doi: 10.3934/dcdss.2020061

Bifurcation revisited along footprints of Jürgen Scheurle

Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D50931 Köln, Germany

Received  September 2017 Revised  June 2018 Published  April 2019

Actual research concerning, in particular, the occurrence of "gap-solitons" bifurcating from the continuous spectrum confirms that this part of Bifurcation Theory that started around 40 years ago flourishes. In this lecture we review the origins of "Bifurcation from the continuous spectrum" with regard to the achievements of Jürgen Scheurle and sketch how the early results dealing with the bifurcation of singular solutions have prepared the ground for present and further developments.

Citation: Tassilo Küpper. Bifurcation revisited along footprints of Jürgen Scheurle. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020061
References:
[1]

S. Alama and Y. Li, Existence of solutions for semilinear elliptic equations with indefinite linear part, J. Diff. Equat., 96 (1992), 89-115. doi: 10.1016/0022-0396(92)90145-D. Google Scholar

[2]

N. Bazley and T. Küpper, Branches of solutions in nonlinear eigenvalue problems, in Applications of Nonlinear Analysis in the Physical Sciences (eds. H. Amann, N. Bazley and K. Kirchgässner), Pitman, (1981), 24-40. Google Scholar

[3]

V. Benci and D. Fortunato, Does bifurcation from the continuous spectrum occur?, Comm. Partial. Diff. Equat., 6 (1981), 249-272. doi: 10.1080/03605308108820176. Google Scholar

[4]

A. BongersH. Heinz and T. Küpper, Existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains, J. Diff. Equat., 47 (1983), 327-357. doi: 10.1016/0022-0396(83)90040-2. Google Scholar

[5]

R. Chiapinelli and C. Stuart, Bifurcation when the linearized problem has no eigenvalues, J. Diff. Equat., 30 (1978), 296-307. doi: 10.1016/0022-0396(78)90002-5. Google Scholar

[6]

H. Chiba, A spectral theory of linear operators on rigged Hilbert spaces under analytical conditions, Adv. In Math., 273 (2015), 324-379. doi: 10.1016/j.aim.2015.01.001. Google Scholar

[7]

H. Chiba, A proof of the Kuramoto conjecture for a bifurcation structure of the infinite dimensional Kuramoto model, Ergo. Theo. Dyn. Syst., 35 (2015), 762-834. doi: 10.1017/etds.2013.68. Google Scholar

[8]

Y. Demay, Bifurcation d'' un solution pour une equation de la physique de plasma, C.R. Acad. Sci. Paris, 285 (1977), 769-772. Google Scholar

[9]

T. DohnalD. Pelinovsky and G. Schneider, Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential, J. Nonl. Sci., 19 (2009), 95-131. doi: 10.1007/s00332-008-9027-9. Google Scholar

[10]

T. Dohnal and H. Uecker, Bifurcation of nonlinear Bloch waves from the spectrum in the Gross-Pitaevskii equation, J. Nonl. Sci., 26 (2016), 581-618. doi: 10.1007/s00332-015-9281-6. Google Scholar

[11]

B. Fiedler, Egodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer-Verlag, Berlin 2001. doi: 10.1007/978-3-642-56589-2. Google Scholar

[12]

M. Golubitsky and D. Schaeffer, Singularities and Groups in Bifurcation Theory Vol.I, Springer-Verlag, 1985. doi: 10.1007/978-1-4612-5034-0. Google Scholar

[13]

M. Golubitsky, D. Schaeffer and I. Stewart, Singularities and Groups in Bifurcation Theory Vol.II, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-4574-2. Google Scholar

[14]

H. Heinz, Über das Verzweigungsverhalten eines nichtlinearen Eigenwertproblems in der Nhe eines unendlich vielfachen Eigenwerts der Linearisierung, Manuscripta Math., 19 (1976), 105-132. doi: 10.1007/BF01275416. Google Scholar

[15]

H. Heinz, Existence and gap-bifurcation of multiple solutions to certain nonlinear eigenvalue problem, Nonlin. Anal., 21 (1993), 457-484. doi: 10.1016/0362-546X(93)90128-F. Google Scholar

[16]

