Advertisement

Hill Equation: From 1 to 2 Degrees of Freedom

  • M. Joaquin ColladoEmail author
Chapter

Abstract

After the introduction, in the first part of the chapter, we review some properties of the scalar Hill equation, a second-order linear ordinary differential equation with periodic coefficients. In the second part, we extend and compare the vectorial Hill equation; most of the results are confined to the case of two degrees of freedom (DOF). In both cases, we describe the equations with parameters \( \left( \alpha ,\beta \right) \), the zones of instability in the \(\alpha -\beta \) plane are called Arnold Tongues. We graphically illustrate the properties wherever it is possible with the aid of the Arnold Tongues.

Notes

Acknowledgements

The author wishes to thank to A. Rodriguez, C. Franco and M. Ramirez for a detailed revision of the first draft and also to G. Rodriguez for the computational work.

References

  1. 1.
    Adrianova, L.Y.: Introduction to Linear Systems of Differential Equations. American Mathematical Society, Providence (1995)zbMATHGoogle Scholar
  2. 2.
    Arnol’d, V.I.: Remarks on the perturbation theory for problems of Mathieu type. Russ. Math. Surv. 38(4), 215–233 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arnold, V.I.: Mathematical Methods of Classical Mechanics. Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein. Corrected reprint of the second (1989) edition. Graduate Texts in Mathematics, vol. 60 (1989)Google Scholar
  4. 4.
    Atkinson, F.V.: Discrete and Continuous Boundary Problems. Academic Press, New York (1964)zbMATHGoogle Scholar
  5. 5.
    Belokolos, E.D., Gesztesy, F., Makarov, K.A., Sakhnovich, L.A.: Matrix-valued generalizations of the theorems of Borg and Hochstadt. Lect. Notes Pure Appl. Math. 234, 1–34 (2003)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bolotin, V.V.: The Dynamic Stability of Elastic Systems. Holden- Day lnc, San Francisco (1964)zbMATHGoogle Scholar
  7. 7.
    Borg, G.: Eine umkehrung der Sturm-Liouvilleschen eigenwertaufgabe. An inverse problem of the self-adjoint Sturm-Liouville equation. Acta Math. 78(1), 1–96 (1946) (In German)Google Scholar
  8. 8.
    Brockett, R.: Finite Dimensional Linear Systems. Wiley, New York (1969)zbMATHGoogle Scholar
  9. 9.
    Broer, H., Levi, M., Simo, C.: Large scale radial stability density of Hill’s equation. Nonlinearity 26(2), 565–589 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brown, B.M., Eastham, M.S., Schmidt, K.M.: Periodic Differential Operators, vol. 228. Springer Science & Business Media, New York (2012)zbMATHGoogle Scholar
  11. 11.
    Champneys, A.: Dynamics of parametric excitation. Mathematics of Complexity and Dynamical Systems, pp. 183–204. Springer, New York (2012)CrossRefGoogle Scholar
  12. 12.
    Chen, C.T.: Linear System Theory and Design. Oxford University Press, Oxford (1998)Google Scholar
  13. 13.
    Chulaevsky, V.A.: Almost Periodic Operators and Related Nonlinear Integrable Systems. Manchester University Press, Manchester (1989)zbMATHGoogle Scholar
  14. 14.
    Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. Tata McGraw-Hill Education, New York (1955)zbMATHGoogle Scholar
  15. 15.
    Collado, J.: Critical lines in vectorial Hill’s equations. Under preparation (2017)Google Scholar
  16. 16.
    Collado, J., Jardón-Kojakhmetov, H.: Vibrational stabilization by reshaping Arnold Tongues: a numerical approach. Appl. Math. 7, 2005–2020 (2016)CrossRefGoogle Scholar
  17. 17.
    Collado, J., Ramirez, M., Franco, C.: Hamiltonian systems with dissipation and its application to attenuate vibrations. Under preparation (2017)Google Scholar
  18. 18.
    Corduneaunu, C.: Almost Periodic Oscillations and Waves. Springer, Berlin (2009)CrossRefGoogle Scholar
  19. 19.
    Eastham, M.S.P.: The Spectral Theory of Periodic Differential Equations. Scottish Academic Press, London (1973)zbMATHGoogle Scholar
  20. 20.
    Floquet, G.: Sur les équations différentielles linéaires à coefficients périodiques. Annales scientifiques de l’École normale supérieure 12, 47–88 (1883)CrossRefzbMATHGoogle Scholar
  21. 21.
    Fossen, T., Nijmeijer, H. (eds.): Parametric Resonance in Dynamical Systems. Springer Science & Business Media, New York (2011)Google Scholar
  22. 22.
    Gantmacher, F.R.: The Theory of Matrices, vol. 2. Chelsea, Providence (1959)zbMATHGoogle Scholar
  23. 23.
    Gelfand, I.M., Levitan, B.M.: On the determination of a differential equation from its spectral function. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 15(4), 309–360. English Trans. Am. Math. Soc. Trans. 1(2), 253–304, 1955 (1951)Google Scholar
