Exceptional and diabolical points in stability questions

Kirillov, Oleg N.

doi:10.1002/prop.201200068

Cited by 15 publications

(12 citation statements)

References 59 publications

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“…Mathematically, this is described by the formation of a nondiagonal Jordan block structure in the matrix representation of the (non-self-adjoint) dynamo operator associated with algebraic eigenvectors [46]. A characteristic property of exceptional points with two coinciding eigenvalues is reflected in the time dependence of the field amplitude, which, exactly at the exceptional point, exhibits an additional secular term linear in t [47] so…”

Section: B Impact Of Nonaxisymmetric Velocity Perturbationsmentioning

confidence: 99%

Impact of time-dependent nonaxisymmetric velocity perturbations on dynamo action of von Kármán-like flows

2012

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We present numerical simulations of the kinematic induction equation in order to examine the dynamo efficiency of an axisymmetric von Kármán-like flow subject to time-dependent nonaxisymmetric velocity perturbations. The numerical model is based on the setup of the French von Kármán-sodium dynamo (VKS) and on the flow measurements from a water experiment conducted at the University of Navarra in Pamplona, Spain. The principal experimental observations that are modeled in our simulations are nonaxisymmetric vortexlike structures which perform an azimuthal drift motion in the equatorial plane. Our simulations show that the interactions of these periodic flow perturbations with the fundamental drift of the magnetic eigenmode (including the special case of nondrifting fields) essentially determine the temporal behavior of the dynamo state. We find two distinct regimes of dynamo action that depend on the (prescribed) drift frequency of an (m = 2) vortexlike flow perturbation. For comparatively slowly drifting vortices we observe a narrow window with enhanced growth rates and a drift of the magnetic eigenmode that is synchronized with the perturbation drift. The resonance-like enhancement of the growth rates takes place when the vortex drift frequency roughly equals the drift frequency of the magnetic eigenmode in the unperturbed system. Outside of this small window, the field generation is hampered compared to the unperturbed case, and the field amplitude of the magnetic eigenmode is modulated with approximately twice the vortex drift frequency. The abrupt transition between the resonant regime and the modulated regime is identified as a spectral exceptional point where eigenvalues (growth rates and frequencies) and eigenfunctions of two previously independent modes collapse. In the actual configuration the drift frequencies of the velocity perturbations that are observed in the water experiment are much larger than the fundamental drift frequency of the magnetic eigenmode that is obtained from our numerical simulations. Hence, we conclude that the fulfillment of the resonance condition might be unlikely in present day dynamo experiments. However, a possibility to increase the dynamo efficiency in the VKS experiment might be realized by an application of holes or fingers on the outer boundary in the equatorial plane. These mechanical distortions provoke an anchorage of the vortices at fixed positions thus allowing an adjustment of the temporal behavior of the nonaxisymmetric flow perturbations.

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Section: B Impact Of Nonaxisymmetric Velocity Perturbationsmentioning

confidence: 99%

Impact of time-dependent nonaxisymmetric velocity perturbations on dynamo action of von Kármán-like flows

2012

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“…A simplified discussion how exceptional points arise mathematically can be found in Appendix A1. Exceptional points have been found in coupled dissipative dynamical systems [54], mechanical problems [55], electronic circuits [56], microwave cavities [57], gyrokinetics of plasmas [58], coupled fiber-ring resonators [59], atomic spectra [60,61], coupled quantum cascade microdisk lasers [62], photonic crystals [63], inhomogeneous gain media [64], optical lattices of driven cold atoms [65] and plasmonic waveguides [66]. We also point to literature on exceptional points with a degeneracy larger than two [63,[67][68][69].…”

Section: @ Rrlmentioning

confidence: 80%

Optically anisotropic media: New approaches to the dielectric function, singular axes, microcavity modes and Raman scattering intensities

Grundmann

Sturm

Kranert

et al. 2016

Physica Rapid Research Ltrs

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Six out of seven crystal systems are optically anisotropic and birefringent. We review recent insight that biaxial crystals generally exhibit four singular axes (or exceptional points) which can pairwise degenerate for special cases. Planar anisotropic microcavities are discussed as effectively biaxial systems and we predict exceptional points and demonstrate experimentally partially coalesced eigenstates. Also the general form of the dielectric function of anisotropic crystals based on individual dipole oscillators for phonon and electronic resonance is discussed. The impact of birefringence on Raman scattering intensities has been historically either ignored or modeled incorrectly. A recent theory for uniaxial and biaxial crystals explains experimental Raman scattering intensities for excitation off the principal directions without free parameters, allowing the unambiguous determination of the Raman tensor components. The above points are demonstrated and relevant in particular for the currently technologically important materials GaN, ZnO (uniaxial) and β‐Ga2O3 (biaxial).

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“…saddles possessing different radii of curvature in two directions) would again lose stability as the rotating speed Ω exceeds an upper bound. 15 For symmetric saddle, however, either the mass-point model or the rigid-body model till now are only able to predict a lower bound for stabilizing the balls. But we do observe that the trapping time of a polyfoam ball becomes considerably shorter as the rotating speed of our symmetric saddle becomes high (see Fig.…”

Section: Critical Angular Velocitymentioning

confidence: 99%

Confining rigid balls by mimicking quadrupole ion trapping

Fan¹,

Du²,

Wang³

et al. 2017

American Journal of Physics

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The rotating saddle not only is an interesting system that is able to trap a ball near its saddle point, but can also intuitively illustrate the operating principles of quadrupole ion traps in modern physics. Unlike the conventional models based on the mass-point approximation, we study the stability of a ball in a rotating-saddle trap using rigid-body dynamics. The stabilization condition of the system is theoretically derived and subsequently verified by experiments. The results are compared with the previous mass-point model, giving large discrepancy as the curvature of the ball is comparable to that of the saddle. We also point out that the spin angular velocity of the ball is analogous to the cyclotron frequency of ions in an external magnetic field utilized in many prevailing ion-trapping schemes

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Exceptional and diabolical points in stability questions

Cited by 15 publications

References 59 publications

Impact of time-dependent nonaxisymmetric velocity perturbations on dynamo action of von Kármán-like flows

Impact of time-dependent nonaxisymmetric velocity perturbations on dynamo action of von Kármán-like flows

Optically anisotropic media: New approaches to the dielectric function, singular axes, microcavity modes and Raman scattering intensities

Confining rigid balls by mimicking quadrupole ion trapping

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