# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems, 2013, 33 (10) : 4435-4471. doi: 10.3934/dcds.2013.33.4435
##### 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.  Google Scholar [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)., ().   Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] P. Gaspard, Local birth of homoclinic chaos, Phys. D, 62 (1993), 94-122. doi: 10.1016/0167-2789(93)90276-7.  Google Scholar [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.  Google Scholar [20] P. Glendinning and C. Sparrow, Local and global behavior near homoclinic orbits, J. Statist. Phys., 35 (1984), 645-696. doi: 10.1007/BF01010828.  Google Scholar [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).  Google Scholar [22] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields," $3^{rd}$ edition, Springer-Verlag, New York, 1990.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] P. Kent and J. Elgin, Noose bifurcation of periodic orbits, Nonlinearity, 4 (1991), 1045-1061. doi: 10.1088/0951-7715/4/4/002.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," $3^{rd}$ edition, Springer-Verlag, New York, 2004.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [36] D. Michelson, Steady solutions of the Kuramoto-Sivashinsky equation, Phys. D, 19 (1986), 89-111. doi: 10.1016/0167-2789(86)90055-2.  Google Scholar [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.  Google Scholar [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.  Google Scholar [39] C. Simó, On the Hénon-Pomeau attractor, J. Statist. Phys., 21 (1979), 465-494. doi: 10.1007/BF01009612.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [43] C. Simó and A. Vieiro, Planar radial weakly dissipative diffeomorphisms, Chaos, 20 (2010), 043138. doi: 10.1063/1.3515168.  Google Scholar [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.  Google Scholar [45] F. Takens, Singularities of vector fields, Inst.Hautes Etudes Sci. Publ. Math., 43 (1974), 47-100.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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)., ().   Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] P. Gaspard, Local birth of homoclinic chaos, Phys. D, 62 (1993), 94-122. doi: 10.1016/0167-2789(93)90276-7.  Google Scholar [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.  Google Scholar [20] P. Glendinning and C. Sparrow, Local and global behavior near homoclinic orbits, J. Statist. Phys., 35 (1984), 645-696. doi: 10.1007/BF01010828.  Google Scholar [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).  Google Scholar [22] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields," $3^{rd}$ edition, Springer-Verlag, New York, 1990.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] P. Kent and J. Elgin, Noose bifurcation of periodic orbits, Nonlinearity, 4 (1991), 1045-1061. doi: 10.1088/0951-7715/4/4/002.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," $3^{rd}$ edition, Springer-Verlag, New York, 2004.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [36] D. Michelson, Steady solutions of the Kuramoto-Sivashinsky equation, Phys. D, 19 (1986), 89-111. doi: 10.1016/0167-2789(86)90055-2.  Google Scholar [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.  Google Scholar [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.  Google Scholar [39] C. Simó, On the Hénon-Pomeau attractor, J. Statist. Phys., 21 (1979), 465-494. doi: 10.1007/BF01009612.  Google Scholar [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.  Google Scholar [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.  Google Scholar [42] C. Simó and A. 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