuramoto-Sivashinsky equation, J. Engrg. Math., 45 (2003), 197-226. doi: 10.1023/A:1022600411856.

[2]

I. Baldomá, O. Castejón and T. M. Seara, Exponentially small heteroclinic breakdown in the generic Hopf-Zero singularity,, Journal of Dynamics and Differential Equations (to appear)., (). 

[3]

P. Bonckaert and E. Fontich, Invariant manifolds of maps close to a product of rotations: Close to the rotation axis, J. Differential Equations, 191 (2003), 490-517.

[4]

P. Bonckaert and E. Fontich, Invariant manifolds of dynamical systems close to a rotation: Transverse to the rotation axis, J. Differential Equations, 214 (2005), 128-155. doi: 10.1016/j.jde.2005.02.012.

[5]

H. W. Broer and G. Vegter, Subordinate Sil'nikov bifurcations near some singularities of vector fields having low codimension, Ergodic Theory Dynam. Systems, 4 (1984), 509-525. doi: 10.1017/S0143385700002613.

[6]

H. W. Broer, C. Simó and R. Vitolo, Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing, Nonlinearity, 15 (2002), 1205-1267. doi: 10.1088/0951-7715/15/4/312.

[7]

H. W. Broer, C. Simó and R. Vitolo, Quasi-periodic Hénon-like attractors in the Lorenz-84 climate model with seasonal forcing, in "Proceedings Equadiff 2003" (eds. F. Dumortier et al.), World Sci. Publ., Hackensack, NJ, (2005), 601-606. doi: 10.1142/9789812702067_0100.

[8]

H. W. Broer, C. Simó and R. Vitolo, Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance bubble, Phys. D, 237 (2008), 1773-1999. doi: 10.1016/j.physd.2008.01.026.

[9]

H. W. Broer, C. Simó and R. Vitolo, The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: The Arnol'd resonance web, Bull. Belgian Math. Soc. Simon Stevin, 15 (2008), 769-787.

[10]

H. W. Broer, C. Simó and R. Vitolo, Chaos and quasi-periodicity in diffeomorphisms of the solid torus, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 871-905. doi: 10.3934/dcdsb.2010.14.871.

[11]

A. R. Champneys and V. Kirk, The entwined wiggling of homoclinic curves emerging from saddle-node/Hopf instabilities, Phys. D, 195 (2004), 77-105. doi: 10.1016/j.physd.2004.03.004.

[12]

F. Dumortier and S. Ibáñez, Nilpotent singularities in generic 4-parameter families of 3-dimensional vector fields, J. Differential Equations, 127 (1996), 590-647. doi: 10.1006/jdeq.1996.0085.

[13]

F. Dumortier and S. Ibáñez, Singularities of vector fields on $\mathbbR^3$, Nonlinearity, 11 (1998), 1037-1047. doi: 10.1088/0951-7715/11/4/015.

[14]

F. Dumortier, S. Ibáñez and H. Kokubu, New aspects in the unfolding of the nilpotent singularity of codimension three, Dyn. Syst., 16 (2001), 63-95. doi: 10.1080/02681110010017417.

[15]

F. Dumortier, S. Ibáñez and H. Kokubu, Cocoon bifurcations in three-dimensional reversible vector fields, Nonlinearity, 19 (2006), 305-328. doi: 10.1088/0951-7715/19/2/004.

[16]

E. Fontich and C. Simó, The splitting of sepratrices for analytic diffeomorphisms, Ergodic Theory Dynam. Systems, 10 (1990), 295-318. doi: 10.1017/S0143385700005563.

[17]

E. Fontich and C. Simó, Invariant manifolds for near identity differentiable maps and splitting of separatrices, Ergodic Theory Dynam. Systems, 10 (1990), 319-346. doi: 10.1017/S0143385700005575.

[18]

P. Gaspard, Local birth of homoclinic chaos, Phys. D, 62 (1993), 94-122. doi: 10.1016/0167-2789(93)90276-7.

[19]

N. K. Gavrilov, On some bifurcations of equilibria with a zero and a pair of purely imaginary roots, (1978), in "Methods of the Qualitative Theory of Differential Equations (Bifurcations of an equilibrium state with one zero root and a pair of purely imaginary roots and additional degeneration)" (ed. E. A. Leontovich-Andronova), Gor'kov. Gos. Univ., Gorki, (1987), 43-51.

