Occurrence Of Chaotic Motion Research Articles

Sensitivity of actuation dynamics on normal and lateral Casimir forces: Interaction of phase change and topological insulator materials.

We investigated here the influence of the lateral and normal Casimir force on the actuation dynamics between sinusoidal corrugated surfaces undergoing both normal and lateral displacements. The calculations were performed for topological insulators and phase change materials that are of high interest for device applications. The results show that the lateral Casimir force becomes stronger by increasing the material conductivity and the corrugations toward similar sizes producing wider normal separation changes during lateral motion. In a conservative system, bifurcation and Poincaré portrait analysis shows that larger but similar in size corrugations and/or higher material conductivity favor stable motion along the lateral direction. However, in the normal direction, the system shows higher sensitivity on the optical properties for similar in size corrugations leading to reduced stable operation for higher material conductivity. Furthermore, in non-conservative systems, the Melnikov function with the Poincaré portrait analysis was combined to probe the possible occurrence of chaotic motion. During lateral actuation, systems with more conductive materials and/or the same but high corrugations exhibit lower possibility for chaotic motion. By contrast, during normal motion, chaotic behavior leading to stiction of the moving components is more likely to occur for systems with more conductive materials and similar in magnitude corrugations.

Chaotic motion of a nonlinear near resonance centrifuge

The equilibrium point and stability of the motion equation of the nonlinear near resonance centrifuge is studied, and the critical conditions for chaotic motions of the system under external excitation are studied by Melnikov method. The expression of Melnikov function and the boundary value between chaotic and non-chaotic regions are given. According to the range of parameters, the numerical simulations are carried out. The results show that the critical parameters of chaotic motion determined using Melnikov method are consistent with that obtained by the numerical simulation. This method effectively judges the occurrence of chaotic motion.

Resonant Chaotic Dynamics of a Symmetric Cross-Ply Composite Laminated Plate Under Transverse and In-Plane Excitations

The resonant chaotic dynamics of a symmetric cross-ply composite laminated plate are studied using the exponential dichotomies and an averaging procedure for the first time. The partial differential governing equations of motion for the symmetric cross-ply composite laminated plate are derived by using Reddy’s third-order shear deformation plate theory and von Karman type equation. The partial differential governing equations of motion are discretized into two-degree-of-freedom nonlinear systems including the quadratic and cubic nonlinear terms by using Galerkin method. There exists a fixed point of saddle-focus in the linear part for two-degree-of-freedom nonlinear system. The Melnikov method containing the terms of the nonhyperbolic mode is developed to investigate the resonant chaotic motions of the symmetric cross-ply composite laminated plate. The obtained results indicate that the nonhyperbolic mode of the symmetric cross-ply composite laminated plate does not affect the critical conditions in the occurrence of chaotic motions in the resonant case. When the resonant chaotic motion occurs, we can draw a conclusion that the resonant chaotic motions of the hyperbolic subsystem are shadowed for the full nonlinear system of the symmetric cross-ply composite laminated plate.

Investigation of chaotic instabilities in railway wheel squeal

The occurrence of chaotic motion in a railway instability phenomenon known as wheel squeal is investigated. The analysis is motivated and applied to predicting the large amplitude friction excited oscillations of the coupled wheel and rail motion. The equations of motion reduce to two autonomous coupled nonlinear second-order systems. Instabilities of the wheel and rail motions are shown to be due to the friction coupling which at low amplitudes causes limit cycle behaviour via a Hopf bifurcation. When the amplitude grows large enough, full nonlinear creep oscillations are shown to occur causing oscillations about positive and negative sliding conditions. Chaos is shown to occur when the motion meanders and jumps between large positive and negative creep and is characterised by a Poincare map with fractal nature. The route to chaos is shown to be via quasiperiodicity. Based on this insight, necessary analytical conditions for wheel squeal chaos are developed and verified using numerical simulations over a range of angles of attack and wheel/rail contact angles. Conditions under which chaotic instability is more likely to occur are identified and discussed, including mode coupling and parametric excitation. The results may describe why some very loud occurrences of wheel squeal are not characterised by a pure tone.

