Skip to main content
SpringerLink
Log in

Center Manifolds

  • Reference work entry
Mathematics of Complexity and Dynamical Systems

Article Outline

Glossary

Definition of the Subject

Introduction

Center Manifold in Ordinary Differential Equations

Center Manifold in Discrete Dynamical Systems

Normally Hyperbolic Invariant Manifolds

Applications

Center Manifold in Infinite-Dimensional Space

Future Directions

Bibliography

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 600.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Abbreviations

Bifurcation:

Bifurcation is a qualitative change of the phase portrait. The term “bifurcation” was introduced by H. Poincaré.

Continuous and discrete dynamical systems:

A dynamical system is a mapping \({X(t,x)}\), \({t\in R}\) or \({t \in Z}\), \({x \in E}\) which satisfies the group property \({X(t+s,x)} {=X(t,X(s,x))}\). The dynamical system is continuous or discrete when t takes real or integer values, respectively. The continuous system is generated by an autonomous system of ordinary differential equations

$$ \dot x\equiv \frac{\mathrm{d}x}{\mathrm{d}t}=F(x) $$
(1)

as the solution X(t, x) with the initial condition \({X(0,x)} {=x}\). The discrete system generated by a system of difference equations

$$ x_{m+1}=G(x_{m}) $$
(2)

as \({X(n,x)=G^n(x)}\). The phase space E is the Euclidean or Banach.

Critical part of the spectrum :

Critical part of the spectrum for a differential equation \({\dot x=Ax}\) is σ c  = {eigenvalues of A with zero real part}. Critical part of the spectrum for a diffeomorphism \({x\to Ax}\) is σ c  = {eigenvalues of A with modulus equal to 1}.

Eigenvalue and spectrum:

If for a matrix (linear mapping) A the equality \({Av=\lambda v}\), \({v\ne 0}\) holds then v and λ are called eigenvector and eigenvalue of A. The set of the eigenvalues is the spectrum of A. If there exists k such that \({(A-\lambda I)^kv=0}\), v is said to be generating vector.

Equivalence of dynamical systems:

Two dynamical systems f and g are topologically equivalent if there is a continuous one-to-one correspondence (homeomorphism) that maps trajectories (orbits) of f on trajectories of \({g.}\) It should be emphasized that the homeomorphism need not be differentiable.

Invariant manifold:

In applications an invariant manifold arises as a surface such that the trajectories starting on the surface remain on it under the system evolution.

Local properties:

If \({F(p)=0}\), the point p is an equilibrium of (1). If \({G(q)=q, \, q}\) is a fixed point of (2). We study dynamics near an equilibrium or a fixed point of the system. Thus we consider the system in a neighborhood of the origin which is supposed to be an equilibrium or a fixed point. In this connection we use terminology local invariant manifold or local topological equivalence.

Reduction principle:

In accordance with this principle a locally invariant (center) manifold corresponds to the critical part of the spectrum of the linearized system. The behavior of orbits on a center manifold determines the dynamics of the system in a neighborhood of its equilibrium or fixed point. The term “reduction principle” was introduced by V. Pliss [59].

Bibliography

Primary Literature

  1. Arnold VI (1973) Ordinary Differential Equations. MIT, Cambridge

    MATH  Google Scholar 

  2. Auchmuty J, Nicolis G (1976) Bifurcation analysis of reaction-diffusion equation (III). Chemical Oscilations. Bull Math Biol 38:325–350

    MathSciNet  MATH  Google Scholar 

  3. Aulbach B (1982) A reduction principle for nonautonomous differential equations. Arch Math 39:217–232

    Article  MathSciNet  MATH  Google Scholar 

  4. Aulbach B, Garay B (1994) Partial linearization for noninvertible mappings. J Appl Math Phys (ZAMP) 45:505–542

    Article  MathSciNet  MATH  Google Scholar 

  5. Aulbach B, Colonius F (eds) (1996) Six Lectures on Dynamical Systems. World Scientific, New York

    MATH  Google Scholar 

  6. Balakrishnan AV (1976) Applied Functional Analysis. Springer, New York, Heidelberg

    MATH  Google Scholar 

  7. Bates P, Jones C (1989) Invariant manifolds for semilinear partial differential equations. In: Dynamics Reported 2. Wiley, Chichester, pp 1–38

    Chapter  Google Scholar 

  8. Bogoliubov NN, Mitropolsky YUA (1963) The method of integral manifolds in non-linear mechanics. In: Contributions Differential Equations 2. Wiley, New York, pp 123–196