[20]

P. Glendinning and C. Sparrow, Local and global behavior near homoclinic orbits, J. Statist. Phys., 35 (1984), 645-696. doi: 10.1007/BF01010828.

[21]

J. Guckenheimer, On a codimension two bifurcation, in "Dynamical Systems and Turbulence" (eds. D. A. Rand and L. A. Young), Lecture Notes in Math., 898, Springer, Berlin-New York, (1981).

[22]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields," $3^{rd}$ edition, Springer-Verlag, New York, 1990.

[23]

A. J. Homburg, Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbits to saddle-focus equilibria, Nonlinearity, 15 (2002), 1029-1050. doi: 10.1088/0951-7715/15/4/304.

[24]

S. Ibáñez and J. A. Rodríguez, Shil'nikov configurations in any generic unfolding of the nilpotent singularity of codimension three on $\mathbbR^3$, J. Differential Equations, 208 (2005), 147-175. doi: 10.1016/j.jde.2003.08.006.

[25]

N. Ishimura, Remarks on third-order ODEs relevant to the Kuramoto-Sivashinsky equation, J. Differential Equations, 178 (2002), 466-477. doi: 10.1006/jdeq.2001.4018.

[26]

J. Jones, W. C. Troy and A. D. McGillivary, Steady solutions of the Kuramoto-Sivashinsky equation for small wave speed, J. Differential Equations, 96 (1992), 28-55. doi: 10.1016/0022-0396(92)90143-B.

[27]

P. Kent and J. Elgin, A Shil'nikov-type analysis in a system with symmetry, Phys. Lett. A, 152 (1991), 28-32. doi: 10.1016/0375-9601(91)90623-G.

[28]

P. Kent and J. Elgin, Noose bifurcation of periodic orbits, Nonlinearity, 4 (1991), 1045-1061. doi: 10.1088/0951-7715/4/4/002.

[29]

P. Kent and J. Elgin, Travelling-waves of the Kuramoto-Sivashinsky equation: Period multiplyng bifurcations, Nonlinearity, 5 (1992), 899-919. doi: 10.1088/0951-7715/5/4/004.

[30]

Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55 (1976), 356-369. doi: 10.1143/PTP.55.356.

[31]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," $3^{rd}$ edition, Springer-Verlag, New York, 2004.

[32]

J. S. W. Lamb, M.-A. Teixeira and K. N. Webster, Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in $\mathbbR^3$, J. Differential Equations, 219 (2005), 78-115. doi: 10.1016/j.jde.2005.02.019.

[33]

Y.-T. Lau, The "cocoon" bifurcations in three-dimensional systems with two fixed points, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2 (1992), 543-558. doi: 10.1142/S0218127492000690.

[34]

F. Ledrappier, M. Shub, C. Simó and A. Wilkinson, Random versus deterministic exponents in a rich family of diffeomorphisms, J. Statist. Phys., 113 (2003), 85-149. doi: 10.1023/A:1025770720803.

[35]

C. K. McCord, Uniqueness of connecting orbits in the equation $Y^{(3)}=Y^2-1$, J. Math. Anal. Appl., 114 (1986), 584-592. doi: 10.1016/0022-247X(86)90110-1.

[36]

D. Michelson, Steady solutions of the Kuramoto-Sivashinsky equation, Phys. D, 19 (1986), 89-111. doi: 10.1016/0167-2789(86)90055-2.

[37]

J. Puig and C. Simó, Resonance tongues in the quasi-periodic Hill-Schrödinger equation with three frequencies, Regul. Chaotic Dyn., 16 (2011), 61-78. doi: 10.1134/S1560354710520047.

[38]

S. V. Raghavan, J. B. McLeod and W. C. Troy, A singular perturbation problem arising from the Kuramoto-Sivashinsky equation, Differential Integral Equations, 10 (1997), 1-36.

[39]

C. Simó, On the Hénon-Pomeau attractor, J. Statist. Phys., 21 (1979), 465-494. doi: 10.1007/BF01009612.

[40]

C. Simó, Global dynamics and fast indicators, in "Global Analysis of Dynamical Systems" (eds. H. W. Broer, B. Krauskopf and G. Vegter), Inst. Phys., Bristol, (2001), 373-389.

[41]

C. Simó, Some properties of the global behaviour of conservative low-dimensional systems, in "Foundations of Computational Mathematics: Hong Kong 2008" (eds. F. Cucker et al.), London Math. Soc. Lecture Note Ser., 363, Cambridge Univ. Press, (2009), 163-189.