Orbital stability and homoclinic bifurcation in a parametrized deformable double-well potential

Abstract The motion of a particle in a one-dimensional deformable double-well potential is studied. In a case study related deformable potential that permits theoretical adaptation of the model to various physical situations, the existence and the stability of periodic motion as well as the routes toward chaos upon the potential parameter are investigated. It is demonstrated that the shape of the potential is crucial in controlling the orbital stability of the system. The occurrence of chaotic motions is found to be quite sensitive to the shape parameter of the potential. These instability and chaotic behaviors result from a delicate balance between the damping, the periodic force and the total nonlinearity induced by the variable shape potential.

Chaotic Threshold for a Class of Power System Model

This paper deals with the bifurcation and chaotic dynamic characteristic of a single‐machine infinite‐bus (SMIB) power system under two kinds of harmonic excitation disturbance, which are induced by the external periodic load and the outer mechanical disturbance. By applying Melnikov’s method, the threshold value for the occurrence of chaotic motion is provided. In addition, the chaotic boundary surface is given. The efficiency of the criteria for chaotic motion obtained in this paper is verified by bifurcation diagram, phase portraits, Poincaré section, and frequency spectrum. The results obtained in this paper will provide a better understanding of the nonlinear dynamic behaviors for this class of SMIB power system subjected to two kinds of harmonic excitation components.

Non-linear dynamics and control of an automotive suspension system based on local and global bifurcation analysis

This paper details the non-linear dynamic behaviours and control of a non-linear semi-active suspension system using a quarter-car model under kinematic excitation by a road surface profile. The results of local and global bifurcation analysis indicate that the hysteretic non-linear characteristics of damping force cause the suspension system to exhibit codimension-two bifurcation, resulting in homoclinic orbits and a pitchfork bifurcation. The complex dynamic behaviour of automotive suspension systems was examined using a bifurcation diagram, phase portraits, a Poincare map, and frequency spectra. We also used Lyapunov exponent to identify the occurrence of chaotic motion and verify our analysis. Finally, a dither signal control was used to convert chaotic behaviours into periodic motion. Simulation results verify the effectiveness of the proposed control method.

Synchronization with particular application to the control of a chaotic electromagnetic system

This study confirms the chaotic motion of an electromagnetic dynamic system and then elucidates the application of synchronization to an electromagnetic system to control the chaos. A bifurcation diagram reveals rich non-linear behaviour over a range of parameter values. Next, the Lyapunov exponent procedure is employed to show the occurrence of chaotic motion and to verify the results of numerical simulation. Finally, a continuous feedback control approach based on the synchronization characteristics is developed to control a chaotic electromagnetic system. Simulation results demonstrate the feasibility of the proposed method.

Chaos of a beam on a nonlinear elastic foundation under moving loads

Chaos of a beam on a nonlinear elastic foundation under moving vehicle loads is investigated. A nonlinear vibration equation is obtained using Galerkin method. Occurrence of chaotic motions is analyzed using the method of Melnikov. Effects of system parameters on chaotic region are studied. Poincare maps, phase trajectories and time history curves are also obtained to verify the theoretical result from Melnikov method.

Non-linear dynamics and chaos control for an electromagnetic system

A non-linear mathematical model has been obtained by applying a modified conventional identification technique based on the principle of harmonic balance. In this study, analytical work is carried out on this identified non-linear model by applying the first-harmonic approximation solution and the Floquet theory. The resulting criteria for bifurcations can be used to evaluate the operational range of a system employing such a non-linear actuator. We also employ the method of Lyapunov exponents to show the occurrence of chaotic motion and to verify the above analyses. Finally, various methods, such as the state feedback control and injection of dither signal control are used to control chaos effectively.

Chaotification of Induction Motor Drives under Periodic Speed Command

This paper proposes an approach to chaotify a practical induction motor drive system in which field-oriented control is adopted. The key is to tune the gain of the compensator for the compensation of the increase of rotor resistance due to prolonged operation. Hence, the occurrence of chaotic motion becomes controllable when the drive system is under a periodic speed command. The validity of the proposed chaotification is verified by both computer simulation and experimental measurement.

Noise-induced chaotic motions in Hamiltonian systems with slow-varying parameters

This paper studies chaotic motions in quasi-integrable Hamiltonian systems with slow-varying parameters under both harmonic and noise excitations. Based on the dynamic theory and some assumptions of excited noises, an extended form of the stochastic Melnikov method is presented. Using this extended method, the homoclinic bifurcations and chaotic behavior of a nonlinear Hamiltonian system with weak feed-back control under both harmonic and Gaussian white noise excitations are analyzed in detail. It is shown that the addition of stochastic excitations can make the parameter threshold value for the occurrence of chaotic motions vary in a wider region. Therefore, chaotic motions may arise easily in the system. By the Monte-Carlo method, the numerical results for the time-history and the maximum Lyapunov exponents of an example system are finally given to illustrate that the presented method is effective.

