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We study a perturbation theory for embedding gravity equations in a background for which corrections to the embedding function are linear with respect to corrections to the flat metric. The arbitrariness remaining after solving the linearized field equations is fixed by an assumption that the solution is static in the second order. A nonlinear differential equation is obtained, which makes it possible to find the gravitational potential for a spherically symmetric case if a background embedding is given. An explicit form of a spherically symmetric background parameterized by one function of radius is proposed. It is shown that this function can be chosen in such a way that the gravitational potential is in a good agreement with the observed distribution of dark matter in a galactic halo., Comment: LaTeX, 14 pages. This version corresponds to the published one

We consider Hodgkin-Huxley-type model that is a stiff ODE system with two fast and one slow variables. For the parameter ranges under consideration the original version of the model has unstable fixed point and the oscillating attractor that demonstrates bifurcation from bursting to spiking dynamics. Also a modified version is considered where the bistability occurs such that an area in the parameter space appears where the fixed point becomes stable and coexists with the bursting attractor. For these two systems we create artificial neural networks that are able to reproduce their dynamics. The created networks operate as recurrent maps and are trained on trajectory cuts sampled at random parameter values within a certain range. Although the networks are trained only on oscillatory trajectory cuts, it also discover the fixed point of the considered systems. The position and even the eigenvalues coincide very well with the fixed point of the initial ODEs. For the bistable model it means that the network being trained only on one brunch of the solutions recovers another brunch without seeing it during the training. These results, as we see it, are able to trigger the development of new approaches to complex dynamics reconstruction and discovering. From the practical point of view reproducing dynamics with the neural network can be considered as a sort of alternative method of numerical modeling intended for use with contemporary parallel hard- and software., Comment: 17 pages, 12 figures, 2 tables

Faddeev variant of embedding theory is an example of using the embedding approach for the description of gravity. In the original form of the embedding approach, the gravity is described by an embedding function of a four-dimensional surface representing our spacetime. In Faddeev variant, the independent variable is a non-square vielbein, which is a derivative of embedding function in embedding theory. We study the possibility of the existence of extra solutions in Faddeev variant, which makes this theory non-equivalent to GR. To separate the degrees of freedom corresponding to extra matter, we propose a formulation of this theory as GR with an additional contribution to the action. We analyze the equations of motion for a specific class of solutions corresponding to a weak gravitational field. We construct a simple exact solution corresponding to arbitrary matter and nontrivial torsion, which is an extra solution in Faddeev variant in the absence of real matter., Comment: LaTeX, 7 pages, based on a talk given at the International Conference PhysicA.SPb/2021, Saint Petersburg, Russia, 18-22 October 2021

A system of three non-identical Josephson junctions connected via an RLC circuit is considered. The method of Lyapunov exponents charts is used, which makes it possible to identify the main types of dynamics of the system and to analyze the dependence of its properties on parameters. The possibility of both two and three-frequency invariant tori is demonstrated. Saddle-node bifurcations of resonant tori are studied with the use of instantaneous Lyapunov exponents. The dependence of the charts on the type of coupling in the system is discussed., Comment: 17 pages, 9 figures, 1 table

We suggest a universal map capable to recover a behavior of a wide range of dynamical systems given by ODEs. The map is built as an artificial neural network whose weights encode a modeled system. We assume that ODEs are known and prepare training datasets using the equations directly without computing numerical time series. Parameter variations are taken into account in the course of training so that the network model captures bifurcation scenarios of the modeled system. Theoretical benefit from this approach is that the universal model admits using common mathematical methods without needing to develop a unique theory for each particular dynamical equations. Form the practical point of view the developed method can be considered as an alternative numerical method for solving dynamical ODEs suitable for running on contemporary neural network specific hardware. We consider the Lorenz system, the Roessler system and also Hindmarch-Rose neuron. For these three examples the network model is created and its dynamics is compared with ordinary numerical solutions. High similarity is observed for visual images of attractors, power spectra, bifurcation diagrams and Lyapunov exponents., Comment: 17 pages, 10 figures