    Google Scholar 

  9. Caraballo T, Chueshov I, Landa J (2005) Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations. Nonlinearity 18:747–767

    Article  MathSciNet  MATH  Google Scholar 

  10. Carr J (1981) Applications of Center Manifold Theory. In: Applied Mathematical Sciences, vol 35. Springer, New York

    Google Scholar 

  11. Carr J, Muncaster RG (1983) Applications of center manifold theory to amplitude expansions. J Diff Equ 59:260–288

    Article  MathSciNet  Google Scholar 

  12. Chicone C, Latushkin Yu (1999) Evolution Semigroups in Dynamical Systems and Differential Equations. Math Surv Monogr 70. Amer Math Soc, Providene

    Google Scholar 

  13. Chow SN, Lu K (1995) Invariant manifolds and foliations for quasiperiodic systems. J Diff Equ 117:1–27

    Article  MathSciNet  MATH  Google Scholar 

  14. Chueshov I (2007) Invariant manifolds and nonlinear master-slave synchronization in coupled systems. In: Applicable Analysis, vol 86, 3rd edn. Taylor and Francis, London, pp 269–286

    Google Scholar 

  15. Chueshov I (2004) A reduction principle for coupled nonlinesr parabolic-hyperbolic PDE. J Evol Equ 4:591–612

    Article  MathSciNet  MATH  Google Scholar 

  16. Diliberto S (1960) Perturbation theorem for periodic surfaces I, II. Rend Cir Mat Palermo 9:256–299; 10:111–161

    Google Scholar 

  17. Du A, Duan J (2006) Invariant manifold reduction for stochastic dynamical systems. http://arXiv:math.DS/0607366

    Google Scholar 

  18. Engel K, Nagel R (2000) One-parameter Semigroups for Linear Evolution Equations. Springer, New York

    MATH  Google Scholar 

  19. Fenichel N (1971) Persistence and smoothness of invariant manifolds for flows. Ind Univ Math 21:193–226

    Article  MathSciNet  MATH  Google Scholar 

  20. Fenichel N (1974) Asymptotic stability with rate conditions. Ind Univ Math 23:1109–1137

    Article  MathSciNet  MATH  Google Scholar 

  21. Gallay T (1993) A center-stable manifold theorem for differential equations in Banach space. Commun Math Phys 152:249–268

    Google Scholar 

  22. Gesztesy F, Jones C, Latushkin YU, Stanislavova M (2000) A spectral mapping theorem and invariant manifolds for nonlinear Schrodinger equations. Ind Univ Math 49(1):221–243

    MathSciNet  MATH  Google Scholar 

  23. Grobman D (1959) Homeomorphism of system of differential equations. Dokl Akad Nauk SSSR 128:880 (in Russian)

    MathSciNet  Google Scholar 

  24. Grobman D (1962) The topological classification of the vicinity of a singular point in n‑dimensional space. Math USSR Sbornik 56:77–94; in Russian

    MathSciNet  Google Scholar 

  25. Guckenheimer J, Holmes P (1993) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vectors Fields. Springer, New York

    Google Scholar 

  26. Hadamard J (1901) Sur l’etaration et les solution asymptotiques des equations differentielles. Bull Soc Math France 29:224–228

    MATH  Google Scholar 

  27. Haken H (2004) Synergetics: Introduction and Advanced topics. Springer, Berlin

    Google Scholar 

  28. Hale J (1961) Integral manifolds of perturbed differential systems. Ann Math 73(2):496–531

    Article  MathSciNet  MATH  Google Scholar 

  29. Hartman P (1960) On local homeomorphisms of Euclidean spaces. Bol Soc Mat Mex 5:220

    MathSciNet  MATH  Google Scholar 

  30. Hartman P (1964) Ordinary Differential Equations. Wiley, New York

    MATH  Google Scholar 

  31. Hassard BD, Kazarinoff ND, Wan YH (1981) Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge

    Google Scholar 

  32. Henry D (1981) Geometric theory of semilinear parabolic equations. Lect Notes Math 840:348

    MathSciNet  Google Scholar 

  33. Hirsch M, Smale S (1974) Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, Orlando

    MATH  Google Scholar 

  34. Hirsch M, Pugh C, Shub M (1977) Invariant manifolds. Lect Notes Math 583:149

    MathSciNet  Google Scholar 

  35. Hopf E (1942) Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber Verh Sachs Akad Wiss Leipzig Math-Nat 94:3–22

    Google Scholar 

  36. Iooss G (1979) Bifurcation of Maps and Application. N-Holl Math Stud 36:105

    Article  MathSciNet  Google Scholar 

  37. Jolly MS, Rosa R (2005) Computation of non-smooth local centre manifolds. IMA J Numer Anal 25(4):698–725

    Article  MathSciNet  MATH  Google Scholar 

  38. Kelley A (1967) The stable, center stable, center, center unstable and unstable manifolds. J Diff Equ 3:546–570

    Article  MathSciNet  MATH  Google Scholar 

  39. Kirchgraber U, Palmer KJ (1990) Geometry in the Neighborhood of Invariant Manifolds of the Maps and Flows and Linearization. In: Pitman Research Notes in Math, vol 233. Wiley, New York