  24. 24.
    Gelfand, I.M., Lidskii, V.B.: On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients. Am. Math. Soc. Transl. 2(8), 143–181 (1958)MathSciNetGoogle Scholar
  25. 25.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31. Springer Science & Business Media, New York (2006)Google Scholar
  26. 26.
    Hansen, J.: Stability diagrams for coupled Mathieu-equations. Ingenieur-Archiv 55(6), 463–473 (1985)CrossRefzbMATHGoogle Scholar
  27. 27.
    Hayashi, C.: Forced Oscillations in Non-Linear Systems. Nippon Printing and Publishing Co, Osaka (1953)Google Scholar
  28. 28.
    Hill, G.W.: On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon. Acta Math. 8(1), 1–36 (1886)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hochstadt, H.: Function theoretic properties of the discriminant of Hill’s equation. Mathematische Zeitschrift 82(3), 237–242 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Howard, J.E., MacKay, R.S.: Linear stability of symplectic maps. J. Math. Phys. 28(5), 1036–1051 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kalman, D.: Uncommon Mathematical Excursions: Polynomial and Related Realms. Dolciani Mathematical Expositions, vol. 35. Mathematical Association of America (2009)Google Scholar
  32. 32.
    Kapitsa, P.L.: Dynamic stability of the pendulum when the point of suspension is oscillating. Sov. Phys. JETP 21, 588 (In Russian). English Translation: In Collected works of P. Kapitsa (1951)Google Scholar
  33. 33.
    Khalil, H.K.: Nonlinear Systems. Prentice-Hall, Upper Saddle River (2001)Google Scholar
  34. 34.
    Krein, M.G.: Fundamental aspects of the theory of $\lambda -$ zones of stability of a canonical system of linear differential equations with periodic coefficients. To the memory of AA Andronov [in Russian], Izd. Akad. Nauk SSSR, Moscow, 414–498. English Translation: Am. Math. Soc. Transl. 120(2), 1–70 (1955)Google Scholar
  35. 35.
    Lyapunov, A.M.: The general problem of the stability of motion. Int. J. Control 55(3), 531–534. Also: Taylor and Francis (1992). From the original Ph.D. thesis submitted in Kharkov (1882)Google Scholar
  36. 36.
    Magnus, W., Winkler, S.: Hill’s Equation. Dover Phoenix Editions. Originally published by Wiley, New York 1966 and 1979 (2004)Google Scholar
  37. 37.
    Marchenko, V.A.: Sturm-Liouville operators and their applications. Kiev Izdatel Naukova Dumka. Engl. Transl. (2011). Sturm-Liouville Operators and Applications, vol. 373. American Mathematical Society (1977)Google Scholar
  38. 38.
    McKean, H.P., Van Moerbeke, P.: The spectrum of Hill’s equation. Inventiones mathematicae 30(3), 217–274 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Meyer, K., Hall, G., Offin, D.: Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, vol. 90. Springer Science & Business Media, New York (2008)zbMATHGoogle Scholar
  40. 40.
    Richards, J.A.: Modeling parametric processes–a tutorial review. Proc. IEEE 65(11), 1549–1557 (1977)CrossRefGoogle Scholar
  41. 41.
    Rodriguez, A., Collado, J.: Periodically forced Kapitza’s pendulum. In: American Control Conference (ACC), pp. 2790–2794. IEEE (2016)Google Scholar
  42. 42.
    Sanmartín Losada, J.R.: O Botafumeiro: parametric pumping in the middle age. Am. J. Phys. 52, 937–945 (1984)CrossRefGoogle Scholar
  43. 43.
    Seyranian, A.P.: Parametric resonance in mechanics: classical problems and new results. J. Syst. Des. Dyn. 2(3), 664–683 (2008)Google Scholar
  44. 44.
    Seyranian, A.P., Mailybaev, A.A.: Multiparameter Stability Theory with Mechanical Applications, vol. 13. World Scientific, Singapore (2003)Google Scholar
  45. 45.
    Staržhinskiĭ, V.M.: Survey of works on conditions of stability of the trivial solution of a system of linear differential equations with periodic coefficients. Prik. Mat. Meh. 18, 469–510 (Russian). Engl. Transl. Am. Math. Soc. Transl. Ser. 1(2) (1955)Google Scholar
  46. 46.
    Stephenson, A.: On induced stability. Lond. Edinb. Dublin Philos. Mag. J. Sci. 15(86), 233–236 (1908)CrossRefzbMATHGoogle Scholar
  47. 47.
    van der Pol, B., Strutt, M.J.O.: On the stability of the solutions of Mathieu’s equation. Lond. Edinb. Dublin Philos. Mag. J. Sci. 5(27), 18–38 (1928)CrossRefzbMATHGoogle Scholar
  48. 48.
    Yakubovich, V.A., Starzhinskii, V.M.: Linear Differential Equations with Periodic Coefficients, vol. 1 & 2. Halsted, Jerusalem (1975)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Automatic Control DepartmentCINVESTAVMexico CityMexico

Personalised recommendations