(Abridged) Modern studies and industrial applications related to the design, radiation protection, and reliability of nuclear facilities, medical applications, as well as space research and exploration are relying on extensive simulations and modeling. Computer codes realizing semi-classical and quantum molecular dynamics (QMD) approaches are often employed to make up for the lack of accelerator data on many nuclear reactions at intermediate and high energies (>10s of MeV/n) and are in high demand. This contribution focuses on the methodology of generating reliable proton-induced cross sections in the energy range below 1 GeV. We developed a problem-oriented computer framework based on MCNPX and CASCADE/INPE codes to calculate activation cross section data at intermediate and high energies. Goodness of the fits of nucleon-nucleus interaction models to the existing data is evaluated based on elaborated algorithms. The method is based on the analysis of a large set of data and calculated cross sections for different targets and residual nuclei in a wide range of proton energies using numerous criteria. In practice, this could be done by tuning the model parameters and algorithms to fit required experimental data subset, or through achieving unification and consistency of fundamental parameters for all considered reactions. The presented framework is pursuing the latter approach. We use proton-induced reactions on Si and Fe nuclei to illustrate the application of the developed multi-criteria algorithm, where we use all data below 1 GeV available from the EXFOR data library and the IAEA CRP "Benchmark of Spallation Models." We show that the analysis of the predictive power of various intermediate and high-energy models based on the multi-criteria algorithm allows more sophisticated selection of appropriate models in a given energy range and residual nuclei domain., Comment: A poster E1.16-0085-21 presented at an event E1.16 "Origin of Cosmic Rays," 43th COSPAR Scientific Assembly (hybrid), 28 Jan - 4 Feb 2021, Sydney, Australia. For a full agenda of the event E1.16, see https://www.cospar-assembly.org/admin/session_cospar.php?session=903

We consider a self-oscillator whose excitation parameter is varied. Frequency of the variation is much smaller then the natural frequency of the oscillator so that oscillations in the system are periodically excited and decay. Also a time delay as added such that when the oscillations start to grow at a new excitation stage they are influenced via the delay line by the oscillations at the penultimate excitation stage. Due to a nonlinearity the seeding from the past arrives with a doubled phase so that oscillation phase changes from stage to stage according to chaotic Bernoulli-type map. As a result, the system operates as two coupled hyperbolic chaotic subsystems. Varying the relation between the delay time and the excitation period we affect a coupling strength between these subsystems as well as intensity of the phase doubling mechanism responsible for the hyperbolicity. Due to this, a transition from non-hyperbolic to hyperbolic hyperchaos occurs. The following steps of the transition scenario are revealed and analyzed: (a) an intermittency as alternation of long staying near a fixed point at the origin and short chaotic bursts; (b) chaotic oscillations with often visits to the fixed point; (c) plain hyperchaos without hyperbolicity after termination visiting the fixed point; (d) transformation of hyperchaos to hyperbolic form., Comment: 13 pages, 11 figures

Dimension of an inertial manifold for a chaotic attractor of spatially distributed system is estimated using autoencoder neural network. The inertial manifold is a low dimensional manifold where the chaotic attractor is embedded. The autoencoder maps system state vectors onto themselves letting them pass through an inner state with a reduced dimension. The training processes of the autoencoder is shown to depend dramatically on the reduced dimension: a learning curve saturates when the dimension is too small and decays if it is sufficient for a lossless information transfer. The smallest sufficient value is considered as a dimension of the inertial manifold, and the autoencoder implements a mapping onto the inertial manifold and back. The correctness of the computed dimension is confirmed by its remarkable coincidence with the one obtained as a number of covariant Lyapunov vectors with vanishing pairwise angles. These vectors are called physical modes. Unlike never having zero angles residual ones they are known to span a tangent subspace for the inertial manifold., Comment: 10 pages and 9 figures

In this paper we analyze local structure of several chaotic attractors recently suggested in literature as pseudohyperbolic. The absence of tangencies and thus the presence of the pseudohyperbolicity is verified using the method of angles that includes computation of distributions of the angles between the corresponding tangent subspaces. Also we analyze how volume expansion in the first subspace and the contraction in the second one occurs locally. For this purpose we introduce a family of instant Lyapunov exponents. Unlike the well known finite time ones, the instant Lyapunov exponents show expansion or contraction on infinitesimal time intervals. Two types of instant Lyapunov exponents are defined. One is related to ordinary finite time Lyapunov exponents computed in the course of standard algorithm for Lyapunov exponents. Their sums reveal instant volume expanding properties. The second type of instant Lyapunov exponents shows how covariant Lyapunov vectors grow or decay on infinitesimal time. Using both instant and finite time Lyapunov exponents we demonstrate that specific to the pseudohyperbolicity average expanding and contracting properties are typically violated on infinitesimal time. Instantly volumes from the first subspace can sometimes be contacted, directions in the second subspace can sometimes be expanded, and the instant contraction in the first subspace can sometimes be stronger than the contraction in the second subspace., Comment: 25 pages, 26 figures