H. HeinzT. Küpper and C. Stuart, Existence and bifurcation for nonlinear perturbations of the periodic Schrödinger equation, J. Diff. Equat., 100 (1992), 341-354. doi: 10.1016/0022-0396(92)90118-7. Google Scholar

[17]

C. Jones and T. Küpper, On the infinitely many solutions of a semilinear elliptic equation, SIAM J. Math. Anal., 17 (1986), 803-835. doi: 10.1137/0517059. Google Scholar

[18]

C. Jones and T. Küpper, Characterisation of bifurcation from the continuous spectrum by nodal properties, J. Diff. Equat., 54 (1984), 196-220. doi: 10.1016/0022-0396(84)90158-X. Google Scholar

[19]

K. Kirchgässner and J. Scheurle, Verzweigung und Stabilitaet von Loesungen semilinearer elliptischer Randwertprobleme, Jahresberichte Deutsche Mathematiker Vereinigung, 77 (1975), 39-54. Google Scholar

[20]

K. Kirchgässner and J. Scheurle, On the bounded solutions of a semilinear elliptic equation in a strip, J. Diff. Equat., 32 (1979), 119-148. doi: 10.1016/0022-0396(79)90055-X. Google Scholar

[21]

K. Kirchgässner and J. Scheurle, Bifurcation from the continuous spectrum and singular solutions, in Trends in Applications of pure Mathematics to Mechanics, Vol. III (ed. R.Knops), Pitman, London, 11 (1981), 138-154. Google Scholar

[22]

K. Kirchgässner and J. Scheurle, Global branches of periodic solutions of reversible systems, in Contributions to Nonlinear Partial Differential Equations (eds. H.Brezis), Pitman, London, 50 (1981), 103-130. Google Scholar

[23]

M. KunzeT. KüpperV. Shapiro and S. Turitsyn, Nonlinearity solitary waves with Gaussian tails, Physica D, 128 (1999), 273-295. doi: 10.1016/S0167-2789(98)00297-8. Google Scholar

[24]

T. Küpper, On minimal nonlinearities which permit bifurcation from the continuous spectrum, Math. Meth. Appl. Sci., 1 (1979), 572-580. doi: 10.1002/mma.1670010414. Google Scholar

[25]

T. Küpper, Singuläre Verzweigungsprobleme, Habilitationsschrift, Universitaet zu Köln, 1979.Google Scholar

[26]

T. Küpper, The lowest point of the continuous spectrum as bifurcation point, J. Diff. Equat., 34 (1979), 212-217. doi: 10.1016/0022-0396(79)90005-6. Google Scholar

[27]

T. Küpper, Verzweigung aus dem wesentlichen Spektrum, GAMM-Mitteilungen, 1 (1991), 11-22. Google Scholar

[28]

T. Küpper, Nonlinear Phenomena: Final Report on Joint Research Program with Chinese Universities, Köln., 2005.Google Scholar

[29]

T. Küpper and T. Mrziglod, On the bifurcation structure of nonlinear perturbations of Hill`s equations at boundary points of the continuous sprectrum, SIAM J. Math. Anal., 26 (1995), 1284-1305. doi: 10.1137/S0036141093250876. Google Scholar

[30]

T. Küpper and D. Riemer, Necessary and sufficient conditions for bifurcation from the continuous spectrum, Nonl. Anal (TMA), 3 (1979), 555-561. doi: 10.1016/0362-546X(79)90073-7. Google Scholar

[31]

T. Küpper and C. Stuart, Bifurcation into gaps in the essential spectrum, J. Reine Angew. Math., 409 (1990), 1-34. doi: 10.1007/978-94-009-0659-4_32. Google Scholar

[32]

T. Küpper and C. Stuart, Gap-bifurcation for nonlinear perturbations of Hill's equation, J. Reine Angew. Math., 410 (1990), 23-52. Google Scholar

[33]

T. Küpper and C. Stuart, Bifurcation at boundary points of the continuous spectrum, in Progress in Nonlinear Differential Equations and their Applications (eds. N.G. Lloyd, W.M. Ni, L.A. Peletier and J. Serrin), 7 (1992), 287-297. Google Scholar

[34]

T. Küpper and C. Stuart, Necessary and sufficient conditions for gap-bifurcation, Nonlin. Anal., 18 (1992), 893-903. doi: 10.1016/0362-546X(92)90230-C. Google Scholar