    Google Scholar 

  40. Li C, Wiggins S (1997) Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrodinger Equations. Springer, New York

    Book  Google Scholar 

  41. Lyapunov AM (1892) Problémé Générale de la Stabilité du Mouvement, original was published in Russia 1892, transtalted by Princeton Univ. Press, Princeton, 1947

    Google Scholar 

  42. Ma T, Wang S (2005) Dynamic bifurcation of nonlinear evolution equations and applications. Chin Ann Math 26(2):185–206

    Article  MATH  Google Scholar 

  43. Marsden J, McCracken M (1976) Hopf bifurcation and Its Applications. Appl Math Sci 19:410

    MathSciNet  Google Scholar 

  44. Mañé R (1978) Persistent manifolds are normally hyperbolic. Trans Amer Math Soc 246:261–284

    Google Scholar 

  45. Mielke A (1988) Reduction of quasilinear elliptic equations in cylindrical domains with application. Math Meth App Sci 10:51–66

    Article  MathSciNet  MATH  Google Scholar 

  46. Mielke A (1991) Locally invariant manifolds for quasilinear parabolic equations. Rocky Mt Math 21:707–714

    Article  MathSciNet  MATH  Google Scholar 

  47. Mielke A (1996) Dynamics of nonlinear waves in dissipative systems: reduction, bifurcation and stability. In: Pitman Research Notes in Mathematics Series, vol 352. Longman, Harlow, pp 277

    Google Scholar 

  48. Mitropolskii YU, Lykova O (1973) Integral Manifolds in Nonlinear Mechanics. Nauka, Moscow

    Google Scholar 

  49. Mizohata S (1973) The Theory of Partial Differential Equations. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  50. Neimark Y (1967) Integral manifolds of differential equations. Izv Vuzov, Radiophys 10:321–334 (in Russian)

    MathSciNet  Google Scholar 

  51. Osipenko G, Ershov E (1993) The necessary conditions of the preservation of an invariant manifold of an autonomous system near an equilibrium point. J Appl Math Phys (ZAMP) 44:451–468

    Article  MathSciNet  MATH  Google Scholar 

  52. Osipenko G (1996) Indestructibility of invariant non-unique manifolds. Discret Contin Dyn Syst 2(2):203–219

    Article  MathSciNet  MATH  Google Scholar 

  53. Osipenko G (1997) Linearization near a locally non-unique invariant manifold. Discret Contin Dyn Syst 3(2):189–205

    Article  MathSciNet  MATH  Google Scholar 

  54. Palis J, Takens F (1977) Topological equivalence of normally hyperbolic dynamical systems. Topology 16(4):336–346

    Article  MathSciNet  Google Scholar 

  55. Palmer K (1975) Linearization near an integral manifold. Math Anal Appl 51:243–255

    Article  MathSciNet  MATH  Google Scholar 

  56. Palmer K (1987) On the stability of center manifold. J Appl Math Phys (ZAMP) 38:273–278

    Article  MathSciNet  MATH  Google Scholar 

  57. Perron O (1928) Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystem. Math Z 29:129–160

    Article  MathSciNet  MATH  Google Scholar 

  58. Pillet CA, Wayne CE (1997) Invariant manifolds for a class of dispersive. Hamiltonian, partial differential equations. J Diff Equ 141:310–326

    Article  MathSciNet  MATH  Google Scholar 

  59. Pliss VA (1964) The reduction principle in the theory of stability of motion. Izv Acad Nauk SSSR Ser Mat 28:1297–1324; translated (1964) In: Soviet Math 5:247–250

    MATH  Google Scholar 

  60. Pliss VA (1966) On the theory of invariant surfaces. In: Differential Equations, vol 2. Nauka, Moscow pp 1139–1150

    Google Scholar 

  61. Poincaré H (1885) Sur les courbes definies par une equation differentielle. J Math Pure Appl 4(1):167–244

    Google Scholar 

  62. Pugh C, Shub M (1970) Linearization of normally hyperbolic diffeomorphisms and flows. Invent Math 10:187–198

    Article  MathSciNet  MATH  Google Scholar 

  63. Reinfelds A (1974) A reduction theorem. J Diff Equ 10:645–649

    Google Scholar 

  64. Reinfelds A (1994) The reduction principle for discrete dynamical and semidynamical systems in metric spaces. J Appl Math Phys (ZAMP) 45:933–955