We develop an extension of the fast method of angles for hyperbolicity verification in chaotic systems with an arbitrary number of time-delay feedback loops. The adopted method is based on the theory of covariant Lyapunov vectors and provides an efficient algorithm applicable for systems with high-dimensional phase space. Three particular examples of time-delay systems are analyzed and in all cases the expected hyperbolicity is confirmed., Comment: 16 pages, 3 figures

After having announced the statistically significant observation (5.6~$\sigma$) of the new exotic $\pi K$ atom, the DIRAC experiment at the CERN proton synchrotron presents the measurement of the corresponding atom lifetime, based on the full $\pi K$ data sample: $\tau = (5.5^{+5.0}_{-2.8}) \cdot 10^{-15}s$. By means of a precise relation ($<1\%$) between atom lifetime and scattering length, the following value for the S-wave isospin-odd $\pi K$ scattering length $a_0^{-}~=~\frac{1}{3}(a_{1/2}-a_{3/2})$ has been derived: $\left|a_0^-\right| = (0.072^{+0.031}_{-0.020}) M_{\pi}^{-1}$., Comment: 18 pages, 17 figures

We consider extended starlike networks where the hub node is coupled with several chains of nodes representing star rays. Assuming that nodes of the network are occupied by nonidentical self-oscillators we study various forms of their cluster synchronization. Radial cluster emerges when the nodes are synchronized along a ray, while circular cluster is formed by nodes without immediate connections but located on identical distances to the hub. By its nature the circular synchronization is a new manifestation of so called remote synchronization [Phys. Rev. E 85 (2012), 026208]. We report its long-range form when the synchronized nodes interact through at least three intermediate nodes. Forms of long-range remote synchronization are elements of scenario of transition to the total synchronization of the network. We observe that the far ends of rays synchronize first. Then more circular clusters appear involving closer to hub nodes. Subsequently the clusters merge and, finally, all network become synchronous. Behavior of the extended starlike networks is found to be strongly determined by the ray length, while varying the number of rays basically affects fine details of a dynamical picture. Symmetry of the star also extensively influences the dynamics. In an asymmetric star circular cluster mainly vanish in favor of radial ones, however, long-range remote synchronization survives., Comment: 16 pages, 13 figures

Let $G$ be a connected graph; denote by $\tau(G)$ the set of its spanning trees. Let $\mathbb F_q$ be a finite field, $s(\alpha,G)=\sum_{T\in\tau(G)} \prod_{e \in E(T)} \alpha_e$, where ${\alpha_e\in \mathbb F_q}$. Kontsevich conjectured in 1997 that the number of nonzero values of $s(\alpha, G)$ is a polynomial in $q$ for all graphs. This conjecture was disproved by Brosnan and Belkale. In this paper, using the standard technique of the Fourier transformation of Feynman amplitudes, we express the flow polynomial $F_G(q)$ in terms of the "correct" Kontsevich formula. Our formula represents $F_G(q)$ as a linear combination of Legendre symbols of $s(\alpha, H)$ with coefficients $\pm 1/q^{(|V(H)|-1)/2}$, where $H$ is a contracted graph of $G$ depending on $\alpha\in \left(\mathbb F^*_q\right)^{E(G)}$, and $|V(H)|$ is odd. The case $q=5$ corresponds to the least number with which all coefficients in the linear combination are positive. This allows us to hope that the obtained result can be applied to prove the Tutte 5-flow conjecture., Comment: 18 pages

We develop a numerical test of hyperbolicity of chaotic dynamics in time-delay systems. The test is based on the angle criterion and includes computation of angle distributions between expanding, contracting and neutral manifolds of trajectories on the attractor. Three examples are tested. For two of them previously predicted hyperbolicity is confirmed. The third one provides an example of a time-delay system with nonhyperbolic chaos., Comment: 7 pages, 5 figures