[35]

T. Mrziglod, Untersuchungen zur Loesungsstruktur nichtlinearer Hill'scher Gleichungen, Habilitationsschrift Univ.- Köln, 1996.Google Scholar

[36]

D. Pelinovsky, A. A. Sukhorukov and Y. S. Kivshar, Bifurcations and stability of gap solitons in periodic potentials, Physical Review E, 70 (2004), 036618, 17pp. doi: 10.1103/PhysRevE.70.036618. Google Scholar

[37]

A. Plate, Verzweigung aus dem wesentlichen Spektrum, Dissertation Universitt Hannover, 1992.Google Scholar

[38]

J. Scheurle, Newton iterations without inverting the derivative, Math. Meth. Appl. Sci., 1 (1979), 514-529. doi: 10.1002/mma.1670010409. Google Scholar

[39]

J. Scheurle, Ein selektives Iterationsverfahren und Verzweigungsprobleme, Dissertation, Universitt Stuttgart, 1975.Google Scholar

[40]

J. Scheurle, A selective iteration procedure for Taylor's stability problem, Proc. of the 2nd GAMM-conference on Numerical Methods in Fluid Mech., (1977), 176-183.Google Scholar

[41]

J. Scheurle, Ein selektives Projektions-Iterationsverfahren und Anwendungen auf Verzweigungsprobleme, Num. Math., 29 (1977), 11-35. doi: 10.1007/BF01389310. Google Scholar

[42]

J. Scheurle, Selective iteration and applications, J. Math.Anal. Appl., 59 (1977), 596-616. doi: 10.1016/0022-247X(77)90084-1. Google Scholar

[43]

J. Scheurle, Bifurcation of a stationary solution of a dynamical system into n-dimensional tori of quasiperiodic, in Functional Differential Equations and Approximation of Fixed Points, Lecture Notes in Math., 730 (1979), 442-454. Google Scholar

[44]

J. Scheurle, Über die Konstruktion invarianter Tori, welche von einer stationaeren Grundloesung eines reversiblen dynamischen Systems abzweigen, in Constructive Methods for Nonlinear Boundary Value Problems and Nonlinear Oscillations (eds. J. Albrecht, L. Collatz and K. Kirchgässner) Google Scholar

[45]

J. Scheurle, Verzweigung Quasiperiodischer L Sungen bei Reversiblen Dynamischen Systemen, Habilitationsschrift Universität Stuttgart, 1980.Google Scholar

[46]

J. Scheurle, Quasiperiodic solutions of a semilinear equation in a two-dimensional strip, in Dynamical Problems in Mathematical Physics (eds. B. Brosowski and E. Martensen), 26, Lang-Verlag, (1983), 201-223. Google Scholar

[47]

J. Scheurle and K. Kirchgässner, Bifurcation of non-periodic solutions of some semi-linear equations in unbounded domains, in Applications of Nonlinear Analysis in the Physical Sciences (eds. H. Amann, N. Bazley and K. Kirchg ssner), London: Pitman, (1981), 41-59. Google Scholar

[48]

C. A. Stuart, Bifurcation for Dirichlet problems without eigenvalues, Proc. London Math. Soc. Ser., 45 (1982), 169-192. doi: 10.1112/plms/s3-45.1.169. Google Scholar

[49]

C. A. Stuart, Bifurcation for Neumann problems without eigenvalues, J. Diff. Equat., 36 (1980), 391-407. doi: 10.1016/0022-0396(80)90057-1. Google Scholar

[50]

C. A. Stuart, Bifurcation when the linearisation has no eigenvalues, J. Funct. Anal., 38 (1980), 169-187. doi: 10.1016/0022-1236(80)90063-4. Google Scholar

[51]

C. A. Stuart, Bifurcation from the continuous spectrum in LP(R), in Bifurcation: Analysis, Algorithms, Applications ISNM (eds. T. Küpper, R. Seydel and H. Troger), Birkhäuser, 79 (1987), 306-318. Google Scholar

[52]

C. A. Stuart, Bifurcation of homoclinic orbits and bifurcation from the essential spectrum, SIAM J. Math. Anal., 20 (1989), 1145-1171. doi: 10.1137/0520076. Google Scholar

