2011, 29(4): 1309-1344. doi: 10.3934/dcds.2011.29.1309

Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields

1. 

Bristol Centre for Applied Nonlinear Mathematics, Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, United Kingdom

2. 

Department of Computer Science, Concordia University, 1455 Boulevard de Maisonneuve O., Montréal, Québec H3G 1M8, Canada

3. 

Bristol Center for Applied Nonlinear Mathematics, Department of Engineering Mathematics, University of Bristol, Queen's Building, Bristol BS8 1TR, United Kingdom

Received  December 2009 Revised  July 2010 Published  December 2010

We consider a homoclinic bifurcation of a vector field in $\R^3$, where a one-dimensional unstable manifold of an equilibrium is contained in the two-dimensional stable manifold of this same equilibrium. How such one-dimensional connecting orbits arise is well understood, and software packages exist to detect and follow them in parameters.
   In this paper we address an issue that it is far less well understood: how does the associated two-dimensional stable manifold change geometrically during the given homoclinic bifurcation? This question can be answered with the help of advanced numerical techniques. More specifically, we compute two-dimensional manifolds, and their one-dimensional intersection curves with a suitable cross-section, via the numerical continuation of orbit segments as solutions of a boundary value problem. In this way, we are able to explain how homoclinic bifurcations may lead to quite dramatic changes of the overall dynamics. This is demonstrated with two examples. We first consider a Shilnikov bifurcation in a semiconductor laser model, and show how the associated change of the two-dimensional stable manifold results in the creation of a new basin of attraction. We then investigate how the basins of the two symmetrically related attracting equilibria change to give rise to preturbulence in the first homoclinic explosion of the Lorenz system.
Citation: Pablo Aguirre, Eusebius J. Doedel, Bernd Krauskopf, Hinke M. Osinga. Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1309-1344. doi: 10.3934/dcds.2011.29.1309
References:
[1]

R. H. Abraham and C. D. Shaw, "Dynamics -- The Geometry Of Behavior, Part Three: Global Behavior,", Aerial Press, (1985).

[2]

U. M. Ascher, J. Christiansen and R. D. Russell, Colsys -- A collocation code for boundary-value problems,, Lecture Notes in Computer Science, 76 (1979), 164.

[3]

U. M. Ascher and R. J. Spiteri, Collocation software for boundary value differential-algebraic equations,, SIAM J. Sci. Comput., 15 (1994), 938. doi: 10.1137/0915056.

[4]

M. R. Bassett and J. L. Hudson, Shil'nikov chaos during copper electrodissolution,, J. Phys. Chem., 92 (1988), 6963. doi: 10.1021/j100335a025.

[5]

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

[6]

C. J. Budd and J. P. Wilson, Bogdanov-Takens bifurcation points and Shilnikov homoclinicity in a simple power system model of voltage collapse,, IEEE Trans. Circuits Systems I, 43 (2002), 575. doi: 10.1109/TCSI.2002.1001947.

[7]

A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman and J. Sneyd, When Shil'nikov meets Hopf in excitable systems,, SIAM J. Appl. Dyn. Syst., 6 (2007), 663. doi: 10.1137/070682654.

[8]

A. R. Champneys, Y. Kuznetsov and B. Sandstede, A numerical toolbox for homoclinic bifurcation analysis,, Int. J. Bifurc. Chaos, 6 (1996), 867. doi: 10.1142/S0218127496000485.

[9]

B. Deng and G. Hines, Food chain chaos due to Shilnikov's orbit,, Chaos, 12 (2002), 533. doi: 10.1063/1.1482255.

[10]

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.

[11]

E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems,, Congr. Numer., 30 (1981), 265.

[12]

E. J. Doedel and B. E. Oldeman, with major contributions from A. R. Champneys, F. Dercole, T. F. Fairgrieve, Yu. A. Kuznetsov, R. C. Paffenroth, B. Sandstede, X. J. Wang, and C. H. Zhang, AUTO-07p Version 0.7: Continuation and bifurcation software for ordinary differential equations,, Department of Computer Science, (2010).

[13]

E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations,, in, (2007), 1. doi: 10.1007/978-1-4020-6356-5_1.

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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.

[15]

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.

[16]

E. J. Doedel, R. C. Paffenroth, H. B. Keller, D. Dichmann, J. Galán-Vioque and A. Vanderbauwhede, Computation of periodic solutions of conservative systems with application to the 3-body problem,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1353. doi: 10.1142/S0218127403007291.

[17]

E. J. Doedel, V. Romanov, R. C. Paffenroth, H. B. Keller, D. Dichmann, J. Galán-Vioque and A. Vanderbauwhede, Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 2625. doi: 10.1142/S0218127407018671.