A generalized model of starlike network is suggested that takes into account non-additive coupling and nonlinear transformation of coupling variables. For this model a method of analysis of synchronized cluster stability is developed. Using this method three starlike networks based on Ikeda, predator-prey and H\'enon maps are studied., Comment: 15 pages, 8 figures

In this paper we categorize dynamical regimes demonstrated by star-like networks with chaotic nodes. This analysis is done in view of further studying of chaotic scale-free networks, since a star-like structure is the main motif of them. We analyze star-like networks of Henon maps. They are found to demonstrate a huge diversity of regimes. Varying the coupling strength we reveal chaos, quasiperiodicity, and periodicity. The nodes can be both fully- and phase-synchronized. The hub node can be either synchronized with the subordinate nodes or oscillate separately from fully synchronized subordinates. There is a range of wild multistability where the zoo of regimes is the most various. One can hardly predict here even a qualitative nature of the expected solution, since each perturbation of the coupling strength or initial conditions results in a new character of dynamics., Comment: 9 pages, 9 figures

The results of a search for hydrogen-like atoms consisting of $\pi^{\mp}K^{\pm}$ mesons are presented. Evidence for $\pi K$ atom production by 24 GeV/c protons from CERN PS interacting with a nickel target has been seen in terms of characteristic $\pi K$ pairs from their breakup in the same target ($178 \pm 49$) and from Coulomb final state interaction ($653 \pm 42$). Using these results the analysis yields a first value for the $\pi K$ atom lifetime of $\tau=(2.5_{-1.8}^{+3.0})$ fs and a first model-independent measurement of the S-wave isospin-odd $\pi K$ scattering length $\left|a_0^-\right|=\frac{1}{3}\left|a_{1/2}-a_{3/2}\right|= \left(0.11_{-0.04}^{+0.09} \right)M_{\pi}^{-1}$ ($a_I$ for isospin $I$)., Comment: 14 pages, 8 figures

Covariant Lyapunov vectors for scale-free networks of Henon maps are highly localized. We revealed two mechanisms of the localization related to full and phase cluster synchronization of network nodes. In both cases the localization nodes remain unaltered in course of the dynamics, i.e., the localization is nonwandering. Moreover this is predictable: the localization nodes are found to have specific dynamical and topological properties and they can be found without computing of the covariant vectors. This is an example of explicit relations between the system topology, its phase space dynamics, and the associated tangent space dynamics of covariant Lyapunov vectors., Comment: largely revised and extended version, 12 pages, 13 figures

We study an ensemble of identical noisy phase oscillators with a blinking mean-field coupling, where one-cluster and two-cluster synchronous states alternate. In the thermodynamic limit the population is described by a nonlinear Fokker-Planck equation. We show that the dynamics of the order parameters demonstrates hyperbolic chaos. The chaoticity manifests itself in phases of the complex mean field, which obey a strongly chaotic Bernoulli map. Hyperbolicity is confirmed by numerical tests based on the calculations of relevant invariant Lyapunov vectors and Lyapunov exponents. We show how the chaotic dynamics of the phases is slightly smeared by finite-size fluctuations., Comment: 8 pages, 4 figures

Despite much theoretical work, different modifications of backoff protocols in 802.11 networks lack empirical evidence demonstrating their real-life performance. To fill the gap we have set out to experiment with performance of exponential backoff by varying its backoff factor. Despite the satisfactory results for throughput, we have witnessed poor fairness manifesting in severe capture effect. The design of standard backoff protocol allows already successful nodes to remain successful, giving little chance to those nodes that failed to capture the channel in the beginning. With this at hand, we ask a conceptual question: Can one improve the performance of wireless backoff by introducing a mechanism of self-penalty, when overly successful nodes are penalized with big contention windows? Our real-life measurements using commodity hardware demonstrate that in many settings such mechanism not only allows to achieve better throughput, but also assures nearly perfect fairness. We further corroborate these results with simulations and an analytical model. Finally, we present a backoff factor selection protocol which can be implemented in access points to enable deployment of the penalty backoff protocol to consumer devices., Comment: 14 pages, 21 figures, 3 tables

