Possibility Of Chaotic Motion Research Articles

Rydberg excitons in cuprous oxide: A two-particle system with classical chaos.

When an electron in a semiconductor gets excited to the conduction band, the missing electron can be viewed as a positively charged particle, the hole. Due to the Coulomb interaction, electrons and holes can form a hydrogen-like bound state called the exciton. For cuprous oxide, a Rydberg series up to high principle quantum numbers has been observed by Kazimierczuk et al. [Nature 514, 343 (2014)] with the extension of excitons up to the μm-range. In this region, the correspondence principle should hold and quantum mechanics turn into classical dynamics. Due to the complex valence band structure of Cu2O, classical dynamics deviates from a purely hydrogen-like behavior. The uppermost valence band in cuprous oxide splits into various bands resulting in yellow and green exciton series. Since the system exhibits no spherical symmetry, the angular momentum is not conserved. Thus, the classical dynamics becomes non-integrable, resulting in the possibility of chaotic motion. Here, we investigate the classical dynamics of the yellow and green exciton series in cuprous oxide for two-dimensional orbits in the symmetry planes as well as fully three-dimensional orbits. Our analysis reveals substantial differences between the dynamics of the yellow and green exciton series. While it is mostly regular for the yellow series, large regions in phase space with classical chaos do exist for the green exciton series.

Dynamical sensitivity on thermal fluctuations of torsional homogeneous and nonhomogeneous microsystems under the influence of Casimir and electrostatic torques: Nonlinear actuation towards chaotic motion

We have studied how the mechanical Casimir torque between the components of a torsional double beam microdevice under the influence of thermal fluctuations could play significant role on the dynamics of the device, and under what conditions can even lead to chaotic behavior. The influence of thermal fluctuations on the dynamical behavior of torsional microsystems, whether homogeneous or nonhomogeneous, has been revealed through bifurcation and phase diagrams, and Poincare maps. For nonhomogeneous microsystems, the device with low conductivity reveals significant sensitivity to thermal fluctuations. Moreover, it is shown the significant dependence of the optimal voltage distribution, which leads to a significant increase in the stable operation range of the device, on the ambient temperature conditions and the optical properties of the constituent materials. For nonconservative torsional microsystem, the use of the Melnikov method and Poincaré plots have shown how thermal effects increase the probability of chaotic behavior. This is because different temperature enhances the effective Casimir torque and the possibility of chaotic motion. Furthermore, it is illustrated how the use of the optimal voltage distribution, by taking into consideration thermal fluctuations and optical properties, can keep the device away from chaotic behavior. This is because by dividing the electrostatic torques the influence of the effective Casimir torque is reduced. Finally, it is demonstrated that by transforming a homogeneous system into a heterogeneous one, the response of the system to thermal fluctuations is reduced making it possible to avoid to increase the possibility for chaotic behavior by changing the temperature.

Numerical analysis of grasshopper escapement.

The dynamics of a driven, damped pendulum as used in mechanical clocks is numerically investigated. In addition to the analysis of well-known mechanisms such as chronometer escapement, the unusual properties of Harrison's grasshopper escapement are explored, giving some insights regarding the dynamics of this system. Both the steady-state operation and transient effects are discussed, indicating the optimal condition for stable long-term clock accuracy. The possibility of chaotic motion is investigated.

Threshold for horseshoe chaos in fractional-order hysteretic nonlinear suspension system of vehicle

The chaotic behavior of a nonlinear vehicle suspension system with a fractional-order derivative is considered. A hysteretic nonlinear model of vehicle suspension with a fractional-order derivative term is established and the analytically necessary condition for heterotopic chaos is derived based on the Melnikov method. The largest Lyapunov exponents are compared. Then, the necessary condition is numerically verified by various simulation factors. The possibility of chaotic motion in vehicles should be higher for larger amplitude of road excitation. It is found that the coefficient and order of the fractional differential term, the stiffness coefficient, and the damping coefficient of the system all affect the necessary condition, and analysis on the effects of these parameters is presented individually. It has been shown that the larger the coefficient of the fractional differential term, or the stiffness and damping coefficients of the system, the lower the possibility of chaos in the system. Meanwhile, without considering the fractional order, the integer order suspension model obviously reduces the actual area where chaos may occur, so designing suspensions according to the fractional order model can avoid chaos more accurately than the integer order model.

Analysis on Chaotic Vibrations of the Magneto-Rheological Suspension System

The magneto-rheological(MR) suspension system has been established, by employing the modified Bouc-wen force-velocity (F-v) model of magneto-rheological damper(MRD). The possibility of chaotic motions in MR suspension is discovered by employing nonlinear systems stability theory. With the bifurcation diagram and corresponding Lyapunov exponent spectrum diagrams detected through numerical calculation, we can observe the complex dynamical behaviors and oscillating mechanism of alternating periodic oscillations, quasi-periodic oscillations and chaotic oscillations with different profiles of road excitation, as well as the dynamical evolution to chaos by period-doubling bifurcations, saddle-node bifurcations and reverse period-doubling bifurcations.