[18]

J. P. England, B. Krauskopf and H. M. Osinga, Computing one-dimensional global manifolds of Poincaré maps by continuation,, SIAM J. Appl. Dyn. Sys., 4 (2005), 1008. doi: 10.1137/05062408X.

[19]

J. P. England, B. Krauskopf and H. M. Osinga, Computing two-dimensional global invariant manifolds in slow-fast systems,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 805. doi: 10.1142/S0218127407017562.

[20]

J. A. Feroe, Homoclinic orbits in a parametrized saddle-focus system,, Physica D, 62 (1993), 254. doi: 10.1016/0167-2789(93)90285-9.

[21]

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.

[22]

P. Glendinning and C. Sparrow, Local and global behavior near homoclinic orbits,, J. Statist. Phys., 35 (1984), 645.

[23]

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii and G. Huyet, Excitability in a quantum dot semiconductor laser with optical injection,, Phys. Rev. Lett., 98 (2007).

[24]

G. Gómez, W. S. Koon, M. W. Lo, J. E. Marsden, J. Masdemont and S. D. Ross, "Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design,", Astrodynamics Specialist Meeting, (2001), 01.

[25]

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

[26]

M. E. Henderson, Multiple parameter continuation: Computing implicitly defined $k$-manifolds,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 451.

[27]

M. E. Henderson, Computing invariant manifolds by integrating fat trajectories,, SIAM J. Appl. Dyn. Sys., 4 (2005), 832.

[28]

M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Mathematics, 583 (1977).

[29]

A. J. Homburg and B. Krauskopf, Resonant homoclinic flip bifurcations,, J. Dynam. Diff. Eqs, 12 (2000), 807.

[30]

A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields,, in B. Fiedler (Ed.), ().

[31]

J. L. Kaplan and J. A. Yorke, Preturbulence: A regime observed in a fluid flow model of Lorenz,, Commun. Math. Phys., 67 (1979), 93.

[32]

B. Krauskopf and H. M. Osinga, Growing 1D and quasi-2D unstable manifolds of maps,, J. Comput. Phys., 146 (1998), 406.

[33]

B. Krauskopf and H. M. Osinga, Two-dimensional global manifolds of vector fields,, Chaos, 9 (1999), 768.

[34]

B. Krauskopf and H. M. Osinga, Computing geodesic level sets on global (un)stable manifolds of vector fields,, SIAM J. Appl. Dyn. Sys., 2 (2003), 546.

[35]

B. Krauskopf and H. M. Osinga, Computing invariant manifolds via the continuation of orbit segments,, in, (2007), 117.

[36]

B. Krauskopf, H. M. Osinga and E. J. Doedel, Visualizing global manifolds during the transition to chaos in the Lorenz system,, in, (2009), 115. doi: 10.1007/978-3-540-88606-8_9.

[37]

B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763. doi: 10.1142/S0218127405012533.

[38]

B. Krauskopf, K. Schneider, J. Sieber, S. M. Wieczorek and M. Wolfrum, Excitability and self-pulsations near homoclinic bifurcations in semiconductor lasers,, Optics Communications, 215 (2003), 230. doi: 10.1016/S0030-4018(02)02239-3.

[39]

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.

[40]

Yu. A. Kuznetsov, "CONTENT - Integrated Environment for Analysis of Dynamical Systems. Tutorial,", École Normale Supérieure de Lyon, (1998), 98.

[41]

Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", 3nd edition, (2004).

[42]

C. M. Lee, P. J. Collins, B. Krauskopf and H. M. Osinga, Tangency bifurcations of global Poincaré maps,, SIAM J. Appl. Dyn. Syst., 7 (2008), 712. doi: 10.1137/07069972X.

[43]

X.-B. Lin, Using Melnikov's method to solve Shilnikov's problems,, Proc. R. Soc. Edinb. A, 116 (1990), 295.

[44]

E. N. Lorenz, Deterministic nonperiodic flows,, J. Atmosph. Sci., 20 (1963), 130. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

[45]

T. Noh, Shilnikov's chaos in the oxidation of formic acid with bismuth ion on Pt ring electrode,, Electrochimica Acta, 54 (2009), 3657. doi: 10.1016/j.electacta.2009.01.043.

[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]

H. M. Osinga and B. Krauskopf, Visualizing the structure of chaos in the Lorenz system,, Computers and Graphics, 25 (2002), 815. doi: 10.1016/S0097-8493(02)00136-X.

[48]

H. M. Osinga and B. Krauskopf, Crocheting the Lorenz manifold,, The Mathematical Intelligencer, 26 (2004), 25. doi: 10.1007/BF02985416.

[49]

H. M. Osinga and B. Krauskopf, Visualizing curvature on the Lorenz manifold,, Journal of Mathematics and the Arts, 1 (2007), 113. doi: 10.1080/17513470701503632.

