Stability and branching of normal oscillation forms of nonlinear systems

Non-Invertible Temporal Transformations and Power Series Periodic Solutions

“…Namely, di+erentiation, integration or any su.ciently smooth function of representation (3) gives an element of the same two-component structure as equation (3). These properties are simply dictated by the fact that none of the operations listed above will destroy the periodicity of the function, and hence, Proposition 1 can be applied to the result of the operations as well.…”

“…Let us represent a periodic solution in the form (3). By taking into account trigonometric identities (7) on each step of the transformation, and collecting separately terms with common factor e, one obtains…”

“…X O #(a#b!2b ) X"0, which admits periodic solutions in terms of the power series with respect to the new temporal argument " ")1 [2]. Transformation (1) with the power series methods were employed for non-linear vibrating systems as well [3,4]. Non-harmonic time transformations dealing with Jacobian functions can also be found in the literature [5].…”

Normal vibrations of a general class of conservative oscillators

“…Let us assume that the variational equation set may be 'split' into n independent equations by an invertible linear transformation with constant coefficients. The possibility of such 'splitting' is discussed in more detail in [21,22].…”

“…If 0 x 1 x 2 is a homogeneous even function, (35) may even be reduced to a hypergeometric equation by the substitution x p z, where p is the degree of homogeneity [22]. If the potential energy 0 x x 2 contains terms of second and fourth powers of x and x 2 , (35) is the Lame equation [23].…”