We consider a chain of oscillators with hyperbolic chaos coupled via diffusion. When the coupling is strong the chain is synchronized and demonstrates hyperbolic chaos so that there is one positive Lyapunov exponent. With the decay of the coupling the second and the third Lyapunov exponents approach zero simultaneously. The second one becomes positive, while the third one remains close to zero. Its finite-time numerical approximation fluctuates changing the sign within a wide range of the coupling parameter. These fluctuations arise due to the unstable dimension variability which is known to be the source for non-hyperbolicity. We provide a detailed study of this transition using the methods of Lyapunov analysis., Comment: 24 pages, 13 figures

We consider time evolution of Turing patterns in an extended system governed by an equation of the Swift-Hohenberg type, where due to an external periodic parameter modulation long-wave and short-wave patterns with length scales related as 1:3 emerge in succession. We show theoretically and demonstrate numerically that the spatial phases of the patterns, being observed stroboscopically, are governed by an expanding circle map, so that the corresponding chaos of Turing patterns is hyperbolic, associated with a strange attractor of the Smale-Williams solenoid type. This chaos is shown to be robust with respect to variations of parameters and boundary conditions., Comment: 4 pages, 4 figures

The effective numerical method is developed performing the test of the hyperbolicity of chaotic dynamics. The method employs ideas of algorithms for covariant Lyapunov vectors but avoids their explicit computation. The outcome is a distribution of a characteristic value which is bounded within the unit interval and whose zero indicate the presence of tangency between expanding and contracting subspaces. To perform the test one needs to solve several copies of equations for infinitesimal perturbations whose amount is equal to the sum of numbers of positive and zero Lyapunov exponents. Since for high-dimensional system this amount is normally much less then the full phase space dimension, this method provide the fast and memory saving way for numerical hyperbolicity test of such systems., Comment: 4 pages and 4 figures

The DIRAC experiment at CERN has achieved a sizeable production of $\pi^+\pi^-$ atoms and has significantly improved the precision on its lifetime determination. From a sample of 21227 atomic pairs, a 4% measurement of the S-wave $\pi\pi$ scattering length difference $|a_0-a_2| = (.0.2533^{+0.0080}_{-0.0078}|_\mathrm{stat}.{}^{+0.0078}_{-0.0073}|_\mathrm{syst})M_{\pi^+}^{-1}$ has been attained, providing an important test of Chiral Perturbation Theory., Comment: 6 pages, 6 figures

Lyapunov exponents are well-known characteristic numbers that describe growth rates of perturbations applied to a trajectory of a dynamical system in different state space directions. Covariant (or characteristic) Lyapunov vectors indicate these directions. Though the concept of these vectors has been known for a long time, they became practically computable only recently due to algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in covariant Lyapunov vectors and their wide range of potential applications, in this article we summarize the available information related to Lyapunov vectors and provide a detailed explanation of both the theoretical basics and numerical algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The angles between these vectors and the original covariant vectors are norm-independent and can be considered as characteristic numbers. Moreover, we present and study in detail an improved approach for computing covariant Lyapunov vectors. Also we describe, how one can test for hyperbolicity of chaotic dynamics without explicitly computing covariant vectors., Comment: 21 pages, 5 figures

In this Letter we show that the analysis of Lyapunov-exponents fluctuations contributes to deepen our understanding of high-dimensional chaos. This is achieved by introducing a Gaussian approximation for the large deviation function that quantifies the fluctuation probability. More precisely, a diffusion matrix $\bf D$ (a dynamical invariant itself) is measured and analysed in terms of its principal components. The application of this method to three (conservative, as well as dissipative) models, allows: (i) quantifying the strength of the effective interactions among the different degrees of freedom; (ii) unveiling microscopic constraints such as those associated to a symplectic structure; (iii) checking the hyperbolicity of the dynamics., Comment: 4 pages, 3 figures. To appear in Phys. Rev. Lett