[42]

C. Simó and A. Vieiro, Resonant zones, inner and outer splittings in generic and low order resonances of area preserving maps, Nonlinearity, 22 (2009), 1191-1245. doi: 10.1088/0951-7715/22/5/012.

[43]

C. Simó and A. Vieiro, Planar radial weakly dissipative diffeomorphisms, Chaos, 20 (2010), 043138. doi: 10.1063/1.3515168.

[44]

C. Simó and A. Vieiro, Dynamics in chaotic zones of area preserving maps: Close to separatrix and global instability zones, Phys. D, 240 (2011), 732-753. doi: 10.1016/j.physd.2010.12.005.

[45]

F. Takens, Singularities of vector fields, Inst.Hautes Etudes Sci. Publ. Math., 43 (1974), 47-100.

[46]

W. C. Troy, The existence of steady solutions of the Kuramoto-Sivashinsky equation, J. Differential Equations, 82 (1989), 269-313. doi: 10.1016/0022-0396(89)90134-4.

[47]

R. Vitolo, H. W. Broer and C. Simó, Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms, Nonlinearity, 23 (2010), 1919-1947. doi: 10.1088/0951-7715/23/8/007.

[48]

R. Vitolo, H. W. Broer and C. Simó, Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems, Regul. Chaotic Dyn., 16 (2011), 154-184. doi: 10.1134/S1560354711010060.

[49]

K. N. Webster and J. Elgin, Asymptotic analysis of the Michelson system, Nonlinearity, 16 (2003), 2149-2162. doi: 10.1088/0951-7715/16/6/316.

[50]

D. Wilczak, Symmetric heteroclinic connections in the Michelson system: A computer assisted proof (electronic), SIAM J. Appl. Dyn. Syst., 4 (2005), 489-514. doi: 10.1137/040611112.

[51]

T.-S. Yang, On traveling wave solutions of the Kuramoto-Sivashinsky equation, Phys. D, 110 (1997), 25-42. doi: 10.1016/S0167-2789(97)00121-8.

show all references

References:
[1]

K. L. Adams, J. R. King and R. H. Tew, Beyond-all-orders effects in multiple-scales asymptotics: Travelling-wave solutions to the Kuramoto-Sivashinsky equation, J. Engrg. Math., 45 (2003), 197-226. doi: 10.1023/A:1022600411856.

[2]

I. Baldomá, O. Castejón and T. M. Seara, Exponentially small heteroclinic breakdown in the generic Hopf-Zero singularity,, Journal of Dynamics and Differential Equations (to appear)., (). 

[3]

P. Bonckaert and E. Fontich, Invariant manifolds of maps close to a product of rotations: Close to the rotation axis, J. Differential Equations, 191 (2003), 490-517.

[4]

P. Bonckaert and E. Fontich, Invariant manifolds of dynamical systems close to a rotation: Transverse to the rotation axis, J. Differential Equations, 214 (2005), 128-155. doi: 10.1016/j.jde.2005.02.012.

[5]

H. W. Broer and G. Vegter, Subordinate Sil'nikov bifurcations near some singularities of vector fields having low codimension, Ergodic Theory Dynam. Systems, 4 (1984), 509-525. doi: 10.1017/S0143385700002613.

[6]

H. W. Broer, C. Simó and R. Vitolo, Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing, Nonlinearity, 15 (2002), 1205-1267. doi: 10.1088/0951-7715/15/4/312.

[7]

H. W. Broer, C. Simó and R. Vitolo, Quasi-periodic Hénon-like attractors in the Lorenz-84 climate model with seasonal forcing, in "Proceedings Equadiff 2003" (eds. F. Dumortier et al.), World Sci. Publ., Hackensack, NJ, (2005), 601-606. doi: 10.1142/9789812702067_0100.

[8]

H. W. Broer, C. Simó and R. Vitolo, Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance bubble, Phys. D, 237 (2008), 1773-1999. doi: 10.1016/j.physd.2008.01.026.

[9]

H. W. Broer, C. Simó and R. Vitolo, The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: The Arnol'd resonance web, Bull. Belgian Math. Soc. Simon Stevin, 15 (2008), 769-787.

[10]

H. W. Broer, C. Simó and R. Vitolo, Chaos and quasi-periodicity in diffeomorphisms of the solid torus, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 871-905. doi: 10.3934/dcdsb.2010.14.871.