[53]

C. A. Stuart, Bifurcation into Spectral Gaps, Bulletin of the Belgian Mathematical Society, 1995. Google Scholar

show all references

References:
[1]

S. Alama and Y. Li, Existence of solutions for semilinear elliptic equations with indefinite linear part, J. Diff. Equat., 96 (1992), 89-115. doi: 10.1016/0022-0396(92)90145-D. Google Scholar

[2]

N. Bazley and T. Küpper, Branches of solutions in nonlinear eigenvalue problems, in Applications of Nonlinear Analysis in the Physical Sciences (eds. H. Amann, N. Bazley and K. Kirchgässner), Pitman, (1981), 24-40. Google Scholar

[3]

V. Benci and D. Fortunato, Does bifurcation from the continuous spectrum occur?, Comm. Partial. Diff. Equat., 6 (1981), 249-272. doi: 10.1080/03605308108820176. Google Scholar

[4]

A. BongersH. Heinz and T. Küpper, Existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains, J. Diff. Equat., 47 (1983), 327-357. doi: 10.1016/0022-0396(83)90040-2. Google Scholar

[5]

R. Chiapinelli and C. Stuart, Bifurcation when the linearized problem has no eigenvalues, J. Diff. Equat., 30 (1978), 296-307. doi: 10.1016/0022-0396(78)90002-5. Google Scholar

[6]

H. Chiba, A spectral theory of linear operators on rigged Hilbert spaces under analytical conditions, Adv. In Math., 273 (2015), 324-379. doi: 10.1016/j.aim.2015.01.001. Google Scholar

[7]

H. Chiba, A proof of the Kuramoto conjecture for a bifurcation structure of the infinite dimensional Kuramoto model, Ergo. Theo. Dyn. Syst., 35 (2015), 762-834. doi: 10.1017/etds.2013.68. Google Scholar

[8]

Y. Demay, Bifurcation d'' un solution pour une equation de la physique de plasma, C.R. Acad. Sci. Paris, 285 (1977), 769-772. Google Scholar

[9]

T. DohnalD. Pelinovsky and G. Schneider, Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential, J. Nonl. Sci., 19 (2009), 95-131. doi: 10.1007/s00332-008-9027-9. Google Scholar

[10]

T. Dohnal and H. Uecker, Bifurcation of nonlinear Bloch waves from the spectrum in the Gross-Pitaevskii equation, J. Nonl. Sci., 26 (2016), 581-618. doi: 10.1007/s00332-015-9281-6. Google Scholar

[11]

B. Fiedler, Egodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer-Verlag, Berlin 2001. doi: 10.1007/978-3-642-56589-2. Google Scholar

[12]

M. Golubitsky and D. Schaeffer, Singularities and Groups in Bifurcation Theory Vol.I, Springer-Verlag, 1985. doi: 10.1007/978-1-4612-5034-0. Google Scholar

[13]

M. Golubitsky, D. Schaeffer and I. Stewart, Singularities and Groups in Bifurcation Theory Vol.II, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-4574-2. Google Scholar

[14]

H. Heinz, Über das Verzweigungsverhalten eines nichtlinearen Eigenwertproblems in der Nhe eines unendlich vielfachen Eigenwerts der Linearisierung, Manuscripta Math., 19 (1976), 105-132. doi: 10.1007/BF01275416. Google Scholar

[15]

H. Heinz, Existence and gap-bifurcation of multiple solutions to certain nonlinear eigenvalue problem, Nonlin. Anal., 21 (1993), 457-484. doi: 10.1016/0362-546X(93)90128-F. Google Scholar

[16]

H. HeinzT. Küpper and C. Stuart, Existence and bifurcation for nonlinear perturbations of the periodic Schrödinger equation, J. Diff. Equat., 100 (1992), 341-354. doi: 10.1016/0022-0396(92)90118-7. Google Scholar

[17]

C. Jones and T. Küpper, On the infinitely many solutions of a semilinear elliptic equation, SIAM J. Math. Anal., 17 (1986), 803-835. doi: 10.1137/0517059. Google Scholar

[18]