    Article  MathSciNet  MATH  Google Scholar 

  65. Renardy M (1994) On the linear stability of hyperbolic PDEs and viscoelastic flows. J Appl Math Phys (ZAMP) 45:854–865

    Article  MathSciNet  MATH  Google Scholar 

  66. Sacker RJ (1967) Hale J, LaSalle J (eds) A perturbation theorem for invariant Riemannian manifolds. Proc Symp Diff Equ Dyn Syst Univ Puerto Rico. Academic Press, New York, pp 43–54

    Google Scholar 

  67. Sandstede B, Scheel A, Wulff C (1999) Bifurcations and dynamics of spiral waves. J Nonlinear Sci 9(4):439–478

    Article  MathSciNet  MATH  Google Scholar 

  68. Shoshitaishvili AN (1972) Bifurcations of topological type at singular points of parameterized vector fields. Func Anal Appl 6:169–170

    Article  MATH  Google Scholar 

  69. Shoshitaishvili AN (1975) Bifurcations of topological type of a vector field near a singular point. Trudy Petrovsky seminar, vol 1. Moscow University Press, Moscow, pp 279–309

    Google Scholar 

  70. Sijbrand J (1985) Properties of center manifolds. Trans Amer Math Soc 289:431–469

    Article  MathSciNet  MATH  Google Scholar 

  71. Van Strien SJ (1979) Center manifolds are not \({C^{\infty}}\). Math Z 166:143–145

    Article  MathSciNet  MATH  Google Scholar 

  72. Vanderbauwhede A (1989) Center Manifolds, Normal Forms and Elementary Bifurcations. In: Dynamics Reported, vol 2. Springer, Berlin, pp 89–169

    Chapter  Google Scholar 

  73. Vanderbauwhede A, Iooss G (1992) Center manifold theory in infinite dimensions. In: Dynamics Reported, vol 1. Springer, Berlin, pp 125–163

    Chapter  Google Scholar 

  74. Wan YH (1977) On the uniqueness of invariant manifolds. J Diff Equ 24:268–273

    Article  MATH  Google Scholar 

  75. Wang W, Duan J (2006) Invariant manifold reduction and bifurcation for stochastic partial differential equations. http://arXiv:math.DS/0607050

    Google Scholar 

  76. Wiggins S (1992) Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York

    Google Scholar 

  77. Wiggins S (1994) Normally Hyperbolic Invariant Manifolds of Dynamical Systems. Springer, New York

    Google Scholar 

  78. Wulff C (2000) Translation from relative equilibria to relative periodic orbits. Doc Mat 5:227–274

    MathSciNet  Google Scholar 

Books and Reviews

  1. Bates PW, Lu K, Zeng C (1998) Existence and persistence of invariant manifolds for semiflows in Banach spaces. Mem Amer Math Soc 135:129

    MathSciNet  Google Scholar 

  2. Bates PW, Lu K, Zeng C (1999) Persistence of overflowing manifolds for semiflow. Comm Pure Appl Math 52(8):983–1046

    Article  MathSciNet  Google Scholar 

  3. Bates PW, Lu K, Zeng C (2000) Invariant filiations near normally hyperbolic invariant manifolds for semiflows. Trans Amer Math Soc 352:4641–4676

    Article  MathSciNet  MATH  Google Scholar 

  4. Babin AV, Vishik MI (1989) Attractors for Evolution Equations. Nauka. Moscow; English translation (1992). Elsevier Science, Amsterdam

    Google Scholar 

  5. Bylov VF, Vinograd RE, Grobman DM, Nemyskiy VV (1966) The Theory of Lyapunov Exponents. Nauka, Moscow (in Russian)

    Google Scholar 

  6. Chen X-Y, Hale J, Tan B (1997) Invariant foliations for C 1 semigroups in Banach spaces. J Diff Equ 139:283–318

    Article  MathSciNet  MATH  Google Scholar 

  7. Chepyzhov VV, Goritsky AYU, Vishik MI (2005) Integral manifolds and attractors with exponential rate for nonautonomous hyperbolic equations with dissipation. Russ J Math Phys 12(1):17–39

    MathSciNet  MATH  Google Scholar 

  8. Chicone C, Latushkin YU (1997) Center manifolds for infinite dimensional nonautonomous differential equations. J Diff Equ 141:356–399

