Dissipation-induced instabilities in reversible and Hamiltonian systems

For introduction to the intriguing topic of instabilities induced by dissipation in Hamiltonian or reversible systems, it is sufficient to consider a real polynomial
$p=\lambda^4+a_1\lambda^3+a_2\lambda^2+a_3\lambda+1$
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
$a_1$ and $a_3$ containing contributions from the dissipative forces vanish. Then, $\lambda=\lambda_{id}$ with $\lambda_{id}^2=-\frac{a_2}{2}\pm\frac{1}{2}\sqrt{a_2^2-4}.$

The ideal system is thus marginally stable at $a_2>2$ and unstable at $a_2\le 2$ . Given $a_2>2$ , consider a perturbation of simple pure imaginary roots $\lambda_{id}$ with the parameters $a_1\ge 0$ and $a_3\ge 0$

$\lambda=\lambda_{id}-a_1\frac{\partial_{a_1}p}{\partial_{\lambda}p}-a_3\frac{\partial_{a_3}p}{\partial_{\lambda}p}+o(a_1,a_3),$

where the derivatives are taken at $a_1=0$ and $a_3=0$ . Equating the linear part to zero, we find $a_1\lambda_{id}^2(a_2)+a_3=0$ , which yields the exact threshold of asymptotic stability

$a_2=2+\frac{(a_1-a_3)^2}{a_1a_3}.$

Let us use the scaling $a_1=\varepsilon \hat a_1$ and $a_3=\varepsilon \hat a_3$ where $0<\varepsilon\ll 1$ and $\hat a_1>0$ and $\hat a_3>0$ are given. Then

$\lim_{\varepsilon \rightarrow 0} a_2(\varepsilon)-2=\lim_{\varepsilon \rightarrow 0}\frac{\varepsilon^2(\hat a_1-\hat a_3)^2}{\varepsilon^2 \hat a_1\hat a_3}=O(1).$

The fact that the threshold of instability for the bifurcation parameter $a_2(a_1,a_3)$ of the damped system does not tend at $\hat a_1 \ne \hat a_3$ to the ideal threshold $a_2=2$ in the limit $\varepsilon \rightarrow 0$ 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.