C. Jones and T. Küpper, Characterisation of bifurcation from the continuous spectrum by nodal properties, J. Diff. Equat., 54 (1984), 196-220. doi: 10.1016/0022-0396(84)90158-X. Google Scholar

[19]

K. Kirchgässner and J. Scheurle, Verzweigung und Stabilitaet von Loesungen semilinearer elliptischer Randwertprobleme, Jahresberichte Deutsche Mathematiker Vereinigung, 77 (1975), 39-54. Google Scholar

[20]

K. Kirchgässner and J. Scheurle, On the bounded solutions of a semilinear elliptic equation in a strip, J. Diff. Equat., 32 (1979), 119-148. doi: 10.1016/0022-0396(79)90055-X. Google Scholar

[21]

K. Kirchgässner and J. Scheurle, Bifurcation from the continuous spectrum and singular solutions, in Trends in Applications of pure Mathematics to Mechanics, Vol. III (ed. R.Knops), Pitman, London, 11 (1981), 138-154. Google Scholar

[22]

K. Kirchgässner and J. Scheurle, Global branches of periodic solutions of reversible systems, in Contributions to Nonlinear Partial Differential Equations (eds. H.Brezis), Pitman, London, 50 (1981), 103-130. Google Scholar

[23]

M. KunzeT. KüpperV. Shapiro and S. Turitsyn, Nonlinearity solitary waves with Gaussian tails, Physica D, 128 (1999), 273-295. doi: 10.1016/S0167-2789(98)00297-8. Google Scholar

[24]

T. Küpper, On minimal nonlinearities which permit bifurcation from the continuous spectrum, Math. Meth. Appl. Sci., 1 (1979), 572-580. doi: 10.1002/mma.1670010414. Google Scholar

[25]

T. Küpper, Singuläre Verzweigungsprobleme, Habilitationsschrift, Universitaet zu Köln, 1979.Google Scholar

[26]

T. Küpper, The lowest point of the continuous spectrum as bifurcation point, J. Diff. Equat., 34 (1979), 212-217. doi: 10.1016/0022-0396(79)90005-6. Google Scholar

[27]

T. Küpper, Verzweigung aus dem wesentlichen Spektrum, GAMM-Mitteilungen, 1 (1991), 11-22. Google Scholar

[28]

T. Küpper, Nonlinear Phenomena: Final Report on Joint Research Program with Chinese Universities, Köln., 2005.Google Scholar

[29]

T. Küpper and T. Mrziglod, On the bifurcation structure of nonlinear perturbations of Hill`s equations at boundary points of the continuous sprectrum, SIAM J. Math. Anal., 26 (1995), 1284-1305. doi: 10.1137/S0036141093250876. Google Scholar

[30]

T. Küpper and D. Riemer, Necessary and sufficient conditions for bifurcation from the continuous spectrum, Nonl. Anal (TMA), 3 (1979), 555-561. doi: 10.1016/0362-546X(79)90073-7. Google Scholar

[31]

T. Küpper and C. Stuart, Bifurcation into gaps in the essential spectrum, J. Reine Angew. Math., 409 (1990), 1-34. doi: 10.1007/978-94-009-0659-4_32. Google Scholar

[32]

T. Küpper and C. Stuart, Gap-bifurcation for nonlinear perturbations of Hill's equation, J. Reine Angew. Math., 410 (1990), 23-52. Google Scholar

[33]

T. Küpper and C. Stuart, Bifurcation at boundary points of the continuous spectrum, in Progress in Nonlinear Differential Equations and their Applications (eds. N.G. Lloyd, W.M. Ni, L.A. Peletier and J. Serrin), 7 (1992), 287-297. Google Scholar

[34]

T. Küpper and C. Stuart, Necessary and sufficient conditions for gap-bifurcation, Nonlin. Anal., 18 (1992), 893-903. doi: 10.1016/0362-546X(92)90230-C. Google Scholar

[35]

T. Mrziglod, Untersuchungen zur Loesungsstruktur nichtlinearer Hill'scher Gleichungen, Habilitationsschrift Univ.- Köln, 1996.Google Scholar

[36]

D. Pelinovsky, A. A. Sukhorukov and Y. S. Kivshar, Bifurcations and stability of gap solitons in periodic potentials, Physical Review E, 70 (2004), 036618, 17pp. doi: 10.1103/PhysRevE.70.036618. Google Scholar