In the tangent space of some spatially extended dissipative systems one can observe "physical" modes which are highly involved in the dynamics and are decoupled from the remaining set of hyperbolically "isolated" degrees of freedom representing strongly decaying perturbations. This mode splitting is studied for the Ginzburg-Landau equation at different strength of the spatial coupling. We observe that isolated modes coincide with eigenmodes of the homogeneous steady state of the system; that there is a local basis where the number of non-zero components of the state vector coincides with the number of "physical" modes; that in a system with finite number of degrees of freedom the strict mode splitting disappears at finite value of coupling; that above this value a fussy mode splitting is observed., Comment: 6 pages, 5 figures

We present evidence for the first observation of electromagnetically bound $\pi^\pm K^\mp$-pairs ($\pi K$-atoms) with the DIRAC experiment at the CERN-PS. The $\pi K$-atoms are produced by the 24 GeV/c proton beam in a thin Pt-target and the $\pi^\pm$ and $K^\mp$-mesons from the atom dissociation are analyzed in a two-arm magnetic spectrometer. The observed enhancement at low relative momentum corresponds to the production of 173 $\pm$ 54 $\pi K$-atoms. The mean life of $\pi K$-atoms is related to the s-wave $\pi K$-scattering lengths, the measurement of which is the goal of the experiment. From these first data we derive a lower limit for the mean life of 0.8 fs at 90% confidence level., Comment: 15 pages, 9 figures

Departing from a system of two non-autonomous amplitude equations, demonstrating hyperbolic chaotic dynamics, we construct a 1D medium as ensemble of such local elements introducing spatial coupling via diffusion. When the length of the medium is small, all spatial cells oscillate synchronously, reproducing the local hyperbolic dynamics. This regime is characterized by a single positive Lyapunov exponent. The hyperbolicity survives when the system gets larger in length so that the second Lyapunov exponent passes zero, and the oscillations become inhomogeneous in space. However, at a point where the third Lyapunov exponent becomes positive, some bifurcation occurs that results in violation of the hyperbolicity due to the emergence of one-dimensional intersections of contracting and expanding tangent subspaces along trajectories on the attractor. Further growth of the length results in two-dimensional intersections of expanding and contracting subspaces that we classify as a stronger type of the violation. Beyond of the point of the hyperbolicity loss, the system demonstrates an extensive spatiotemporal chaos typical for extended chaotic systems: when the length of the system increases the Kaplan-Yorke dimension, the number of positive Lyapunov exponents, and the upper estimate for Kolmogorov-Sinai entropy grow linearly, while the Lyapunov spectrum tends to a limiting curve., Comment: 11 pages, 11 figures, results reproduced with higher precision, new figures added, text revised

Recently, a system with uniformly hyperbolic attractor of Smale-Williams type has been suggested [Kuznetsov, Phys. Rev. Lett., 95, 144101, 2005]. This system consists of two coupled non-autonomous van der Pol oscillators and admits simple physical realization. In present paper we introduce amplitude equations for this system and prove that the attractor of the system of amplitude equations is also uniformly hyperbolic. Also we represent qualitative illustrations as well as quantitative characteristics of a chaotic dynamics on this attractor., Comment: 17 pages, 8 figures

Flow and Diffusion Distributed Structures (FDS) are stationary spatially periodic patterns that can be observed in reaction-diffusion-advection systems. These structures arise when the flow rate exceeds a certain bifurcation point provided that concentrations of interacting species at the inlet differ from steady state values and the concentrations are held constant. Normally, theoretical studies of these patterns are developed without concerning a noise. In this paper we consider FDS for a more realistic conditions and assume that the inlet concentrations are perturbed by a small noise. When the flow rate is small, the FDS is linearly sensitive to noise at the inlet. Even weak fluctuations destroy the stationary pattern and an oscillatory solution appears instead. For higher flow rates the instability becomes nonlinear: the pattern remains unaltered for a weak noise and undergoes the destruction when the noise amplitude passes a certain threshold. We represent a detailed description of these effects and examine two scenarios for the stabilization., Comment: 31 pages, 14 figures, to appear in MatCom

We introduce here a new universality conjecture for levels of random Hamiltonians, in the same spirit as the local REM conjecture made by S. Mertens and H. Bauke. We establish our conjecture for a wide class of Gaussian and non-Gaussian Hamiltonians, which include the $p$-spin models, the Sherrington-Kirkpatrick model and the number partitioning problem. We prove that our universality result is optimal for the last two models by showing when this universality breaks down., Comment: 34 pages