39 research outputs found
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
The Cauchy-Lagrangian method for numerical analysis of Euler flow
A novel semi-Lagrangian method is introduced to solve numerically the Euler
equation for ideal incompressible flow in arbitrary space dimension. It
exploits the time-analyticity of fluid particle trajectories and requires, in
principle, only limited spatial smoothness of the initial data. Efficient
generation of high-order time-Taylor coefficients is made possible by a
recurrence relation that follows from the Cauchy invariants formulation of the
Euler equation (Zheligovsky & Frisch, J. Fluid Mech. 2014, 749, 404-430).
Truncated time-Taylor series of very high order allow the use of time steps
vastly exceeding the Courant-Friedrichs-Lewy limit, without compromising the
accuracy of the solution. Tests performed on the two-dimensional Euler equation
indicate that the Cauchy-Lagrangian method is more - and occasionally much more
- efficient and less prone to instability than Eulerian Runge-Kutta methods,
and less prone to rapid growth of rounding errors than the high-order Eulerian
time-Taylor algorithm. We also develop tools of analysis adapted to the
Cauchy-Lagrangian method, such as the monitoring of the radius of convergence
of the time-Taylor series. Certain other fluid equations can be handled
similarly.Comment: 30 pp., 13 figures, 45 references. Minor revision. In press in
Journal of Scientific Computin
Generation of multiscale magnetic field by parity-invariant time-periodic flows
We study generation of magnetic fields involving large spatial scales by time- and space-periodic short-scale parity-invariant flows. The anisotropic magnetic eddy diffusivity tensor is calculated by the standard procedure involving expansion of magnetic modes and their growth rates in power series in the scale ratio. Our simulations, conducted for flows with random harmonic composition and exponentially decaying energy spectra, demonstrate that for a substantial part of time-periodic flows magnetic eddy diffusivity is negative for molecular diffusivity above the instability threshold for short-scale magnetic field generation. Thus, like it was in the case of steady flows, enlargement of the spatial scale of magnetic field is shown to be beneficial for generation by time-periodic flows. However, they are less efficient dynamos, than steady flows
Existence, uniqueness and analyticity of space-periodic solutions to the regularised long-wave equation
We consider space-periodic evolutionary and travelling-wave solutions to the
regularised long-wave equation (RLWE) with damping and forcing. We establish
existence, uniqueness and smoothness of the evolutionary solutions for smooth
initial conditions, and global in time spatial analyticity of such solutions
for analytical initial conditions. The width of the analyticity strip decays at
most polynomially. We prove existence of travelling-wave solutions and
uniqueness of travelling waves of a sufficiently small norm. The importance of
damping is demonstrated by showing that the problem of finding travelling-wave
solutions to the undamped RLWE is not well-posed. Finally, we demonstrate the
asymptotic convergence of the power series expansion of travelling waves for a
weak forcing.Comment: 29 pp., 4 figures, 44 reference
Dependence of magnetic field generation by thermal convection on the rotation rate: a case study
Dependence of magnetic field generation on the rotation rate is explored by
direct numerical simulation of magnetohydrodynamic convective attractors in a
plane layer of conducting fluid with square periodicity cells for the Taylor
number varied from zero to 2000, for which the convective fluid motion halts
(other parameters of the system are fixed). We observe 5 types of hydrodynamic
(amagnetic) attractors: two families of two-dimensional (i.e. depending on two
spatial variables) rolls parallel to sides of periodicity boxes of different
widths and parallel to the diagonal, travelling waves and three-dimensional
"wavy" rolls. All types of attractors, except for one family of rolls, are
capable of kinematic magnetic field generation. We have found 21 distinct
nonlinear convective MHD attractors (13 steady states and 8 periodic regimes)
and identified bifurcations in which they emerge. In addition, we have observed
a family of periodic, two-frequency quasiperiodic and chaotic regimes, as well
as an incomplete Feigenbaum period doubling sequence of bifurcations of a torus
followed by a chaotic regime and subsequently by a torus with 1/3 of the
cascade frequency. The system is highly symmetric. We have found two novel
global bifurcations reminiscent of the SNIC bifurcation, which are only
possible in the presence of symmetries. The universally accepted paradigm,
whereby an increase of the rotation rate below a certain level is beneficial
for magnetic field generation, while a further increase inhibits it (and halts
the motion of fluid on continuing the increase) remains unaltered, but we
demonstrate that this "large-scale" picture lacks many significant details.Comment: 39 pp., 22 figures (some are low quality), 5 tables. Accepted in
Physica
The Monge-Ampere equation: various forms and numerical methods
We present three novel forms of the Monge-Ampere equation, which is used,
e.g., in image processing and in reconstruction of mass transportation in the
primordial Universe. The central role in this paper is played by our Fourier
integral form, for which we establish positivity and sharp bound properties of
the kernels. This is the basis for the development of a new method for solving
numerically the space-periodic Monge-Ampere problem in an odd-dimensional
space. Convergence is illustrated for a test problem of cosmological type, in
which a Gaussian distribution of matter is assumed in each localised object,
and the right-hand side of the Monge-Ampere equation is a sum of such
distributions.Comment: 24 pages, 2 tables, 5 figures, 32 references. Submitted to J.
Computational Physics. Times of runs added, multiple improvements of the
manuscript implemented