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Lyapunov–Schmidt Method for Dynamical Systems

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

The mathematical description of the state and the evolution of systems used to model all kind of applications requires in many cases a large or even an infinite number of variables. In contrast, a number of basic phenomena observed in such systems (in particular bifurcation phenomena) are known to depend only on a very small number of variables. A simple example is the Hopf bifurcation, where the system changes from an equilibrium state to a periodic regime when the parameters cross a certain critical boundary, a transition which in a typical system essentially only involves two of the state variables. There are several ways in which the original system can be reduced to a much smaller system which captures the phenomenon one wants to analyze. There is a dynamical approach, usually in the form of some kind of center manifold reduction, where one tries to find a subsystem (invariant under the full system flow) which contains the core of the bifurcation under...

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Abbreviations

Bifurcation:

A change in the local or global qualitative behavior of a system under the change of one or more parameters; the critical parameter values at which the change takes place are called bifurcation values.

Saddle-node bifurcation:

A bifurcation where two equilibria, usually one stable and the other unstable, merge and then disappear. A similar bifurcation can also happen with periodic orbits.

Pitchfork bifurcation:

A bifurcation where an equilibrium loses stability and at the same time two new stable equilibria emerge; this type of bifurcation frequently appears in systems with symmetry.

Period-doubling bifurcation:

A bifurcation where a fixed point of a mapping (discrete system) loses stability and a stable period-two point emerges. In a continuous system this corresponds to a periodic orbits losing stability while a new periodic orbit, with approximately twice the period of the original one, bifurcates.

Equivariant system:

A system with symmetries; these symmetries form a group of (usually linear) transformations commuting with the vectorfield or the mapping generating the system.

Reversible system:

A symmetric system where some of the symmetries involve a reversal of the time.

Bibliography

Primary Literature

  1. Andronov A, Leontovich FA (1938) Sur la théorie de la variation de la structure qualitative de la division du plan en trajectoires. Dokl Akad Nauk 21:427–430

    Google Scholar 

  2. Antosiewicz HA (1966) Boundary value problems for nonlinear ordinary differential equations. Pacific J Math 17:191–197

    MathSciNet  MATH  Google Scholar 

  3. Ashwin P, Böhmer K, Mei Z (1993) A numerical Lyapunov–Schmidt method for finitely determined systems. In: Allgower E, Georg K, Miranda R (eds) Exploiting Symmetry in Applied and Numerical Analysis. Lectures in Appl Math, vol 29. AMS, Providence, pp 49–69

    Google Scholar 

  4. Bancroft S, Hale JK, Sweet D (1968) Alternative problems for nonlinear functional equations. J Differ Equ 4:40–56

    MathSciNet  ADS  MATH  Google Scholar 

  5. Bartle R (1953) Singular points of functional equations. Trans Amer Math Soc 75:366–384

    MathSciNet  MATH  Google Scholar 

  6. Böhmer K (1993) On a numerical Lyapunov–Schmidt method for operator equations. Computing 53:237–269

    Google Scholar 

  7. Bourgain J (1998) Quasi-periodic Solutions of Hamiltonian Perturbations of 2D Linear Schrödinger Equations. Ann Math 148(2):363–439

    MathSciNet  MATH  Google Scholar 

  8. Cacciopoli R (1936) Sulle correspondenze funzionali inverse diramata: teoria generale e applicazioni ad alcune equazioni funzionali non lineari e al problema di Plateau. Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur Sez I 24:258–263, 416–421

    Google Scholar 

  9. Cesari L (1940) Sulla stabilità delle soluzioni dei sistemi di equazioni differenziali lineari a coefficienti periodici. Atti Accad Ital Mem Cl Fis Mat Nat 6(11):633–692

    MathSciNet  Google Scholar 

  10. Cesari L (1963) Periodic solutions of hyperbolic partial differential equations. In: Intern. Symposium on Nonlinear Differential Equations and Nonlinear Mechanics. Academic Press, New York, pp 33–57

