Dynamics around Non-Spherical Symmetric Bodies: I. The case of a spherical body with mass anomaly

The space missions designed to visit small bodies of the Solar System boosted the study of the dynamics around non-spherical bodies. In this vein, we study the dynamics around a class of objects classified by us as Non-Spherical Symmetric Bodies, including contact binaries, triaxial ellipsoids, spherical bodies with a mass anomaly, among others. In the current work, we address the results for a body with a mass anomaly. We apply the pendulum model to obtain the width of the spin-orbit resonances raised by non-asymmetric gravitational terms of the central object. The Poincaré surface of section technique is adopted to confront our analytical results and to study the system’s dynamics by varying the parameters of the central object. We verify the existence of two distinct regions around an object with a mass anomaly: a chaotic inner region that extends beyond the corotation radius and a stable outer region. In the latter, we identify structures remarkably similar to those of the classical restrict and planar 3-body problem in the Poincaré surface of sections, including asymmetric periodic orbits associated with 1:1+p resonances. We apply our results to a Chariklo with a mass anomaly, obtaining that Chariklo rings are probably related to first kind periodic orbits and not with 1:3 spin-orbit resonance, as proposed in the literature. We believe that our work presents the first tools for studying mass anomaly systems.

Interaction Between a Robot and Bunimovich Stadium Billiards

The robot–environment–task triad provides many opportunities to revisit physical problems with fresh eyes. Hence, we develop a simple experiment to observe chaos in classical billiards with a macroscopic 3.38-meter long setup. Using a digital video camera, one records the dynamic time evolution of the interaction between a robot and Bunimovich stadium billiards with specular reflection. From the experimental time series, we calculate the Lyapunov exponent λ as a function of a geometric parameter. The results are in concordance with theoretical predictions. In addition, we determine the Poincaré surface of section from the experimental data and check its sensitivity to the initial conditions as a function of time.

Analytical Studies of SWCNTs Embedded in Nonlinear Viscous Elastic Media and the Chaotic Effect of Its Various Parameters

The analytical studies and chaotic behavior of forced vibration on Single-wall Carbon Nanotubes (SWCNTs) embedded in nonlinear viscous elastic medium subjected to parametric excitation are investigated. The analytical solution of the amplitude of nonlinear vibration is studied using Krylov–Bogoliubov–Mitropolsky (KBM) method. Both resonant and nonresonant cases are deduced. The computational techniques are used to draw graphs of time series, phase plot and Poincaré surface of section to analyze the chaotic behavior of the system considered. The plots are drawn for various values of different parameters like linear damping, nonlinear damping and amplitude of external forces in the considered model of SWCNTs. This work could be helpful in differentiating various elements of Carbon Nanotubes (CNTs) into the chaotic elements and controlling elements. The chaotic elements contributes to increase in the aging of CNTs while controlling elements can be used to control the irregular behavior of CNTs.

Dynamic Analysis and FPGA Implementation of New Chaotic Neural Network and Optimization of Traveling Salesman Problem

A neural network is a model of the brain’s cognitive process, with a highly interconnected multiprocessor architecture. The neural network has incredible potential, in the view of these artificial neural networks inherently having good learning capabilities and the ability to learn different input features. Based on this, this paper proposes a new chaotic neuron model and a new chaotic neural network (CNN) model. It includes a linear matrix, a sine function, and a chaotic neural network composed of three chaotic neurons. One of the chaotic neurons is affected by the sine function. The network has rich chaotic dynamics and can produce multiscroll hidden chaotic attractors. This paper studied its dynamic behaviors, including bifurcation behavior, Lyapunov exponent, Poincaré surface of section, and basins of attraction. In the process of analyzing the bifurcation and the basins of attraction, it was found that the network demonstrated hidden bifurcation phenomena, and the relevant properties of the basins of attraction were obtained. Thereafter, a chaotic neural network was implemented by using FPGA, and the experiment proved that the theoretical analysis results and FPGA implementation were consistent with each other. Finally, an energy function was constructed to optimize the calculation based on the CNN in order to provide a new approach to solve the TSP problem.

