Skip to main content
SpringerLink
Log in

Uniform Weak Disconjugacy and Principal Solutions for Linear Hamiltonian Systems

  • Conference paper
  • First Online:
Recent Advances in Delay Differential and Difference Equations

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 94))

Abstract

The paper analyzes the property of (uniform) weak disconjugacy for nonautonomous linear Hamiltonian systems, showing that it is a convenient replacement for the more restrictive property of disconjugacy. In particular, its occurrence ensures the existence of principal solutions. The analysis of the properties of these solutions provides ample information about the dynamics induced by the Hamiltonian system on the Lagrange bundle.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
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

References

  1. Coppel, W.A.: Disconjugacy. Lecture Notes in Mathematics 220, Springer, Berlin/Heidelberg, New York (1971)

    Google Scholar 

  2. Fabbri, R., Johnson, R., Novo, S., Núñez, C.: Some remarks concerning weakly disconjugate linear Hamiltonian systems. J. Math. Anal. Appl. 380, 853–864 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Fabbri, R., Johnson, R., Núñez, C.: Rotation number for non-autonomous linear Hamiltonian systems I: Basic properties. Z. Angew. Math. Phys. 53, 484–502 (2002)

    Google Scholar 

  4. Fabbri, R., Johnson, R., Núñez, C.: On the Yakubovich Frequency Theorem for linear non-autonomous control processes. Discrete Contin. Dynam. Systems, Ser. A 9(3), 677–704 (2003)

    Google Scholar 

  5. Fabbri, R., Johnson, R., Núñez, C.: Disconjugacy and the rotation number for linear, non-autonomous Hamiltonian systems. Ann. Mat. Pura App. 185, S3–S21 (2006)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  7. Johnson, R., Nerurkar, M.: Exponential dichotomy and rotation number for linear Hamiltonian systems. J. Differ. Equat. 108, 201–216 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  8. Johnson, R., Nerurkar, M.: Controllability, stabilization, and the regulator problem for random differential systems. Mem. American Mathematical Society, vol. 646, American Mathematical Society, Providence, RI (1998)

    Google Scholar 

  9. Johnson, R., Novo, S., Obaya, R.: Ergodic properties and Weyl M-functions for linear Hamiltonian systems. Proc. Roy. Soc. Edinburgh 130A, 803–822 (2000)

    MathSciNet  Google Scholar 

  10. Johnson, R., Novo, S., Obaya, R.: An ergodic and topological approach to disconjugate linear Hamiltonian systems. Illinois J. Math. 45, 1045–1079 (2001)

    MathSciNet  Google Scholar 

  11. Johnson, R., Núñez, C.: Remarks on linear-quadratic dissipative control systems. Discrete Contin. Dynam. Systems, Ser. B, to appear.

    Google Scholar 

  12. Johnson, R., Núñez, C., Obaya, R.: Dynamical methods for linear Hamiltonian systems with applications to control processes. J. Dynam. Differ. Equat. 25(3), 679–713 (2013)

    Article  MATH  Google Scholar 

  13. Kato, T.: Perturbation theory for linear operators. Corrected Printing of the Second Edition, Springer, Berlin/Heidelberg (1995)

    MATH  Google Scholar 

  14. Lidskiĭ, V.B.: Oscillation theorems for canonical systems of differential equations. Dokl. Akad. Nank. SSSR 102, 877–880 (1955) (English translation in: NASA Technical Translation, P-14, 696)

    Google Scholar 

  15. Mañé, R.: Ergodic Theory and Differentiable Dynamics. Springer, Berlin/Heidelberg, New York (1987)

    Book  MATH  Google Scholar 

  16. Novo, S., Núñez, C., Obaya, R.: Ergodic properties and rotation number for linear Hamiltonian systems. J. Differ. Equat. 148, 148–185 (1998)

    Article  MATH  Google Scholar 

  17. Reid, W.T.: Sturmian theory for ordinary differential equations. Applied Mathematical Sciences, vol. 31, Springer, New York (1980)

    Google Scholar 

  18. Yakubovich, V.A.: Arguments on the group of symplectic matrices. Mat. Sb. 55 (97), 255–280 (1961) (Russian)

    MathSciNet  Google Scholar 

  19. Yakubovich, V.A.: Oscillatory properties of the solutions of canonical equations. Amer. Math. Soc. Transl. Ser. 2 42, 247–288 (1964)

    Google Scholar 

  20. Yakubovich, V.A., Starzhinskii, V.M.: Linear Differential Equations with Periodic Coefficients. John Wiley and Sons, Inc., New York (1975)

    MATH  Google Scholar 

Download references

Acknowledgements

Partly supported by MIUR (Italy), by MEC (Spain) under project MTM2012-30860, and by JCyL (Spain) under project VA118A12-1.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Firenze, Via di Santa Marta 3, 50139, Firenze, Italy

    Russell Johnson

  2. Departamento de Matemática Aplicada, Escuela de Ingenierías Industriales, Paseo del Cauce 59, Universidad de Valladolid, 47011, Valladolid, Spain

    Sylvia Novo, Carmen Núñez & Rafael Obaya

Authors
  1. Russell Johnson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sylvia Novo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Carmen Núñez
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Rafael Obaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rafael Obaya .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Pannonia, Veszprém, Hungary

    Ferenc Hartung

  2. Department of Mathematics, University of Pannonia, Veszprém, Hungary

    Mihály Pituk

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Johnson, R., Novo, S., Núñez, C., Obaya, R. (2014). Uniform Weak Disconjugacy and Principal Solutions for Linear Hamiltonian Systems. In: Hartung, F., Pituk, M. (eds) Recent Advances in Delay Differential and Difference Equations. Springer Proceedings in Mathematics & Statistics, vol 94. Springer, Cham. https://doi.org/10.1007/978-3-319-08251-6_5

Download citation

Publish with us

Policies and ethics

Search

Navigation