[11]

A. R. Champneys and V. Kirk, The entwined wiggling of homoclinic curves emerging from saddle-node/Hopf instabilities, Phys. D, 195 (2004), 77-105. doi: 10.1016/j.physd.2004.03.004.

[12]

F. Dumortier and S. Ibáñez, Nilpotent singularities in generic 4-parameter families of 3-dimensional vector fields, J. Differential Equations, 127 (1996), 590-647. doi: 10.1006/jdeq.1996.0085.

[13]

F. Dumortier and S. Ibáñez, Singularities of vector fields on $\mathbbR^3$, Nonlinearity, 11 (1998), 1037-1047. doi: 10.1088/0951-7715/11/4/015.

[14]

F. Dumortier, S. Ibáñez and H. Kokubu, New aspects in the unfolding of the nilpotent singularity of codimension three, Dyn. Syst., 16 (2001), 63-95. doi: 10.1080/02681110010017417.

[15]

F. Dumortier, S. Ibáñez and H. Kokubu, Cocoon bifurcations in three-dimensional reversible vector fields, Nonlinearity, 19 (2006), 305-328. doi: 10.1088/0951-7715/19/2/004.

[16]

E. Fontich and C. Simó, The splitting of sepratrices for analytic diffeomorphisms, Ergodic Theory Dynam. Systems, 10 (1990), 295-318. doi: 10.1017/S0143385700005563.

[17]

E. Fontich and C. Simó, Invariant manifolds for near identity differentiable maps and splitting of separatrices, Ergodic Theory Dynam. Systems, 10 (1990), 319-346. doi: 10.1017/S0143385700005575.

[18]

P. Gaspard, Local birth of homoclinic chaos, Phys. D, 62 (1993), 94-122. doi: 10.1016/0167-2789(93)90276-7.

[19]

N. K. Gavrilov, On some bifurcations of equilibria with a zero and a pair of purely imaginary roots, (1978), in "Methods of the Qualitative Theory of Differential Equations (Bifurcations of an equilibrium state with one zero root and a pair of purely imaginary roots and additional degeneration)" (ed. E. A. Leontovich-Andronova), Gor'kov. Gos. Univ., Gorki, (1987), 43-51.

[20]

P. Glendinning and C. Sparrow, Local and global behavior near homoclinic orbits, J. Statist. Phys., 35 (1984), 645-696. doi: 10.1007/BF01010828.

[21]

J. Guckenheimer, On a codimension two bifurcation, in "Dynamical Systems and Turbulence" (eds. D. A. Rand and L. A. Young), Lecture Notes in Math., 898, Springer, Berlin-New York, (1981).

[22]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields," $3^{rd}$ edition, Springer-Verlag, New York, 1990.

[23]

A. J. Homburg, Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbits to saddle-focus equilibria, Nonlinearity, 15 (2002), 1029-1050. doi: 10.1088/0951-7715/15/4/304.

[24]

S. Ibáñez and J. A. Rodríguez, Shil'nikov configurations in any generic unfolding of the nilpotent singularity of codimension three on $\mathbbR^3$, J. Differential Equations, 208 (2005), 147-175. doi: 10.1016/j.jde.2003.08.006.

[25]

N. Ishimura, Remarks on third-order ODEs relevant to the Kuramoto-Sivashinsky equation, J. Differential Equations, 178 (2002), 466-477. doi: 10.1006/jdeq.2001.4018.

[26]

J. Jones, W. C. Troy and A. D. McGillivary, Steady solutions of the Kuramoto-Sivashinsky equation for small wave speed, J. Differential Equations, 96 (1992), 28-55. doi: 10.1016/0022-0396(92)90143-B.

[27]

P. Kent and J. Elgin, A Shil'nikov-type analysis in a system with symmetry, Phys. Lett. A, 152 (1991), 28-32. doi: 10.1016/0375-9601(91)90623-G.

[28]

P. Kent and J. Elgin, Noose bifurcation of periodic orbits, Nonlinearity, 4 (1991), 1045-1061. doi: 10.1088/0951-7715/4/4/002.

[29]

P. Kent and J. Elgin, Travelling-waves of the Kuramoto-Sivashinsky equation: Period multiplyng bifurcations, Nonlinearity, 5 (1992), 899-919. doi: 10.1088/0951-7715/5/4/004.