    Article  MathSciNet  MATH  Google Scholar 

  9. Chow SN, Lu K (1988) Invariant manifolds for flows in Banach spaces. J Diff Equ 74:285–317

    Article  MathSciNet  MATH  Google Scholar 

  10. Chow SN, Lin XB, Lu K (1991) Smooth invariant foliations in infinite dimensional spaces. J Diff Equ 94:266–291

    Article  MathSciNet  MATH  Google Scholar 

  11. Chueshov I (1993) Global attractors for non-linear problems of mathematical physics. Uspekhi Mat Nauk 48(3):135–162; English translation in: Russ Math Surv 48:3

    MathSciNet  Google Scholar 

  12. Chueshov I (1999) Introduction to the Theory of Infinite‐Dimensional Dissipative Systems. Acta, Kharkov (in Russian); English translation (2002) http://www.emis.de/monographs/Chueshov/. Acta, Kharkov

  13. Constantin P, Foias C, Nicolaenko B, Temam R (1989) Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations. Appl Math Sci, vol 70. Springer, New York

    Book  Google Scholar 

  14. Gonçalves JB (1993) Invariant manifolds of a differentiable vector field. Port Math 50(4):497–505

    Google Scholar 

  15. Goritskii AYU, Chepyzhov VV (2005) Dichotomy property of solutions of quasilinear equations in problems on inertial manifolds. SB Math 196(4):485–511

    Article  MathSciNet  Google Scholar 

  16. Hassard B, Wan Y (1978) Bifurcation formulae derived from center manifold theory. J Math Anal Appl 63:297–312

    Article  MathSciNet  MATH  Google Scholar 

  17. Hsia C, Ma T, Wang S (2006) Attractor bifurcation of three‐dimensional double-diffusive convection. http://arXiv:nlin.PS/0611024

    Google Scholar 

  18. Knobloch HW (1990) Construction of center manifolds. J Appl Math Phys (ZAMP) 70(7):215–233

    MathSciNet  MATH  Google Scholar 

  19. Latushkin Y, Li Y, Stanislavova M (2004) The spectrum of a linearized 2D Euler operator. Stud Appl Math 112:259

    Article  MathSciNet  MATH  Google Scholar 

  20. Leen TK (1993) A coordinate independent center manifold reduction. Phys Lett A 174:89–93

    Article  MathSciNet  Google Scholar 

  21. Li Y (2005) Invariant manifolds and their zero-viscosity limits for Navier–Stokes equations. http://arXiv:math.AP/0505390

    Google Scholar 

  22. Osipenko G (1989) Examples of perturbations of invariant manifolds. Diff Equ 25:675–681

    MathSciNet  MATH  Google Scholar 

  23. Osipenko G (1985, 1987, 1988) Perturbation of invariant manifolds I, II, III, IV. Diff Equ 21:406–412, 21:908–914, 23:556–561, 24:647–652

    Google Scholar 

  24. Podvigina OM (2006) The center manifold theorem for center eigenvalues with non-zero real parts. http://arXiv:physics/0601074

    Google Scholar 

  25. Sacker RJ, Sell GR (1974, 1976, 1978) Existence of dichotomies and invariant splitting for linear differential systems. J Diff Equ 15:429-458, 22:478–522, 27:106–137

    Google Scholar 

  26. Scarpellini B (1991) Center manifolds of infinite dimensional. Main results and applications. J Appl Math Phys (ZAMP) 43:1–32

    Google Scholar 

  27. Sell GR (1983) Vector fields on the vicinity of a compact invariant manifold. Lect Notes Math 1017:568–574

    Article  MathSciNet  Google Scholar 

  28. Swanson R (1983) The spectral characterization of normal hyperbolicity. Proc Am Math Soc 89(3):503–508

    Article  MathSciNet  MATH  Google Scholar 

  29. Temam R (1988) Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, Berlin

    Book  MATH  Google Scholar 

  30. Zhenquan Li, Roberts AJ (2000) A flexible error estimate for the application of center manifold theory. http://arXiv.org/abs/math.DS/0002138

  31. Zhu H, Campbell SA, Wolkowicz (2002) Bifurcation analysis of a predator-prey system with nonmonotonic functional response. SIAM J Appl Math 63:636–682

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Polytechnic University, St. Petersburg, Russia

    George Osipenko

Authors
  1. George Osipenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

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

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

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag

About this entry

Cite this entry

Osipenko, G. (2012). Center Manifolds. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_5

Download citation

Publish with us

Policies and ethics

Search

Navigation