Pseudorotations of the -disc and Reeb flows on the -sphere

We use Lerman’s contact cut construction to find a sufficient condition for Hamiltonian diffeomorphisms of compact surfaces to embed into a closed $3$ -manifold as Poincaré return maps on a global surface of section for a Reeb flow. In particular, we show that the irrational pseudorotations of the $2$ -disc constructed by Fayad and Katok embed into the Reeb flow of a dynamically convex contact form on the $3$ -sphere.

Research on Cracking Control of Immersed Tube Tunnels Concrete by Pre-heating treatment

Herein, the pre-heating treatments are researched. With diversity of zone and temperature, highest average temperature of the section 10cm below surface was 14.7°C, which is lower than that of the concrete block. And the distance of 20cm and 40cm to surface of section, the temperature is 11.8°C and 9.8°C， respectively. When the heating water of 60°C, the increase of 48h to 72h, the highest temperature is about 47.2°C and 49.6°C.

Chaotic Dynamics of Piezoelectric MEMS Based on Maximum Lyapunov Exponent and Smaller Alignment Index Computations

We characterize the dynamical states of a piezoelectric micrcoelectromechanical system (MEMS) using several numerical quantifiers including the maximum Lyapunov exponent, the Poincaré Surface of Section and a chaos detection method called the Smaller Alignment Index (SALI). The analysis makes use of the MEMS Hamiltonian. We start our study by considering the case of a conservative piezoelectric MEMS model and describe the behavior of some representative phase space orbits of the system. We show that the dynamics of the piezoelectric MEMS becomes considerably more complex as the natural frequency of the system’s mechanical part decreases. This refers to the reduction of the stiffness of the piezoelectric transducer. Then, taking into account the effects of damping and time-dependent forces on the piezoelectric MEMS, we derive the corresponding nonautonomous Hamiltonian and investigate its dynamical behavior. We find that the nonconservative system exhibits a rich dynamics, which is strongly influenced by the values of the parameters that govern the piezoelectric MEMS energy gain and loss. Our results provide further evidences of the ability of the SALI to efficiently characterize the chaoticity of dynamical systems.

Basis function expansions for galactic dynamics: Spherical versus cylindrical coordinates

Aims. The orbital structure of galaxies is strongly influenced by the accuracy of the force calculation during orbit integration. We explore the accuracy of force calculations for two expansion methods and determine which one is preferable for orbit integration. Methods. We specifically compare two methods, one was introduced by Hernquist & Ostriker (HO), which uses a spherical coordinate system and was built specifically for the Hernquist model, and the other by Vasiliev & Athanassoula (CylSP) has a cylindrical coordinate system. Our comparisons include the Dehnen profile, its triaxial extension (of which the Hernquist profile is a special case) and a multicomponent system including a bar and disk density distributions for both analytical models and N-body realizations. Results. For the generalized Dehnen density, the CylSP method is more accurate than the HO method for nearly all inner power-law indices and shapes at all radii. For N-body realizations of the Dehnen model, or snapshots of an N-body simulation, the CylSP method is more accurate than the HO method in the central region for the oblate, prolate, and triaxial Hernquist profiles if the particle number is more than 5 × 105. For snapshots of the Hernquist models with spherical shape, the HO method is preferred. For the Ferrers bar model, the force from the CylSP method is more accurate than the HO method. The CPU time required for the initialization of the HO method is significantly shorter than that for the CylSP method, while the HO method costs subsequently much more CPU time than the CylSP method if the input corresponds to particle positions. From surface of section analyses, we find that the HO method creates more chaotic orbits than the CylSP method in the bar model. This could be understood to be due to a spurious peak in the central region when the force is calculated with the HO expansion. Conclusions. For an analytical model, the CylSP method with an inner cutoff radius of interpolation Rmin as calculated by the AGAMA software, is preferred due to its accuracy. For snapshots or N-body realizations not including a disk or a bar component, a detailed comparison between these two methods is needed if a density model other than the Dehnen model is used. For multicomponent systems, including a disk and a bar, the CylSP method is preferable.