Classification and stability of simple homoclinic cycles in R^5

The paper presents a complete study of simple homoclinic cycles in R^5. We find all symmetry groups Gamma such that a Gamma-equivariant dynamical system in R^5 can possess a simple homoclinic cycle. We introduce a classification of simple homoclinic cycles in R^n based on the action of the system symmetry group. For systems in R^5, we list all classes of simple homoclinic cycles. For each class, we derive necessary and sufficient conditions for asymptotic stability and fragmentary asymptotic stability in terms of eigenvalues of linearisation near the steady state involved in the cycle. For any action of the groups Gamma which can give rise to a simple homoclinic cycle, we list classes to which the respective homoclinic cycles belong, thus determining conditions for asymptotic stability of these cycles.Comment: 34 pp., 4 tables, 30 references. Submitted to Nonlinearit

Dynamo effect in parity-invariant flow with large and moderate separation of scales

It is shown that non-helical (more precisely, parity-invariant) flows capable of sustaining a large-scale dynamo by the negative magnetic eddy diffusivity effect are quite common. This conclusion is based on numerical examination of a large number of randomly selected flows. Few outliers with strongly negative eddy diffusivities are also found, and they are interpreted in terms of the closeness of the control parameter to a critical value for generation of a small-scale magnetic field. Furthermore, it is shown that, for parity-invariant flows, a moderate separation of scales between the basic flow and the magnetic field often significantly reduces the critical magnetic Reynolds number for the onset of dynamo action.Comment: 44 pages,11 figures, significantly revised versio

Instability of small-amplitude convective flows in a rotating layer with stress-free boundaries

We consider stability of steady convective flows in a horizontal layer with stress-free boundaries, heated below and rotating about the vertical axis, in the Boussinesq approximation (the Rayleigh-Benard convection). The flows under consideration are convective rolls or square cells, the latter being asymptotically equal to the sum of two orthogonal rolls of the same wave number k. We assume, that the Rayleigh number R is close to the critical one, R_c(k), for the onset of convective flows of this wave number: R=R_c(k)+epsilon^2; the amplitude of the flows is of the order of epsilon. We show that the flows are always unstable to perturbations, which are a sum of a large-scale mode not involving small scales, and two large-scale modes, modulated by the original rolls rotated by equal small angles in the opposite directions. The maximal growth rate of the instability is of the order of max(epsilon^{8/5},(k-k_c)^2), where k_c is the critical wave number for the onset of convection.Comment: Latex, 12 pp., 15 refs. An improved version of the manuscript submitted to "Mechanics of fluid and gas", 2006 (in Russian; English translation "Fluid Dynamics"

Magnetic field generation by convective flows in a plane layer

Hydrodynamic and magnetohydrodynamic convective attractors in a plane horizontal layer 0≤z≤1 are investigated numerically. We consider Rayleigh-Bénard convection in Boussinesq approximation assuming stress-free boundary conditions on horizontal boundaries and periodicity with the same period L in the x and y directions. Computations have been performed for the Prandtl number P=1 for $L=2\sqrt2$ and Rayleigh numbers 0<R≤4000, and for L=4, 0<R≤2000. Fifteen different types of hydrodynamic attractors are found, including two types of steady states distinct from rolls, travelling waves, periodic and quasiperiodic flows, and chaotic attractors of heteroclinic nature. Kinematic dynamo problem has been solved for the computed convective attractors. Out of the 15 types of the observed attractors only 6 can act as kinematic dynamos. Nonlinear magnetohydrodynamic regimes have been explored assuming as initial conditions convective attractors capable of magnetic field generation, and a small seed magnetic field. After initial exponential growth, in the saturated regime magnetic energy remains much smaller than the flow kinetic energy. The final magnetohydrodynamic attractors are either quasiperiodic or chaotic