The stability of parametrically forced coupled oscillators in sum resonance

Paradoxes of dissipation‐induced destabilization or who opened Whitney's umbrella?

“…Arnold [4,5], who named it 'deadlock of an edge'. In this respect Bottema's results in [17] can be seen as an early study of bifurcations and structural stability of polynomials and matrices, and therefore of the singularities of their stability boundaries whose systematical treatment was initiated since the beginning of the 1970s in [4,5,70,71] and continued by many authors, see e.g., [38,39,66,97] and references therein. Although Bottema applied his result to nonconservative systems without gyroscopic forces, there are reasons for the singularity to appear in the case when gyroscopic forces are taken into account because the stability is determined by the roots of a similar fourth order characteristic polynomial.…”

“…From (38) it follows that in the vicinity of γ := ν/δ = γ * the limit of the critical value of the gyroscopic parameter Ω cr of the near-Hamiltonian system as δ → 0 exceeds the threshold of gyroscopic stabilization determined by the Krein collision (see [56])…”

“…In [38] a geometrical explanation is presented for damping induced instability in parametric systems using 'all' the parameters of the system as unfolding parameters. It will turn out that, using symmetry and normalization, four parameters are needed to give a complete description in a two degrees of freedom system, or more generally systems where three frequencies are in resonance, but three parameters suffice to visualize the situation.…”

“…These phenomena were actively studied in the 1960's to provide more basic understanding and they have continued to be studied with more sophisticated tools, including early attempts to employ singularity theory [107], until in the mid 1990s it was understood [38,94,95] that the destabilization paradox is related to the Whitney umbrella singularity of the stability boundary [111,112]. After describing in the first sections Whitney's umbrella and Ziegler's paradox, we make in Sect.…”

Tight-binding lattices with an oscillating imaginary gauge field

“…of the propagator over one oscillation period T . A numerical computation of quasi energies for an oscillating field shows rather generally that pseudo Hermitian dynamics can be broken and the quasi energy spectrum can become complex owing to the appearance of resonance tongues, which are the signature of a multiple parametric instability [22][23][24]. As an example, Fig.4 shows the numerically-computed instability domains (regions of complex quasi energies) in the frequency-amplitude plane (ω, h 1 ) for a square-wave ac gauge field…”