    Google Scholar 

  11. Cesari L (1964) Functional analysis and Galerkin's method. Michigan Math J 11:385–414

    MathSciNet  MATH  Google Scholar 

  12. Cicogna C (1981) Symmetry breakdown from bifurcations. Lett Nuovo Cimento 31:600–602

    MathSciNet  Google Scholar 

  13. Chossat P, Lauterbach R (2000) Methods in Equivariant Bifurcations and Dynamical Systems. Advanced Series in Nonlinear Dynamics, vol 15. World Scientific, Singapore

    Google Scholar 

  14. Chow S-N, Hale JK (1982) Methods of Bifurcation Theory. Grundlehren der Mathematischen Wissenschaften, vol 251. Springer, New York

    Google Scholar 

  15. Ciocco M-C, Vanderbauwhede A (2004) Bifurcation of Periodic Points in Reversible Diffeomorphisms. In: Elaydi S, Ladas G, Aulbach B (eds) New Progress in Difference Equations – Proceedings ICDEA2001, Augsburg. Chapman & Hall, CRC Press, pp 75–93

    Google Scholar 

  16. Craig W, Wayne CE (1993) Newton's method and periodic solutions of nonlinear wave equations. Comm Pure Appl Math 46:1409–1498

    MathSciNet  MATH  Google Scholar 

  17. Crandall MG, Rabinowitz PH (1971) Bifurcation from simple eigenvalues. J Funct Anal 8:321–340

    MathSciNet  MATH  Google Scholar 

  18. Crandall MG, Rabinowitz PH (1978) The Hopf bifurcation theorem in infinite dimensions. Arch Rat Mech Anal 67:53–72

    MathSciNet  Google Scholar 

  19. Cronin J (1950) Branch points of solutions of equations in Banach spaces. Trans Amer Math Soc 69:208–231; (1954) 76:207–222

    MathSciNet  MATH  Google Scholar 

  20. Da Prato G, Lunardi A (1986) Hopf bifurcation for fully nonlinear equations in Banach space. Ann Inst Henri Poincaré 3:315–329

    MathSciNet  MATH  Google Scholar 

  21. Friedrichs KO (1953–54) Special Topics in Analysis. Lecture Notes. New York University, New York

    Google Scholar 

  22. Gaines R, Mawhin J (1977) Coincidence Degree, and Nonlinear Differential Equations. Lect Notes in Math, vol 568. Springer, Berlin

    Google Scholar 

  23. Golubitsky M, Schaeffer DG (1985) Singularities and Groups in Bifurcation Theory, vol 1. Appl Math Sci, vol 51. Springer, New York

    Google Scholar 

  24. Golubitsky M, Stewart IN, Schaeffer DG (1988) Singularities and Groups in Bifurcation Theory, vol 2. Appl Math Sci, vol 69. Springer, New York

    Google Scholar 

  25. Golubitsky M, Pivato M, Stewart I (2004) Interior symmetry and local bifurcation in coupled cell networks. Dyn Syst 19:389–407

    MathSciNet  MATH  Google Scholar 

  26. Hale JK (1954) Periodic solutions of non-linear systems of differential equations. Riv Mat Univ Parma 5:281–311

    MathSciNet  MATH  Google Scholar 

  27. Hale JK (1963) Oscillations in nonlinear systems. McGraw-Hill, New York

    MATH  Google Scholar 

  28. Hale JK (1969) Ordinary Differential Equations. Wiley, New York

    MATH  Google Scholar 

  29. Hale JK (1977) Theory of Functional Differential Equations. Appl Math Sciences, vol 3. Springer, New York

    Google Scholar 

  30. Hale JK (1984) Introduction to dynamic bifurcation. In: Salvadori L (ed) Bifurcation Theory and Applications. Lecture Notes in Math, vol 1057. Springer, Berlin, pp 106–151

