Isentropic perturbations of a chaotic domain

Three major properties of the chaotic dynamics of the standard map, namely, the measure \mu of the main connected chaotic domain, the maximum Lyapunov exponent L of the motion in this domain, and the dynamical entropy h = \mu L are studied as functions of the stochasticity parameter K. The perturbations of the domain due to emergence and disintegration of islands of stability, upon small variations of K, are considered in particular. By means of extensive numerical experiments, it is shown that these perturbations are isentropic (at least approximately). In other words, the dynamical entropy does not fluctuate, while local jumps in \mu and L are significant.Comment: 13 pages, 4 figure

Symbolic computation of the Birkhoff normal form in the problem of stability of the triangular libration points

The problem of stability of the triangular libration points in the planar circular restricted three-body problem is considered. A software package, intended for normalization of autonomous Hamiltonian systems by means of computer algebra, is designed so that normalization problems of high analytical complexity could be solved. It is used to obtain the Birkhoff normal form of the Hamiltonian in the given problem. The normalization is carried out up to the 6th order of expansion of the Hamiltonian in the coordinates and momenta. Analytical expressions for the coefficients of the normal form of the 6th order are derived. Though intermediary expressions occupy gigabytes of the computer memory, the obtained coefficients of the normal form are compact enough for presentation in typographic format. The analogue of the Deprit formula for the stability criterion is derived in the 6th order of normalization. The obtained floating-point numerical values for the normal form coefficients and the stability criterion confirm the results by Markeev (1969) and Coppola and Rand (1989), while the obtained analytical and exact numeric expressions confirm the results by Meyer and Schmidt (1986) and Schmidt (1989). The given computational problem is solved without constructing a specialized algebraic processor, i.e., the designed computer algebra package has a broad field of applicability.Comment: 18 page

Lyapunov exponents in resonance multiplets

The problem of estimating the maximum Lyapunov exponents of the motion in a multiplet of interacting nonlinear resonances is considered for the case when the resonances have comparable strength. The corresponding theoretical approaches are considered for the multiplets of two, three, and infinitely many resonances (i.e., doublets, triplets, and "infinitets"). The analysis is based on the theory of separatrix and standard maps. A "multiplet separatrix map" is introduced, valid for description of the motion in the resonance multiplet under certain conditions. In numerical experiments it is shown that, at any given value of the adiabaticity parameter (which controls the degree of interaction/overlap of resonances in the multiplet), the value of the maximum Lyapunov exponent in the multiplet of equally-spaced equally-sized resonances is minimal in the doublet case and maximal in the infinitet case. This is consistent with the developed theory.Comment: 28 pages, 8 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

Phenomenology of scale-dependent space-time dimension

Loop-mediated processes characterized by dynamical scale M indirectly measure space-time dimension d at this scale. Assuming the latter to be scale-dependent d=d(L) and taking as examples B-oscillations and muon (g-2) experimental results we address the question about constraints put by this data on 4-d(L). It is shown that sensitivity is lost for 1/L around 350 GeV, and any value of d(L) between 2 and 5 at this scale is compatible with the data.Comment: RevTeX, 8 pages, 5 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 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

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

Non-Stationary Measurements of Chiral Magnetic Effect

We discuss Chiral Magnetic Effect from quantum theory of measurements point of view for non-stationary measurements. The effect of anisotropy for fluctuations of electric currents in magnetic field is addressed. It is shown that anisotropy caused by nonzero axial chemical potential is indistinguishable in this framework from anisotropy caused by finite measurement time or finite lifetime of the magnetic field, and in all cases it is related to abelian triangle anomaly. Possible P-odd effects for central heavy ions collisions (where Chiral Magnetic Effect is absent) are discussed in this context.Comment: LaTeX, 21 page

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