[50]

J. Palis and W. de Melo, "Geometric Theory of Dynamical Systems,", Springer-Verlag, (1982).

[51]

J. Palis and F. Takens, "Hyperbolicity & Sensitive Chaotic Dynamics at Homoclinic Bifurcations,", Cambridge University Press, (1993).

[52]

T. Peacock and T. Mullin, Homoclinic bifurcations in a liquid crystal flow,, J. Fluid Mech., 432 (2001), 369.

[53]

C. Perelló, Intertwining invariant manifolds and Lorenz attractor,, in, 819 (1979), 375.

[54]

M. Phillips, S. Levy and T. Munzner, Geomview: An interactive geometry viewer,, Not. Am. Math. Soc., 40 (1993), 985.

[55]

A. M. Rucklidge, Chaos in a low-order model of magnetoconvection,, Physica D, 62 (1993), 323. doi: 10.1016/0167-2789(93)90291-8.

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A. L. Shilnikov, On bifurcations of the Lorenz attractor in the Shimizu-Morioka model,, Physica D, 62 (1993), 338. doi: 10.1016/0167-2789(93)90292-9.

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show all references

References:
[1]

R. H. Abraham and C. D. Shaw, "Dynamics -- The Geometry Of Behavior, Part Three: Global Behavior,", Aerial Press, (1985).

[2]

U. M. Ascher, J. Christiansen and R. D. Russell, Colsys -- A collocation code for boundary-value problems,, Lecture Notes in Computer Science, 76 (1979), 164.

[3]

U. M. Ascher and R. J. Spiteri, Collocation software for boundary value differential-algebraic equations,, SIAM J. Sci. Comput., 15 (1994), 938. doi: 10.1137/0915056.

[4]

M. R. Bassett and J. L. Hudson, Shil'nikov chaos during copper electrodissolution,, J. Phys. Chem., 92 (1988), 6963. doi: 10.1021/j100335a025.

[5]

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

[6]

C. J. Budd and J. P. Wilson, Bogdanov-Takens bifurcation points and Shilnikov homoclinicity in a simple power system model of voltage collapse,, IEEE Trans. Circuits Systems I, 43 (2002), 575. doi: 10.1109/TCSI.2002.1001947.

[7]

A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman and J. Sneyd, When Shil'nikov meets Hopf in excitable systems,, SIAM J. Appl. Dyn. Syst., 6 (2007), 663. doi: 10.1137/070682654.

[8]

A. R. Champneys, Y. Kuznetsov and B. Sandstede, A numerical toolbox for homoclinic bifurcation analysis,, Int. J. Bifurc. Chaos, 6 (1996), 867. doi: 10.1142/S0218127496000485.

[9]

B. Deng and G. Hines, Food chain chaos due to Shilnikov's orbit,, Chaos, 12 (2002), 533. doi: 10.1063/1.1482255.

[10]

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.

[11]

E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems,, Congr. Numer., 30 (1981), 265.

[12]

E. J. Doedel and B. E. Oldeman, with major contributions from A. R. Champneys, F. Dercole, T. F. Fairgrieve, Yu. A. Kuznetsov, R. C. Paffenroth, B. Sandstede, X. J. Wang, and C. H. Zhang, AUTO-07p Version 0.7: Continuation and bifurcation software for ordinary differential equations,, Department of Computer Science, (2010).

[13]

E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations,, in, (2007), 1. doi: 10.1007/978-1-4020-6356-5_1.

[14]

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.

[15]

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.

[16]

E. J. Doedel, R. C. Paffenroth, H. B. Keller, D. Dichmann, J. Galán-Vioque and A. Vanderbauwhede, Computation of periodic solutions of conservative systems with application to the 3-body problem,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1353. doi: 10.1142/S0218127403007291.

[17]

E. J. Doedel, V. Romanov, R. C. Paffenroth, H. B. Keller, D. Dichmann, J. Galán-Vioque and A. Vanderbauwhede, Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 2625. doi: 10.1142/S0218127407018671.

[18]

J. P. England, B. Krauskopf and H. M. Osinga, Computing one-dimensional global manifolds of Poincaré maps by continuation,, SIAM J. Appl. Dyn. Sys., 4 (2005), 1008. doi: 10.1137/05062408X.

[19]

J. P. England, B. Krauskopf and H. M. Osinga, Computing two-dimensional global invariant manifolds in slow-fast systems,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 805. doi: 10.1142/S0218127407017562.

[20]

J. A. Feroe, Homoclinic orbits in a parametrized saddle-focus system,, Physica D, 62 (1993), 254. doi: 10.1016/0167-2789(93)90285-9.

[21]

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.

[22]

P. Glendinning and C. Sparrow, Local and global behavior near homoclinic orbits,, J. Statist. Phys., 35 (1984), 645.

[23]

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii and G. Huyet, Excitability in a quantum dot semiconductor laser with optical injection,, Phys. Rev. Lett., 98 (2007).

