10,770 research outputs found
Width of the chaotic layer: maxima due to marginal resonances
Modern theoretical methods for estimating the width of the chaotic layer in
presence of prominent marginal resonances are considered in the perturbed
pendulum model of nonlinear resonance. The fields of applicability of these
methods are explicitly and precisely formulated. The comparative accuracy is
investigated in massive and long-run numerical experiments. It is shown that
the methods are naturally subdivided in classes applicable for adiabatic and
non-adiabatic cases of perturbation. It is explicitly shown that the pendulum
approximation of marginal resonance works good in the non-adiabatic case. In
this case, the role of marginal resonances in determining the total layer width
is demonstrated to diminish with increasing the main parameter \lambda (equal
to the ratio of the perturbation frequency to the frequency of small-amplitude
phase oscillations on the resonance). Solely the "bending effect" is important
in determining the total amplitude of the energy deviations of the
near-separatrix motion at \lambda > 7. In the adiabatic case, it is
demonstrated that the geometrical form of the separatrix cell can be described
analytically quite easily by means of using a specific representation of the
separatrix map. It is shown that the non-adiabatic (and, to some extent,
intermediary) case is most actual, in comparison with the adiabatic one, for
the physical or technical applications that concern the energy jumps in the
near-separatrix chaotic motion.Comment: 17 pages, 2 figure
The width of a chaotic layer
A model of nonlinear resonance as a periodically perturbed pendulum is
considered, and a new method of analytical estimating the width of a chaotic
layer near the separatrices of the resonance is derived for the case of slow
perturbation (the case of adiabatic chaos). The method turns out to be
successful not only in the case of adiabatic chaos, but in the case of
intermediate perturbation frequencies as well.Comment: 27 pages, 8 figure
On Polynomial Solvability of the Hamiltonian Cycle Problem for Graphs of Degree Less Than or Equal To 3
Any graph can be represented pictorially as a figure. Moreover, it can be
represented as two or more figures that can be have different properties to
each other. For the purpose of HCP, we represent a graph by two such figures.
In each of them, there is an exterior part called the contour, and an interior
part. These two figures differ from each other by the constitution of the edges
in the interior part. That is, any edges in the interior part for one figure
are not in the interior for the other figure. We call these two figures basic
objects. We develop rules and algorithms that allow us to represent any graph
of degree d <= 3 by two basic objects. Individually, neither of these
representations possess the features to easily determine the Hamiltonicity of
the graph. However, the combination of these two figures, once certain weights
are assigned to their edges, allows us to determine the Hamiltonicity with a
polynomial-time check. The rules for the assignment of weights are: 1. The
weight of any edge of the interior part is 0, for both objects. 2. In both
figures any common edge of the contour has the same weight. The weights of the
edges allow us to extend the number of parameters of the objects, that is
sufficient to determine the Hamiltonicity of the graph. Then, if the graph is
Hamiltonian, then both figures possess the same set of parameters. If the sets
of parameters for two figures are different, then the graph is not Hamiltonian.
The parameters that determine the Hamiltonicity of the graph are the sums of
weights of edges and windows of contours in the figure. The algorithms of their
construction do not contain a combinatorial number of elements and have
polynomial complexity. We also supply an estimate of the complexity of each
algorithm.Comment: 27 pages, 34 figure
On the maximum Lyapunov exponent of the motion in a chaotic layer
The maximum Lyapunov exponent (referred to the mean half-period of phase
libration) of the motion in the chaotic layer of a nonlinear resonance subject
to symmetric periodic perturbation, in the limit of infinitely high frequency
of the perturbation, has been numerically estimated by two independent methods.
The newly derived value of this constant is 0.80, with precision presumably
better than 0.01.Comment: 15 pages, 3 figure
The Kepler map in the three-body problem
The Kepler map was derived by Petrosky (1986) and Chirikov and Vecheslavov
(1986) as a tool for description of the long-term chaotic orbital behaviour of
the comets in nearly parabolic motion. It is a two-dimensional area-preserving
map, describing the motion of a comet in terms of energy and time. Its second
equation is based on Kepler's third law, hence the title of the map. Since
1980s the Kepler map has become paradigmatic in a number of applications in
celestial mechanics and atomic physics. It represents an important kind of
general separatrix maps. Petrosky and Broucke (1988) used refined methods of
mathematical physics to derive analytical expressions for its single parameter.
