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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 $beta in [0,1]$, 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 3D 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. The Krein space induced geometry of the resonance zones explains the subtleties in finding $alpha$-profiles leading to spectral exceptional points, which are important ingredients in recent theories of polarity reversals of the geomagnetic field.
Nonlinear oscillator systems are ubiquitous in biology and physics, and their control is a practical problem in many experimental systems. Here we study this problem in the context of the two models of spatially-coupled oscillators: the complex Ginzb
Two concepts, very different in nature, have proved to be useful in analytical and numerical studies of spectral stability: (i) the Krein signature of an eigenvalue, a quantity usually defined in terms of the relative orientation of certain subspaces
We study the connection between spherical wedge and full spherical shell geometries using simple mean-field $alpha^2$ dynamos. We solve the equations for a one-dimensional time-dependent mean-field dynamo to examine the effects of varying the polar a
We study magnetic field evolution in flows with fluctuating in time governing parameters in electrically conducting fluid. We use a standard mean-field approach to derive equations for large-scale magnetic field for the fluctuating ABC-flow as well a
In this paper we first discuss observational evidence of longitudinal concentrations of magnetic activity in the Sun and rapidly rotating late-type stars with outer convective envelopes. Scenarios arising from the idea of rotationally influenced anis