Hill Equation: From 1 to 2 Degrees of Freedom

  • M. Joaquin ColladoEmail author


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.



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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Authors and Affiliations

  1. 1.Automatic Control DepartmentCINVESTAVMexico CityMexico

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