[30]

Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55 (1976), 356-369. doi: 10.1143/PTP.55.356.

[31]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," $3^{rd}$ edition, Springer-Verlag, New York, 2004.

[32]

J. S. W. Lamb, M.-A. Teixeira and K. N. Webster, Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in $\mathbbR^3$, J. Differential Equations, 219 (2005), 78-115. doi: 10.1016/j.jde.2005.02.019.

[33]

Y.-T. Lau, The "cocoon" bifurcations in three-dimensional systems with two fixed points, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2 (1992), 543-558. doi: 10.1142/S0218127492000690.

[34]

F. Ledrappier, M. Shub, C. Simó and A. Wilkinson, Random versus deterministic exponents in a rich family of diffeomorphisms, J. Statist. Phys., 113 (2003), 85-149. doi: 10.1023/A:1025770720803.

[35]

C. K. McCord, Uniqueness of connecting orbits in the equation $Y^{(3)}=Y^2-1$, J. Math. Anal. Appl., 114 (1986), 584-592. doi: 10.1016/0022-247X(86)90110-1.

[36]

D. Michelson, Steady solutions of the Kuramoto-Sivashinsky equation, Phys. D, 19 (1986), 89-111. doi: 10.1016/0167-2789(86)90055-2.

[37]

J. Puig and C. Simó, Resonance tongues in the quasi-periodic Hill-Schrödinger equation with three frequencies, Regul. Chaotic Dyn., 16 (2011), 61-78. doi: 10.1134/S1560354710520047.

[38]

S. V. Raghavan, J. B. McLeod and W. C. Troy, A singular perturbation problem arising from the Kuramoto-Sivashinsky equation, Differential Integral Equations, 10 (1997), 1-36.

[39]

C. Simó, On the Hénon-Pomeau attractor, J. Statist. Phys., 21 (1979), 465-494. doi: 10.1007/BF01009612.

[40]

C. Simó, Global dynamics and fast indicators, in "Global Analysis of Dynamical Systems" (eds. H. W. Broer, B. Krauskopf and G. Vegter), Inst. Phys., Bristol, (2001), 373-389.

[41]

C. Simó, Some properties of the global behaviour of conservative low-dimensional systems, in "Foundations of Computational Mathematics: Hong Kong 2008" (eds. F. Cucker et al.), London Math. Soc. Lecture Note Ser., 363, Cambridge Univ. Press, (2009), 163-189.

[42]

C. Simó and A. Vieiro, Resonant zones, inner and outer splittings in generic and low order resonances of area preserving maps, Nonlinearity, 22 (2009), 1191-1245. doi: 10.1088/0951-7715/22/5/012.

[43]

C. Simó and A. Vieiro, Planar radial weakly dissipative diffeomorphisms, Chaos, 20 (2010), 043138. doi: 10.1063/1.3515168.

[44]

C. Simó and A. Vieiro, Dynamics in chaotic zones of area preserving maps: Close to separatrix and global instability zones, Phys. D, 240 (2011), 732-753. doi: 10.1016/j.physd.2010.12.005.

[45]

F. Takens, Singularities of vector fields, Inst.Hautes Etudes Sci. Publ. Math., 43 (1974), 47-100.

[46]

W. C. Troy, The existence of steady solutions of the Kuramoto-Sivashinsky equation, J. Differential Equations, 82 (1989), 269-313. doi: 10.1016/0022-0396(89)90134-4.

[47]

R. Vitolo, H. W. Broer and C. Simó, Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms, Nonlinearity, 23 (2010), 1919-1947. doi: 10.1088/0951-7715/23/8/007.

[48]

R. Vitolo, H. W. Broer and C. Simó, Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems, Regul. Chaotic Dyn., 16 (2011), 154-184. doi: 10.1134/S1560354711010060.

[49]

K. N. Webster and J. Elgin, Asymptotic analysis of the Michelson system, Nonlinearity, 16 (2003), 2149-2162. doi: 10.1088/0951-7715/16/6/316.

[50]

D. Wilczak, Symmetric heteroclinic connections in the Michelson system: A computer assisted proof (electronic), SIAM J. Appl. Dyn. Syst., 4 (2005), 489-514. doi: 10.1137/040611112.

[51]

T.-S. Yang, On traveling wave solutions of the Kuramoto-Sivashinsky equation, Phys. D, 110 (1997), 25-42. doi: 10.1016/S0167-2789(97)00121-8.

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