October  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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[5]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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

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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

[11]

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

[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). Google Scholar

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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

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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. Google Scholar

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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. 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. Dyn. Sys., 4 (2005), 1008. doi: 10.1137/05062408X. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

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

[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). Google Scholar

[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. Google Scholar

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[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. Google Scholar

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

[33]

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

[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. Google Scholar

[35]

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

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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[40]

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[42]

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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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[5]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[11]

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

[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). Google Scholar

[13]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

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

[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). Google Scholar

[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. Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[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. Google Scholar

[32]

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

[33]

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

[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. Google Scholar

[35]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[40]

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

[41]

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

[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. Google Scholar

[43]

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

[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. Google Scholar

[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. 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]

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. Google Scholar

[48]

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

[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. Google Scholar

[50]

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