These methods became available only in the second half of the 20th century, and
it may seem that the map is inherently a very modern mathematical tool. With
the help of the Jacobi integral I show that the Kepler map, including
analytical formulae for its parameter, can be derived by quite elementary
methods. The prehistory and applications of the Kepler map are considered and
discussed.Comment: 18 page
Numeric Deduction in Symbolic Computation. Application to Normalizing Transformations
Algorithms of numeric (in exact arithmetic) deduction of analytical
expressions, proposed and described by Shevchenko and Vasiliev (1993), are
developed and implemented in a computer algebra code. This code is built as a
superstructure for the computer algebra package by Shevchenko and Sokolsky
(1993a) for normalization of Hamiltonian systems of ordinary differential
equations, in order that high complexity problems of normalization could be
solved. As an example, a resonant normal form of a Hamiltonian describing the
hyperboloidal precession of a dynamically symmetric satellite is derived by
means of the numeric deduction technique. The technique provides a considerable
economy, about 30 times in this particular application, in computer memory
consumption. It is naturally parallelizable. Thus the economy of memory
consumption is convertible into a gain in computation speed.Comment: 14 page
Configurations of conjugate permutations
We describe some configurations of conjugate permutations which may be used
as a mathematical model of some genetical processes and crystal growth
Tidal decay of circumbinary planetary systems
It is shown that circumbinary planetary systems are subject to universal
tidal decay (shrinkage of orbits), caused by the forced orbital eccentricity
inherent to them. Circumbinary planets (CBP) are liberated from parent systems,
when, owing to the shrinkage, they enter the circumbinary chaotic zone. On
shorter timescales (less than the current age of the Universe), the effect may
explain, at least partially, the observed lack of CBP of close-enough (with
periods < 5 days) stellar binaries; on longer timescales (greater than the age
of the Universe but well within stellar lifetimes), it may provide massive
liberation of chemically evolved CBP. Observational signatures of the effect
may comprise (1) a prevalence of large rocky planets (super-Earths) in the
whole population of rogue planets (if this mechanism were the only source of
rogue planets); (2) a mass-dependent paucity of CBP in systems of low-mass
binaries: the lower the stellar mass, the greater the paucity.Comment: 21 pages, 2 figure
Chaotic zones around gravitating binaries
The extent of the continuous zone of chaotic orbits of a small-mass tertiary
around a system of two gravitationally bound primaries (a double star, a double
black hole, a binary asteroid, etc.) is estimated analytically, in function of
the tertiary's orbital eccentricity. The separatrix map theory is used to
demonstrate that the central continuous chaos zone emerges (above a threshold
in the primaries mass ratio) due to overlapping of the orbital resonances
corresponding to the integer ratios p:1 between the tertiary and the central
binary periods. In this zone, the unlimited chaotic orbital diffusion of the
tertiary takes place, up to its ejection from the system. The primaries mass
ratio, above which such a chaotic zone is universally present at all initial
eccentricities of the tertiary, is estimated. The diversity of the observed
orbital configurations of biplanetary and circumbinary exosystems is shown to
be in accord with the existence of the primaries mass parameter threshold.Comment: 23 pages, including 4 figure
The quaternion core inverse and its generalizations
In this paper we extend notions of the core inverse, core EP inverse, DMP
inverse, and CMP inverse over the quaternion skew-field and get
their determinantal representations within the framework of the theory of
column-row determinants previously introduced by the author. Since the
Moore-Penrose inverse and the Drazin inverse are necessary tools to represent
these generalized inverses, we use their determinantal representations
previously obtained by using row-column determinants. As the special case, we
give their determinantal representations for matrices with complex entries as
well. A numerical example to illustrate the main result is given.Comment: 34 page
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