Nonlinear Dynamics Analysis of the Semiactive Suspension System with Magneto-Rheological Damper

This paper examines dynamical behavior of a nonlinear oscillator which models a quarter-car forced by the road profile. The magneto-rheological (MR) suspension system has been established, by employing the modified Bouc-Wen force-velocity (F-v) model of magneto-rheological damper (MRD). The possibility of chaotic motions in MR suspension is discovered by employing the method of nonlinear stability analysis. With the bifurcation diagrams and corresponding Lyapunov exponent (LE) spectrum diagrams detected through numerical calculation, we can observe the complex dynamical behaviors and oscillating mechanism of alternating periodic oscillations, quasiperiodic oscillations, and chaotic oscillations with different profiles of road excitation, as well as the dynamical evolutions to chaos through period-doubling bifurcations, saddle-node bifurcations, and reverse period-doubling bifurcations.

Resonance Properties of Extra-Solar Two-Planet Systems

The properties of the distribution of resonance zones in the two-planet problem are analyzed as a function of the semi-major axes and masses of the planets on cosmogonic time scales. A solution to equations averaged using the Hori—DePrit method with accuracy to second order in the small parameter of the problem is used, and the translation from the averaged to the osculating elements taken into account. Conditions for the overlap of resonance zones are obtained. It is shown that motion of pairs of planets in the extrasolar planetary systems HD 73526, 47 UMa, HD 181433 (c-d), GJ 581 (b-c), and HD 155358 can occur in regions of resonance overlap. The orbital evolution of the two-planet system 47 UMa is analyzed in the absence of resonance, in the vicinity of the 1: 2, 3: 7, 2: 5, 3: 8,1: 3, and 1: 7 resonances when the initial data correspond to a region of wide resonance, and in regions of overlap of wide-resonance zones. The possibility of chaotic motions of the planets in the 47 UMa system is demonstrated.

The dynamic analysis of the gas lubricated stage in optical lithography

To identify the model and study the possibility of chaotic motion of the gas lubricated stage, the dynamical behavior of the stage with gas-lubricated bearings in optical lithography is studied theoretically and experimentally. A model of the bearing is established by a single mode approximation using the Navier-stokes equation and the Newton equation. The setup of the stage is dragged by a linear motor, and its positions in three directions are detected by three laser interferometers. Analysis of the dynamics of the stage is carried out too. The results show that the amplitude frequency response of the stage is largely affected by the nonlinear air bearings, and the positioning accuracy is affected by the gap of the air bearings, as well as the control method. So the suitable gap of the air bearing should be taken into account when designing and manufacturing the ultra precision positioning stage.

Experimental Observation of Chaotic Motion in a Rotor with Rubbing

This paper presents an application for chaotic motion identification in a measured signal obtained in an experiment. The method of state space reconstruction with delay co-ordinates with the dynamic evolution described by a map is used. Poincare diagrams, correlation dimensions and Lyapunov exponents are obtained as tools for deciding about the existence of chaotic behaviour. The method is applied to measurements of the lateral displacement of a vertical rotor experiencing rubbing and in some signals chaos is observed. The work concludes that the possibility of chaotic motion is well determined with the observation of Poincare diagrams and computation of Lyapunov exponents. Correlation dimensions computations, strongly influenced by noise, are not convenient tools for investigation of chaotic behaviour in signals generated by mechanical systems.

A Review of Theoretical Models for Fluidelastic Instability of Cylinder Arrays in Cross-Flow

A review of all known theoretical models of fluidelastic instability for cylinder arrays subject to fluid cross-flow is presented. Particular emphasis is given to the physics of the different instability mechanisms, and the assumptions made are analysed and discussed. Comparison are made between the models, and with available experimental data; these comparisons are in terms of the critical flow velocity for instability to occur and of the subcritical response of cylinders in the arrays. Despite the considerable differences in the theoretical models, there is some agreement in the general conclusion obtained. In particular, it is shown that the most important parameter for predicting fluidelastic instability is the unsteady nature of the interstitial flow in the array; specifically, the phase-lag between cylinder motion and fluid forces generated thereby. Methods appropriate for using the analyses, which are predominantly two-dimensional in nature, to predict the stability of three-dimensional heat exchanger spans subject to nonuniform flow are also evaluated. Finally, the effect of nonlinearities, both structural and fluid, on the post-instability behavior is reviewed. For loosely supported tubes with impacting occurring at the loose supports, the possibility of chaotic motion, with a consequent increase in wear rate, is predicted.

Nonlinear dynamics of an electron-beam plasma system

This paper gives a discussion of the analytic properties of an electron-beam plasma dynamical system. The latter is governed by a system three coupled nonlinear ordinary differential equations. Arguments are provided against the possibility of chaotic motions in this system.

Predicting the Autonomous Behaviour of Simple Nonlinear Feedback Systems

The paper examines in depth two approaches, namely the describing function and Tsypkin methods, for predicting the autonomous behaviour of simple nonlinear feedback systems. Both procedures are supported by software which, in the case of the describing function method, allows iteration to the exact limit cycle solution and, for both methods, enables display of resulting limit cycle waveforms. One advantage of the Tsypkin method, which is applicable primarily to relay systems, is that the exact stability of the limit cycle solution can be found. It is shown how this may be helpful in indicating the possibility of chaotic motion. Several examples are given to show the advantages and limitations of the software implementations of the methods.