    Google Scholar 

  31. Hale JK, Sakamoto K (1988) Existence and stability of transition layers. Japan J Appl Math 5:367–405

    MathSciNet  MATH  Google Scholar 

  32. Hale JK, Sakamoto K (2005) A Lyapunov–Schmidt method for transition layers in reaction-diffusion systems. Hiroshima Math J 35:205–249

    MathSciNet  MATH  Google Scholar 

  33. Hall WS (1970) Periodic solutions of a class of weakly nonlinear evolution equations. Arch Rat Mech Anal 39:294–332

    MATH  Google Scholar 

  34. Hall WS (1970) On the existence of periodic solutions of the equation \( { D_{tt}u + (-1)^pD^{2p}_xu = \varepsilon f(t,x,u) } \). J Differ Equ 7:509–526

    ADS  MATH  Google Scholar 

  35. Hopf E (1942) Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber Math Phys Kl Sachs Acad Wiss Leipzig 94:1–22. English translation in: Marsden J, McCracken M (1976) Hopf Bifurcation and its Applications. Springer, New York

    Google Scholar 

  36. Iooss G, Joseph D (1981) Elementary Stability and Bifurcation Theory. Springer, New York

    Google Scholar 

  37. Ize J (1976) Bifurcation theory for Fredholm operators. Mem AMS 174

    Google Scholar 

  38. Jepson A, Spence A (1989) On a reduction process for nonlinear equations. SIAM J Math Anal 20:39–56

    MathSciNet  MATH  Google Scholar 

  39. Kielhöfer H (2004) Bifurcation Theory, An Introduction with Applications to PDEs. Appl Math Sciences, vol 156. Springer, New York

    Google Scholar 

  40. Knobloch J (2000) Lin's method for discrete dynamical systems. J Differ Eq Appl 6:577–623

    MathSciNet  MATH  Google Scholar 

  41. Knobloch J, Vanderbauwhede A (1996) A general reduction method for periodic solutions in conservative and reversible systems. J Dyn Diff Eqn 8:71–102

    MathSciNet  MATH  Google Scholar 

  42. Knobloch J, Vanderbauwhede A (1996) Hopf bifurcation at k-fold resonances in conservative systems. In: Broer HW, van Gils SA, Hoveijn I, Takens F (eds) Nonlinear Dynamical Systems and Chaos. Progress in Nonlinear Differential Equations and Their Applications, vol 19. Birkhäuser, Basel, pp 155–170

    Google Scholar 

  43. Krasnosel'skii MA (1963) Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon Press, Oxford

    Google Scholar 

  44. Krasnosel'skii MA, Vainiko GM, Zabreiko PP, Rutitskii YB, Stesenko VY (1972) Approximate solutions of operator equations. Wolters-Noordhoff, Groningen

    Google Scholar 

  45. Kuznetsov YA (1998) Elements of Applied Bifurcation Theory, 2nd edn. Appl Math Sci, vol 112. Springer, New York

    Google Scholar 

  46. Lewis DC (1956) On the role of first integrals in the perturbations of periodic solutions. Ann Math 63:535–548

    MATH  Google Scholar 

  47. Lin XB (1990) Using Melnikov's method to solve Silnikov's problems. Proc Roy Soc Edinb 116A:295–325

    Google Scholar 

  48. Lyapunov AM (1906) Sur les figures d'équilibre peu différentes des ellipsoï, des d'une masse liquide homogène dotées d'un mouvement de rotation. Zap Akad Nauk St Petersb, pp 1–225; (1908), pp 1–175; (1912), pp 1–228; (1914), pp 1–112

    Google Scholar 

  49. Mawhin J (1969) Degré topologique et solutions périodiques des systèmes différentiels non linéaires. Bull Soc Roy Sci Liège 38:308–398

    MathSciNet  ADS  MATH  Google Scholar 

  50. Nirenberg L (1960–61) Functional Analysis. Lecture Notes. New York University, New York

    Google Scholar 

  51. Rabinowitz PH (1967) Periodic solutions of nonlinear hyperbolic partial differential equations. Comm Pure Appl Math 20:145–206

