Search For Periodic Orbits Research Articles

Search for Periodic Orbits Appearing through Anomalous Rotation Bifurcation

Search for Periodic Orbits Appearing through Anomalous Rotation Bifurcation

Discovering the dynamic behavior of unknown systems using fuzzy logic

To know the dynamic behavior of a system it is convenient to have a good dynamic model of it. However, in many cases it is not possible either because of its complexity or because of the lack of knowledge of the laws involved in its operation. In these cases, obtaining models from input–output data is shown as a highly effective technique. Specifically, intelligent modeling techniques have become important in recent years in this field. Among these techniques, fuzzy logic is especially interesting because it allows to incorporate to the model the knowledge that is possessed of the system, besides offering a more interpretable model than other techniques. A fuzzy model is, formally speaking, a mathematical model. Therefore, this model can be used to analyze the original system using known systems analysis techniques. In this paper a methodology for extract information from unknown systems using fuzzy logic is presented. More precisely, it is presented the exact linearization of a Takagi–Sugeno fuzzy model with no restrictions in use or distribution of its membership functions, as well as obtaining its equilibrium states, the study of its local behavior and the search for periodic orbits by the application of Poincare.

ChemInform Abstract: On the Floquet‐Magnus Expansion: Applications in Solid‐State Nuclear Magnetic Resonance and Physics

AbstractReview: 278 refs.

Search for periodic orbits in Agekyan and Anosova’s region D for the general three-body problem

The goal of this study is to search for close-to-periodic orbits in the general three-body problem with equal masses and zero initial velocities. This search was carried out by scanning region D for all possible configurations of triple systems specified by coordinates (ξ, η). Points in region D whose vicinities could contain the initial conditions for exactly periodic orbits were identified. This was done by requiring that two conditions be satisfied: (1) the minimum of the sum of the squares of the differences between the initial and current coordinates and velocities of the bodies is less than a specified value; (2) the time T when this minimum is attained does not exceed 10τ, where τ is the mean crossing time. This analysis reveals 22 regions with initial coordinates (ξ, η) corresponding to periodic orbits. The trajectories of bodies with initial conditions in these regions are constructed. Some properties of the structure of the orbits found are described.

Search for periodic orbits in the general three-body problem

An original method for searching for regions of initial conditions giving rise to close to periodic orbits is proposed in the framework of the general three-body problem with equal masses and zero angular momentum. Until recently, three stable periodic orbits were known: the Schubart orbit for the rectilinear problem, the Broucke orbit for the isosceles problem, and the Moore eight-figure orbit. Recent studies have also found new periodic orbits for this problem. The proposed method minimizes a functional that calculates the sum of squared differences between the initial and current coordinates and the velocities of the bodies. The search is applied to short-period orbits with periodsT < 10τ, where τ is the mean crossing time for the components of the triple system. Twenty one regions of initial conditions, each corresponding to a particular periodic orbit, have been found in the current study. A criterion for the reliability of the results is that the initial conditions for the previously known stable periodic orbits are located inside the regions found. The trajectories of the bodies with the corresponding initial conditions are presented. The dynamics and geometry of the orbits constructed are described.

Search for Periodic Orbits in Delay Differential Equations

In a previous paper, we developed a new way to apply the Homotopy Analysis Method (HAM) in the search for periodic orbits in dynamical systems modeled by ordinary differential equations. This method differs from the original in the heuristic search of the frequencies of the cycles. In this paper, we show that the method can be extended to the search for periodic orbits in delay differential equations. Herein, this methodology is applied twice, firstly in an equation of van der Pol type and secondly in an anharmonic oscillator, both systems with a delayed feedback.

AN EXACT MAP FOR A CHAOTIC BILLIARD

An exact map for Sinai's billiard with maximal central scatterer has been deduced from geometrical principles. Using this map a simple numerical method has been implemented for the search of periodic orbits. Results that seem to be useful in classical as quantum chaos are shown.

Periodic orbits for anisotropic perturbations of the Kepler problem

We study the Kepler problem perturbed by an anisotropic term, that is a potential conformed by a Newtonian term, 1 / r , plus an anisotropic term, b / ( r 2 [ 1 + ϵ cos 2 θ ] ) β / 2 . Because of the anisotropic term, although the system is conservative the angular momentum is not a constant of motion. In this work we present an analytic and numerical analysis for the periodic orbits of a particle moving under the influence of the above potential. This is a reversible system with two degrees of freedom; thus the technique of symmetry lines can be used in the search for periodic orbits. For the particular case of β = 2 , there is a second constant of motion, so we can define a special kind of Kepler’s third law. We present comparative results for the integrable case β = 2 , and the cases β = 1 and β = 3 .

Pattern recognition using chaotic neural networks

Pattern recognition by chaotic neural networks is studied using a hyperchaotic neural network as model. Virtual basins of attraction are introduced around unstable periodic orbits which are then used as patterns. Search for periodic orbits in dynamical systems is treated as a process of pattern recognition. The role of synapses on patterns in chaotic networks is discussed. It is shown that distorted states having only limited information of the patterns are successfully recognized.

The cycles of Selene

The discovery and use of the Saros, a lunar cycle of 18 years and 10 or 11 days, is reviewed from its earliest origins two millennia ago to the present day, when it is known with precision and enables the accurate prediction of both time and type of solar and lunar eclipses. The theoretical basis for the Saros is discussed, along with other historically known lunar cycles. The geometry of the Sun-Moon-Earth system is found to repeat itself after one Saros, not only at eclipses but also at any phase of the cycle, indicating that the Moon moves in a nearly periodic orbit. The search for periodic orbits using the Saros has led to the discovery of a set of eight periodic orbits of period equal to one Saros whose time evolutions closely resemble that of the real Moon. Finally, the potential of the Saros in studying the dynamics and stability of the Earth-Moon system is examined and the existence of other Saros-like cycles of longer periods in the present, past and future of the Earth-Moon-Sun system is explored.

Numerical models for the study of motion of lunar satellites

The present study deals with numerical modeling of the elliptic restricted three-body problem as well as of the perturbed elliptic restricted three-body (Earth-Moon-Satellite) problem by a fourth body (Sun). Two numerical algorithms are established and investigated. The first is based on the method of the series solution of the differential equations and the second is based on a 5th-order Runge-Kutta method. The applications concern the solution of the equations and integrals of motion of the circular and elliptical restricted three-body problem as well as the search for periodic orbits of the natural satellites of the Moon in the Earth-Moon system in both cases in which the Moon describes circular or elliptical orbit around the Earth before the perturbations induced by the Sun. After the introduction of the perturbations in the Earth-Moon-Satellite system the motions of the Moon and the Satellite are studied with the same initial conditions which give periodic orbits for the unperturbed elliptic problem.