Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Global Analysis on Manifolds

  • Chapter
Linear, Time-varying Approximations to Nonlinear Dynamical Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 411))

  • 1571 Accesses

Introduction

In this chapter we shall consider the non-local theory of systems – i.e. the theory of sections of the tangent bundle of a differentiable manifold which are called vector fields. Most control systems are described in terms of local operating points, i.e. they are linearised about some equilibrium point and then local feedback control is applied to hold this system ‘near’ this point. (Such is the case, for example, with aircraft systems, where the operating point is called a ‘trim condition’.)

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 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Perko, L.: Differential equations and dynamical systems. Springer, New York (1991)

    MATH  Google Scholar 

  2. Banks, S.P., Song, Y.: Elliptic and automorphic dynamical systems on surfaces. Int. J. of Bifurcation and Chaos 16(4), 911–923 (2006)

    Article  MATH  Google Scholar 

  3. Banks, S.P.: Three-dimensional stratifications, knots and bifurcations of two-dimensional dynamical systems. Int. J. of Bifurcation and Chaos 12(1), 1–21 (2002)

    Article  MATH  Google Scholar 

  4. Jost, J.: Dynamical systems - examples of conplex behavior. Springer, Berlin (2005)

    Google Scholar 

  5. Martins, R.: The effect of inversely unstable solutions on the attractor of the forced pendulum equation with friction. J. of Differential Equations 212(2), 351–365 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Song, Y., Banks, S.P.: Inversely Unstable Solutions of Two-Dimensional Systems on Genus-p Surfaces and the Topology of Knotted Attractors. Int. J. of Bifurcation and Chaos (in print)

    Google Scholar 

  7. Markov, A.A.: Insolubility of the problem of homeomorphy. Proc. Inc. Cong. Math., pp. 300–306. Cambridge University Press, Cambridge (1958)

    Google Scholar 

  8. Chari, V., Pressley, A.: A guide to quantum groups. CUP, Cambridge (1994)

    MATH  Google Scholar 

  9. Kassel, C.: Quantum groups. Springer, Heidelberg (1995)

    MATH  Google Scholar 

  10. Lickorish, W.B.R.: A representation of orientable combinatorial 3-manifolds. Ann. of Math. 76, 531–540 (1962)

    Article  MathSciNet  Google Scholar 

  11. Wallace, A.D.: Modifications and cobounding manifolds. Can. J. Math. 12, 503–528 (1960)

    MathSciNet  Google Scholar 

  12. Rolfsen, D.: Knots and links. Publish or Perish, Berkeley (1976)

    MATH  Google Scholar 

  13. Wade, M.: Closed orbits of non-singular Morse-Smale flows on S 3’. J. Math. Soc. Jap. 41(3), 405–413 (1989)

    Article  MathSciNet  Google Scholar 

  14. Franks, J.: Non-singular Smale flows on S 3. Topology 24, 265–282 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hempel, J.: 3-manifolds. Ann. of Math. Studies, vol. 86. Princeton University Press, Princeton (1976)

    MATH  Google Scholar 

  16. Song, Y., Banks, S.P., Diaz, D.: Dynamical Systems on Three Manifolds–Part I: Knots, Links and Chaos. Int. J. of Bifurcation and Chaos 17(6), 2073–2084 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Song, Y., Banks, S.P.: Dynamical Systems On Three Manifolds–Part II: 3-Manifolds, Heegaard Splittings and Three-Dimensional Systems. Int. J. of Bifurcation and Chaos 17(6), 2085–2095 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Jiang, B., Ni, Y., Wang, S.: 3-manifolds that admit knotted solenoids as attractors. Trans. AMS 356(11), 4371–4382 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Song, Y., Banks, S.P.: Generalized Smale solenoids on 3-manifolds (2009) (preprint)

    Google Scholar 

  20. Montesinos, J.: A representation of closed, orientable 3-manifolds as 3-fold branched coverings of S 3. Bull. Amer. Math. Soc. 80, 845–846 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  21. Thurston, W.: On the geometry and dynamics of diffeomrphisms of surfaces. Bull. AMS 19, 417–431 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ghrist, R.W., Holmes, P.J., Sullivan, M.C.: Knots and links in three-dimensional flows. LNM, vol. 1654. Springer, Heidelberg (1997)

    MATH  Google Scholar 

  23. Birman, J., Williams, R.: Knotted periodic orbits in dynamical systems - I: Lorenz’s equations. Topology 22(1), 47–82 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  24. Chen, W., Banks, S.P.: Branched manifolds, knotted surfaces and dynamical systems. Int. J. Bir. and Chaos (2008) (to appear)

    Google Scholar 

  25. Milnor, J.: Lectures on the h-cobordism theorem. Princeton Univ. Press, Princeton (1965)

    MATH  Google Scholar 

  26. Gompf, R.: Three exotic ℝ4’s and other anomalies. J. Diff. Geom. 18, 317–328 (1983)

    MATH  MathSciNet  Google Scholar 

  27. Donaldson, S.: An application of gauge theory to four-dimensional topology. J. Diff. Geom. 18, 279–315 (1988)

    MathSciNet  Google Scholar 

  28. Donaldson, S.: Polynomial invariants for smooth 4-manifolds. Topology 29, 257–315 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  29. Moore, J.D.: Lectures on Seiberg-Witten invariants. Springer, New York (1996)

    MATH  Google Scholar 

  30. Fintushel, R., Stern, R.J.: Knots, links and 4-manifolds. Invent. Math. 134, 363–400 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  31. Etgu, T., Doug Park, B.: Non-isotopic symplectic tori in the same homology class. Trans. AMS 359(9), 3739–3750 (2003)

    Google Scholar 

  32. Gompf, R.: A new construction of symplectic manifolds. Ann. Math. 142(3), 527–595 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  33. Montesinos, J.: A note on moves and irregular coverings of S 4. Contemp. Math. 44, 345–349 (1985)

    MathSciNet  Google Scholar 

  34. Piergallini, R.: Four-manifolds as 4-fold branched covers of S 4. Topology 34(3), 497–508 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  35. Smale, S.: Differentiable dynamical systems. Bull. AMS 73, 747–817 (1967)

    Article  MathSciNet  Google Scholar 

Download references

Authors
  1. María Tomás-Rodríguez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stephen P. Banks
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer London

About this chapter

Cite this chapter

Tomás-Rodríguez, M., Banks, S.P. (2010). Global Analysis on Manifolds. In: Linear, Time-varying Approximations to Nonlinear Dynamical Systems. Lecture Notes in Control and Information Sciences, vol 411. Springer, London. https://doi.org/10.1007/978-1-84996-101-1_11

Download citation

  • DOI: https://doi.org/10.1007/978-1-84996-101-1_11

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-84996-100-4

  • Online ISBN: 978-1-84996-101-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation