Coexistence phenomenon in autoparametric excitation of two degree of freedom systems

Quasi-doubly periodic solutions to a generalized Lamé equation

“…In [13] these recurrence relations were discussed in the context of coexistence of two linearly independent periodic solutions to (3.6).…”

“…If this is the case, the determinant of the infinite matrix in (4.3) separates into the product of determinants of a finite submatrix and an infinite matrix [13]. From setting the determinant of the finite submatrix to zero one determines the allowed values for c and from this the eigenvalues E, see remark 3.1.(i).…”

Trigonometric simplification of a class of conservative nonlinear oscillators

“…It is also well known that a transformation of time from t to τ permits the solution (2) to be written in a simplified form, namely [3] x(τ ) = A cos τ (4) where t and τ are related by the Equation [4] dt = dτ…”

“…As an example, consider the question of the stability of a nonlinear normal mode (NNM) in a two-degree-of-freedom system which is defined by the following expressions for kinetic T and potential V energies [3]:…”

“…However, the v-equation can be simplified by using the transformation (5), replacing t by τ as independent variable. This results in the new v-Equation [3] (…”