[37]

A. Plate, Verzweigung aus dem wesentlichen Spektrum, Dissertation Universitt Hannover, 1992.Google Scholar

[38]

J. Scheurle, Newton iterations without inverting the derivative, Math. Meth. Appl. Sci., 1 (1979), 514-529. doi: 10.1002/mma.1670010409. Google Scholar

[39]

J. Scheurle, Ein selektives Iterationsverfahren und Verzweigungsprobleme, Dissertation, Universitt Stuttgart, 1975.Google Scholar

[40]

J. Scheurle, A selective iteration procedure for Taylor's stability problem, Proc. of the 2nd GAMM-conference on Numerical Methods in Fluid Mech., (1977), 176-183.Google Scholar

[41]

J. Scheurle, Ein selektives Projektions-Iterationsverfahren und Anwendungen auf Verzweigungsprobleme, Num. Math., 29 (1977), 11-35. doi: 10.1007/BF01389310. Google Scholar

[42]

J. Scheurle, Selective iteration and applications, J. Math.Anal. Appl., 59 (1977), 596-616. doi: 10.1016/0022-247X(77)90084-1. Google Scholar

[43]

J. Scheurle, Bifurcation of a stationary solution of a dynamical system into n-dimensional tori of quasiperiodic, in Functional Differential Equations and Approximation of Fixed Points, Lecture Notes in Math., 730 (1979), 442-454. Google Scholar

[44]

J. Scheurle, Über die Konstruktion invarianter Tori, welche von einer stationaeren Grundloesung eines reversiblen dynamischen Systems abzweigen, in Constructive Methods for Nonlinear Boundary Value Problems and Nonlinear Oscillations (eds. J. Albrecht, L. Collatz and K. Kirchgässner) Google Scholar

[45]

J. Scheurle, Verzweigung Quasiperiodischer L Sungen bei Reversiblen Dynamischen Systemen, Habilitationsschrift Universität Stuttgart, 1980.Google Scholar

[46]

J. Scheurle, Quasiperiodic solutions of a semilinear equation in a two-dimensional strip, in Dynamical Problems in Mathematical Physics (eds. B. Brosowski and E. Martensen), 26, Lang-Verlag, (1983), 201-223. Google Scholar

[47]

J. Scheurle and K. Kirchgässner, Bifurcation of non-periodic solutions of some semi-linear equations in unbounded domains, in Applications of Nonlinear Analysis in the Physical Sciences (eds. H. Amann, N. Bazley and K. Kirchg ssner), London: Pitman, (1981), 41-59. Google Scholar

[48]

C. A. Stuart, Bifurcation for Dirichlet problems without eigenvalues, Proc. London Math. Soc. Ser., 45 (1982), 169-192. doi: 10.1112/plms/s3-45.1.169. Google Scholar

[49]

C. A. Stuart, Bifurcation for Neumann problems without eigenvalues, J. Diff. Equat., 36 (1980), 391-407. doi: 10.1016/0022-0396(80)90057-1. Google Scholar

[50]

C. A. Stuart, Bifurcation when the linearisation has no eigenvalues, J. Funct. Anal., 38 (1980), 169-187. doi: 10.1016/0022-1236(80)90063-4. Google Scholar

[51]

C. A. Stuart, Bifurcation from the continuous spectrum in LP(R), in Bifurcation: Analysis, Algorithms, Applications ISNM (eds. T. Küpper, R. Seydel and H. Troger), Birkhäuser, 79 (1987), 306-318. Google Scholar

[52]

C. A. Stuart, Bifurcation of homoclinic orbits and bifurcation from the essential spectrum, SIAM J. Math. Anal., 20 (1989), 1145-1171. doi: 10.1137/0520076. Google Scholar

[53]

C. A. Stuart, Bifurcation into Spectral Gaps, Bulletin of the Belgian Mathematical Society, 1995. Google Scholar

Figure 1.  Phase portrait for $ \lambda >0 $
Figure 2.  Heteroclinic orbit (singular solution)
Figure 3.  (a) λ > 0, (b) λ < 0
Figure 4.  Prof. Tassilo Küpper and Prof. Jürgen Scheurle
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