    MathSciNet  MATH  Google Scholar 

  52. Rabinowitz PH (1976) A Survey of Bifurcation Theory. In: Cesari L, Hale JK, LaSalle J (eds) Dynamical Systems, An International Symposium, vol 1. Academic Press, New York, pp 83–96

    Google Scholar 

  53. Sandstede B (1993) Verzweigungstheorie homokliner Verdopplungen. Ph?D Thesis, University of Stuttgart, IAAS Report No. 7

    Google Scholar 

  54. Sattinger DH (1979) Group Theoretic Methods in Bifurcation Theory. Lecture Notes in Math, vol 762. Springer, Berlin

    Google Scholar 

  55. Schmidt E (1908) Zur Theorie des linearen und nichtlinearen Integralgleichungen, 3 Teil. Über die Auflösung der nichtlinearen Integralgleichungen und die Verzweigung ihrer Lösungen. Math Ann 65:370–399

    MathSciNet  MATH  Google Scholar 

  56. Shimizu T (1948) Analytic operations and analytic operational equations. Math Japon 1:36–40

    MATH  Google Scholar 

  57. Sidorov N, Loginov B, Sinitsyn A, Falaleev M (2002) Lyapunov–Schmidt Methods in Nonlinear Analysis and Applications. Mathematics and Its Applications, vol 550. Kluwer, Dordrecht

    Google Scholar 

  58. Taniguchi M (1995) A Remark on Singular Perturbation Methods via the Lyapunov–Schmidt Reduction. Publ RIMS, Kyoto Univ 31:1001–1010

    MATH  Google Scholar 

  59. Vainberg MM, Trenogin VA (1962) The method of Lyapunov and Schmidt in the theory of nonlinear differential equations and their further development. Russ Math Surv 17:1–60

    MathSciNet  MATH  Google Scholar 

  60. Vainberg MM, Trenogin VA (1974) Theory of Branching of Solutions of Non-Linear Equations. Noordhoff, Leyden

    MATH  Google Scholar 

  61. Vanderbauwhede A (1982) Local bifurcation and symmetry. Research Notes in Mathematics, vol 75. Pitman, London

    Google Scholar 

  62. Vanderbauwhede A (2000) Subharmonic bifurcation at multiple resonances. In: Elaydi S, Allen F, Elkhader A, Mughrabi T, Saleh M (eds) Proceedings of the Mathematics Conference, Birzeit, August 1998. World Scientific, Singapore, pp 254–276

    Google Scholar 

  63. Vanderbauwhede A, Van der Meer JC (1995) A general reduction method for periodic solutions near equilibria in Hamiltonian systems. In: Langford WF, Nagata W (eds) Normal Forms and Homoclinic Chaos. Fields Institute Comm. AMS, Providence, pp 273–294

    Google Scholar 

Books and Reviews

  1. Alligood KT, Sauer T, Yorke JA (1997) Chaos, An Introduction to Dynamical Systems. Springer, New York

    Google Scholar 

  2. Chicone C (2006) Ordinary Differential Equations with Applications, 2nd edn. Texts in Applied Math, vol 34. Springer, New York

    Google Scholar 

  3. Govaerts W (2000) Numerical Methods for Bifurcations of Dynamical Equilibria. SIAM Publications, Philadelphia

    MATH  Google Scholar 

  4. Hale JK, Koçak H (1991) Dynamics and Bifurcations. Texts in Applied Mathematics, vol 3. Springer, New York

    Google Scholar 

  5. Ma T, Wang S (2005) Bifurcation Theory and Applications. World Scientific Series on Nonlinear Science, vol A53. World Scientific, Singapore

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Pure Mathematics and Computer Algebra, Ghent University, Gent, Belgium

    André Vanderbauwhede

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  1. André Vanderbauwhede
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Vanderbauwhede, A. (2009). Lyapunov–Schmidt Method for Dynamical Systems. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_314

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