Ziegler-Bottema dissipation-induced instability and related topics

Abstract: Exactly 60 years ago Ziegler [1] observed (I) that viscous dissipation can move pure imaginary eigenvalues of a Lyapunov stable time-reversible non-conservative mechanical system (Ziegler’s pendulum loaded by a follower force) to the right half of the complex plane and (II) that the threshold of asymptotic stability generically does not converge to the threshold of the Lyapunov stability of the non-damped system when dissipation coefficient tends to zero. In 1956 Bottema [2] related the structurally unstable situation (II) to the Whitney umbrella singularity [3] of the stability boundary. I will show the examples of Hamiltonian, reversible and PT -symmetric systems of physics and mechanics with the similar effects of dissipation-induced instabilities and non-commuting limits of vanishing dissipation. I will discuss the relation of these effects to the multiple non-derogatory eigenvalues occurring both on the stability boundary and inside the domain of asymptotic stability, show the connection to the spectral abscissa minimization [4] and in the Hamiltonian case will demonstrate that a suitable combination of damping and nonconservative positional forces can destabilize the eigenvalues with both positive and negative Krein (symplectic) signature of the unperturbed system [5-7].
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