Locating the Sets of Exceptional Points in Dissipative Systems and the Self-Stability of Bicycles.

Sets in the parameter space corresponding to complex exceptional points (EP) have high codimension, and by this reason, they are difficult objects for numerical location. However, complex EPs play an important role in the problems of the stability of dissipative systems, where they are frequently considered as precursors to instability. We propose to locate the set of complex EPs using the fact that the global minimum of the spectral abscissa of a polynomial is attained at the EP of the highest possible order.

Detecting singular weak-dissipation limit for flutter onset in reversible systems.

A "flutter machine" is introduced for the investigation of a singular interface between the classical and reversible Hopf bifurcations that is theoretically predicted to be generic in nonconservative reversible systems with vanishing dissipation. In particular, such a singular interface exists for the Pflüger viscoelastic column moving in a resistive medium, which is proven by means of the perturbation theory of multiple eigenvalues with the Jordan block. The laboratory setup, consisting of a cantilevered viscoelastic rod loaded by a positional force with nonzero curl produced by dry friction, demonstrates high sensitivity of the classical Hopf bifurcation onset to the ratio between the weak air drag and Kelvin-Voigt damping in the Pflüger column.

Singular diffusionless limits of double-diffusive instabilities in magnetohydrodynamics.

We study local instabilities of a differentially rotating viscous flow of electrically conducting incompressible fluid subject to an external azimuthal magnetic field. In the presence of the magnetic field, the hydrodynamically stable flow can demonstrate non-axisymmetric azimuthal magnetorotational instability (AMRI) both in the diffusionless case and in the double-diffusive case with viscous and ohmic dissipation. Performing stability analysis of amplitude transport equations of short-wavelength approximation, we find that the threshold of the diffusionless AMRI via the Hamilton-Hopf bifurcation is a singular limit of the thresholds of the viscous and resistive AMRI corresponding to the dissipative Hopf bifurcation and manifests itself as the Whitney umbrella singular point.

Destabilization of rotating flows with positive shear by azimuthal magnetic fields.

According to Rayleigh's criterion, rotating flows are linearly stable when their specific angular momentum increases radially outward. The celebrated magnetorotational instability opens a way to destabilize those flows, as long as the angular velocity is decreasing outward. Using a local approximation we demonstrate that even flows with very steep positive shear can be destabilized by azimuthal magnetic fields which are current free within the fluid.

Extending the range of the inductionless magnetorotational instability.

The magnetorotational instability (MRI) can destabilize hydrodynamically stable rotational flows, thereby allowing angular momentum transport in accretion disks. A notorious problem for the MRI is its questionable applicability in regions with low magnetic Reynolds number. Using the WKB method, we extend the range of applicability of the MRI by showing that the inductionless versions of the MRI, such as the helical MRI and the azimuthal MRI, can easily destabilize Keplerian profiles ∝r(-3/2) if the radial profile of the azimuthal magnetic field is only slightly modified from the current-free profile ∝r(-1).

Paradoxical transitions to instabilities in hydromagnetic Couette-Taylor flows.

By methods of modern spectral analysis, we rigorously find distributions of eigenvalues of linearized operators associated with an ideal hydromagnetic Couette-Taylor flow. The transition to instability in the limit of a vanishing magnetic field has a discontinuous change compared to the Rayleigh stability criterion for hydrodynamical flows, which is known as the Velikhov-Chandrasekhar paradox.

Paradoxes of magnetorotational instability and their geometrical resolution.

Magnetorotational instability (MRI) triggers turbulence and enables outward transport of angular momentum in hydrodynamically stable accretion discs. By using the WKB approximation and methods of singular function theory, we resolve two different paradoxes of MRI that appear in the limits of infinite and vanishing magnetic Prandtl number. For the latter case, we derive a strict limit of the critical Rossby number.

Determining role of Krein signature for three-dimensional Arnold tongues of oscillatory dynamos.

Using a homotopic family of boundary eigenvalue problems for the mean-field alpha;{2} dynamo with helical turbulence parameter alpha(r)=alpha_{0}+gammaDeltaalpha(r) and homotopy parameter beta[0,1] , we show that the underlying network of diabolical points for Dirichlet (idealized, beta=0 ) boundary conditions substantially determines the choreography of eigenvalues and thus the character of the dynamo instability for Robin (physically realistic, beta=1 ) boundary conditions. In the (alpha_{0},beta,gamma) space the Arnold tongues of oscillatory solutions at beta=1 end up at the diabolical points for beta=0 . In the vicinity of the diabolical points the space orientation of the three-dimensional tongues, which are cones in first-order approximation, is determined by the Krein signature of the modes involved in the diabolical crossings at the apexes of the cones.