For introduction to the intriguing topic of instabilities induced by dissipation in Hamiltonian or reversible systems, it is sufficient to consider a real polynomial
that may be thought of as a characteristic polynomial of a dissipative mechanical system with 2 degrees of freedom which is a perturbation of an ideal circulatory or a Hamiltonian system. In the ideal case the coefficients
and containing contributions from the dissipative forces vanish. Then, with
The ideal system is thus marginally stable at and unstable at . Given , consider a perturbation of simple pure imaginary roots with the parameters and
where the derivatives are taken at and . Equating the linear part to zero, we find , which yields the exact threshold of asymptotic stability
Let us use the scaling and where and and are given. Then
The fact that the threshold of instability for the bifurcation parameter of the damped system does not tend at to the ideal threshold in the limit is the famous paradox of destabilization by small damping first described by Ziegler in 1952. The discontinuity of the threshold in the limit of vanishing dissipation is a consequence of a qualitative fact established by Arnold that the codimension of a double pure imaginary eigenvalue with the Jordan block increases to 3 in a dissipative system in comparison with the ideal Hamiltonian or reversible system where it is equal to 1. At the origin the stability boundary has the Whitney umbrella singularity corresponding to a double pure imaginary eigenvalue with the Jordan block as it was established first by Bottema in 1956, in full accordance with the Arnold’s classification of generic singularities of 1971.