[24]

G. Gómez, W. S. Koon, M. W. Lo, J. E. Marsden, J. Masdemont and S. D. Ross, "Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design,", Astrodynamics Specialist Meeting, (2001), 01.

[25]

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

[26]

M. E. Henderson, Multiple parameter continuation: Computing implicitly defined $k$-manifolds,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 451.

[27]

M. E. Henderson, Computing invariant manifolds by integrating fat trajectories,, SIAM J. Appl. Dyn. Sys., 4 (2005), 832.

[28]

M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Mathematics, 583 (1977).

[29]

A. J. Homburg and B. Krauskopf, Resonant homoclinic flip bifurcations,, J. Dynam. Diff. Eqs, 12 (2000), 807.

[30]

A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields,, in B. Fiedler (Ed.), ().

[31]

J. L. Kaplan and J. A. Yorke, Preturbulence: A regime observed in a fluid flow model of Lorenz,, Commun. Math. Phys., 67 (1979), 93.

[32]

B. Krauskopf and H. M. Osinga, Growing 1D and quasi-2D unstable manifolds of maps,, J. Comput. Phys., 146 (1998), 406.

[33]

B. Krauskopf and H. M. Osinga, Two-dimensional global manifolds of vector fields,, Chaos, 9 (1999), 768.

[34]

B. Krauskopf and H. M. Osinga, Computing geodesic level sets on global (un)stable manifolds of vector fields,, SIAM J. Appl. Dyn. Sys., 2 (2003), 546.

[35]

B. Krauskopf and H. M. Osinga, Computing invariant manifolds via the continuation of orbit segments,, in, (2007), 117.

[36]

B. Krauskopf, H. M. Osinga and E. J. Doedel, Visualizing global manifolds during the transition to chaos in the Lorenz system,, in, (2009), 115. doi: 10.1007/978-3-540-88606-8_9.

[37]

B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763. doi: 10.1142/S0218127405012533.

[38]

B. Krauskopf, K. Schneider, J. Sieber, S. M. Wieczorek and M. Wolfrum, Excitability and self-pulsations near homoclinic bifurcations in semiconductor lasers,, Optics Communications, 215 (2003), 230. doi: 10.1016/S0030-4018(02)02239-3.

[39]

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.

[40]

Yu. A. Kuznetsov, "CONTENT - Integrated Environment for Analysis of Dynamical Systems. Tutorial,", École Normale Supérieure de Lyon, (1998), 98.

[41]

Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", 3nd edition, (2004).

[42]

C. M. Lee, P. J. Collins, B. Krauskopf and H. M. Osinga, Tangency bifurcations of global Poincaré maps,, SIAM J. Appl. Dyn. Syst., 7 (2008), 712. doi: 10.1137/07069972X.

[43]

X.-B. Lin, Using Melnikov's method to solve Shilnikov's problems,, Proc. R. Soc. Edinb. A, 116 (1990), 295.

[44]

E. N. Lorenz, Deterministic nonperiodic flows,, J. Atmosph. Sci., 20 (1963), 130. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

[45]

T. Noh, Shilnikov's chaos in the oxidation of formic acid with bismuth ion on Pt ring electrode,, Electrochimica Acta, 54 (2009), 3657. doi: 10.1016/j.electacta.2009.01.043.

[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]

H. M. Osinga and B. Krauskopf, Visualizing the structure of chaos in the Lorenz system,, Computers and Graphics, 25 (2002), 815. doi: 10.1016/S0097-8493(02)00136-X.

[48]

H. M. Osinga and B. Krauskopf, Crocheting the Lorenz manifold,, The Mathematical Intelligencer, 26 (2004), 25. doi: 10.1007/BF02985416.

[49]

H. M. Osinga and B. Krauskopf, Visualizing curvature on the Lorenz manifold,, Journal of Mathematics and the Arts, 1 (2007), 113. doi: 10.1080/17513470701503632.

[50]

J. Palis and W. de Melo, "Geometric Theory of Dynamical Systems,", Springer-Verlag, (1982).

[51]

J. Palis and F. Takens, "Hyperbolicity & Sensitive Chaotic Dynamics at Homoclinic Bifurcations,", Cambridge University Press, (1993).

[52]

T. Peacock and T. Mullin, Homoclinic bifurcations in a liquid crystal flow,, J. Fluid Mech., 432 (2001), 369.

[53]

C. Perelló, Intertwining invariant manifolds and Lorenz attractor,, in, 819 (1979), 375.

[54]

M. Phillips, S. Levy and T. Munzner, Geomview: An interactive geometry viewer,, Not. Am. Math. Soc., 40 (1993), 985.

[55]

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