19 research outputs found
On pleated singular points of first order implicit differential equations
We study phase portraits of a first order implicit differential equation in a
neighborhood of its pleated singular point that is a non-degenerate singular
point of the lifted field. Although there is no a visible local classification
of implicit differential equations at pleated singular points (even in the
topological category), we show that there exist only six essentially different
phase portraits, which are presented
Magnetic field generation by pointwise zero-helicity three-dimensional steady flow of incompressible electrically conducting fluid
We introduce six families of three-dimensional space-periodic steady
solenoidal flows, whose kinetic helicity density is zero at any point. Four
families are analytically defined. Flows in four families have zero helicity
spectrum. Sample flows from five families are used to demonstrate numerically
that neither zero kinetic helicity density, nor zero helicity spectrum prohibit
generation of large-scale magnetic field by the two most prominent dynamo
mechanisms: the magnetic -effect and negative eddy diffusivity. Our
computations also attest that such flows often generate small-scale field for
sufficiently small magnetic molecular diffusivity. These findings indicate that
kinetic helicity and helicity spectrum are not the quantities controlling the
dynamo properties of a flow regardless of whether scale separation is present
or not.Comment: 37 pages, 11 figures, 54 reference
Route to hyperchaos in Rayleigh-Benard convection
Transition to hyperchaotic regimes in Rayleigh-Benard convection in a square
periodicity cell is studied by three-dimensional numerical simulations. By
fixing the Prandtl number at P=0.3 and varying the Rayleigh number as a control
parameter, a bifurcation diagram is constructed where a route to hyperchaos
involving quasiperiodic regimes with two and three incommensurate frequencies,
multistability, chaotic intermittent attractors and a sequence of boundary and
interior crises is shown. The three largest Lyapunov exponents exhibit a linear
scaling with the Rayleigh number and are positive in the final hyperchaotic
attractor. Thus, a route to weak turbulence is found
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
Magnetic field generation by intermittent convection
Magnetic field generation in three-dimensional Rayleigh-Benard convection of an electrically conducting fluid is studied numerically by fixing the Prandtl number at P = 0.3 and varying the Rayleigh number (Ra) as a control parameter. A recently reported route to hyperchaos involving quasiperiodic regimes, crises and chaotic intermittent attractors is followed, and the critical magnetic Prandtl number (P-m(c)) for dynamo action is determined as a function of Ra. A mechanism for the onset of intermittency in the magnetic energy is described, the most beneficial convective regimes for dynamo action in this transition to weak turbulence are identified, and the impact of intermittency on the dependence of P-m(c) on Ra is discussed
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
A recursive algorithm based on the maximum principle of pontryagin
This article reports findings in designing a conceptual optimal control algorithm based on the maximum principle of Pontryagin and of the steepest descent type for a relaxed version of the original problem. Allowing the relaxation of initial condition in order to rewrite the two boundary value problem as one with boundary conditions in the same endpoint, key properties of this algorithm are proved. Then, some results are obtained by using an optimization algorithm of the same type for which off-the-shelf routines taking into account the numerical issues, which are always tricky for infinite dimensional problems. © 2018 IEEE