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 ${\mathbb{H}}$ 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