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Hill Equation: From 1 to 2 Degrees of Freedom

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New Perspectives and Applications of Modern Control Theory

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.

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Notes

  1. 1.

    When the pendulum is assumed a mass M hanging of rigid massless rod.

  2. 2.

    More well known as Lord Rayleigh, more correctly Baron Rayleigh because Baron is a higher novelty title than Lord.

  3. 3.

    The name Arnold Tongues was introduced after [2].

  4. 4.

    Matriciant in the russian literature [1]. Also denominated as Cauchy Matrix or Normalized Fundamental Matrix.

  5. 5.

    The necessary and sufficient condition for R to be real is that the real negative eigenvalues of \(\varPhi \left( T,0\right) \), be of algebraic multiplicity even [1].

  6. 6.

    This transformation was introduced by Lyapunov himself [35], other reference is [8].

  7. 7.

    Given a square matrix A, by \(\sigma \left( A\right) \) we denote its spectrum, i.e., the set of all the eigenvalues.

  8. 8.

    Not necessarily periodic.

  9. 9.

    We will assume through the paper that \(q\left( t\right) \) is piecewise continuous, integrable in \(\left[ 0,T\right] \) and of zero average, i.e., \( \mathop {\intop }_0^T q\left( t\right) dt=0\).

  10. 10.

    When the multipliers are \(\pm 1\) and the Monodromy matrix is diagonal, and we say that there is a point of Coexistence, because there are two linearly independent periodic solutions of Hill equation; T-periodic for multipliers \(+1\), and 2T-periodic for the multipliers equal to \(-1\).

  11. 11.

    In Magnus [36] the function that we call \(\phi \left( \alpha ,\beta \right) \), is denoted as \(\varDelta \left( \lambda \right) \), because \(\lambda \) is used instead of our \(\alpha \), and the parameter \(\beta \) is not used in the cited work.

  12. 12.

    Recall that given any skew-hermitian matrix J, then \(\left( iJ\right) \) is an hermitian matrix. [22, 48].

  13. 13.

    Equivalently, if we increase the Hamiltonian, i.e., \(\widetilde{H}\left( t\right) -H\left( t\right) >0,\) and \(\mu \) was an isolated multiplier on the unit circle associated to \(H\left( t\right) \), when \(H\left( t\right) \) is increased to \(\widetilde{H}\left( t\right) ,\) \(\mu \) moves on the unit circle to \(\widetilde{\mu }\); if \(\arg \widetilde{\mu }>\arg \mu \), the multiplier \(\mu \) is said to be a Multiplier of the First Kind, contrarily, i.e., \(\arg \widetilde{\mu }<\arg \mu \), the multiplier \(\mu \) is a Multiplier of the Second Kind [48].

  14. 14.

    In the case that the periodic function \(q\left( t\right) \) is an elliptic function, called Lamé Equation.

  15. 15.

    Chulaevsky [13] justifies the fact that coexistence points are exceptional ones, because: ... ‘From a topological point of view the scalar matrices, which correspond to coexistence points, form a subvariety in the variety of \(2\times 2\) Jordan Cells.’

  16. 16.

    Here \(tr\left[ M(\alpha _{0},\beta _{0},q\left( t\right) +\gamma r\left( t\right) )\right] \) refers to the trace of the Monodromy Matrix associated to \(\overset{\underset{\bullet \,\bullet }{}}{y}+\left[ \overset{}{\alpha _{0}}+\beta _{0}\left( \overset{}{q\left( t\right) }+\gamma r\left( t\right) \right) \right] y=0\).

  17. 17.

    We use \(r\measuredangle \theta \) to represent a complex number with modulus r, and argument \(\theta .\)

  18. 18.

    In [17] this property is extended to Hamiltonian systems with dissipation, strictly speaking this class of systems is not longer Hamiltonian.

  19. 19.

    There is a reduced for of a Floquet theorem, no factorization is possible, but there is a reducibility part.

  20. 20.

    A sequence \(\left\{ \ldots ,x_{k},x_{k+1}\ldots \right\} \) double infinite belongs to \(\ell _{2}\) if \(\sum \limits _{k=-\infty }^{\infty }\left| x_{k}\right| ^{2}=M<\infty \). See for instance [18].

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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.

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  1. Automatic Control Department, CINVESTAV, 07360, Mexico City, Mexico

    M. Joaquin Collado

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  1. Centro de Investigaciones Económicas, Administrativas y Sociales, Instituto Politécnico Nacional, Mexico city, Mexico

    Julio B. Clempner

  2. Department of Automatic Control, Center for Research and Advanced Studies, Mexico city, Mexico

    Wen Yu

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Joaquin Collado, M. (2018). Hill Equation: From 1 to 2 Degrees of Freedom. In: Clempner, J., Yu, W. (eds) New Perspectives and Applications of Modern Control Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-62464-8_3

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