Neimark bifurcation, almost-periodicity and chaos in the forced van der Pol-Duffing system in the neighbourhood of the principal resonance

The occurrence of chaotic motion in the forced van der Pol-Duffing oscillator in close neighbourhood of the principal resonance is considered and interpreted in connection with subcritical Neimark bifurcation. Computer simulations and approximate analytical analysis confirm that the chaotic motion region forms a transition zone between two regular motions: the T-periodic and almost-periodic oscillations.

Bifurcation, Chaos and Suppression of Chaos in FitzHugh-Nagumo Nerve Conduction Model Equation

We study the effect of constant and periodic membrane currents in neuronal axons described by the FitzHugh-Nagumo equation in its wave form. Linear stability analysis is carried out in the absence of periodic membrane current. Occurrence of chaotic motion, (i) in the absence of both constant and periodic membrane currents, (ii) with constant current only, (iii) with periodic membrane current only, and (iv) with both constant and periodic currents is investigated for specific parametric choices. We show how chaos sets in through a cascade of period doubling bifurcations. We then demonstrate the possibility of control of chaos using various control mechanisms. Specifically, we show the control of chaos by (i) adaptive control mechanism, (ii) periodic parametric perturbation and (iii) stabilization of unstable periodic orbits.

Asteroid Proper Elements and the Dynamical Structure of the Asteroid Main Belt

We have computed proper elements for 12,573 asteroids, including all the ones with orbits accurate enough to be useful for family identification. This was done with an upgraded version of our iterative analytical algorithm, resulting in significantly improved accuracy for most asteroids in the low to moderate inclination and eccentricity region of the main belt (typical instability in the proper e and sin I being ≤0.0015 over 5 Myr). This stability has been verified by numerical integrations (within a realistic model) of 35 test cases. In a small percentage of cases, the accuracy of the proper elements computation was degraded by the effects of some resonance, either in mean motion or secular. We have been able to list the resonances responsible for this degradation in almost all the cases, in such a way that these are properly flagged with a "resonance code." This list of resonances, including nine high-degree secular resonances not known before our work, provides a detailed map of the dynamical structure of the asteroid main belt. We investigate the long-term dynamics of some of these secular resonances and find both very large amplitude oscillations of the eccentricity and irregular behavior for asteroids affected by one or more resonances. We show the geometry of these resonances in the proper elements space and their relationship with the asteroid distribution and with the most prominent families. In the second part of the paper, we give a detailed description of the improvements of this version of the proper elements theory with respect to the previously published ones (A. Milani and Z. Knežević, 1990, Celest. Mech. 49, 347-411; 1992, Icarus 98, 211-232). We discuss the success of some of these improvements in cases which were previously of degraded accuracy, and we also comment on the failure of some attempted improvements. We conjecture that our theory is very close to the fundamental limitations to the accuracy of any analytical theory, which result from the fact that there is an infinite web of resonances and from the occurrence of chaotic motions. Thus we think that further improvements of this theory will only be of local nature and will affect only a fraction of the asteroid population.

Dynamics of a harmonically excited impact damper: Bifurcations and chaotic motion

The dynamic behavior of a vibro-impact damper model with two degrees of freedom is investigated. The motions of this model include not only periodic motions but also irregular motions. Instead of employing classical methods for analyzing the dynamic characteristics of this two-degree-of-freedom impact-clearance problem, the concept of using the Poincaré map for observing the bifurcation phenomena is utilized in studying the existence and stability of subharmonic motions. Global bifurcations are discussed by using numerical simulation, and the evolution from the period-doubling sequence to chaotic motions is illustrated. In addition, the occurrence of chaotic motions is examined numerically, on the basis of Devaney's definition for the existence of chaos.

Nonlinear behaviour of an articulated offshore loading platform

An analytical model was developed for the dynamic analysis of an articulated loading platform (ALP) under regular wave excitations. The dynamic response and stability of the ALP tower, treated as a single-degree-of-freedom system, was investigated with direct numerical integration and approximate analytical methods. The highly nonlinear characteristics, including subharmonics, instabilities, period-doubling bifurcations and Lyapunov exponents, were examined. Possible occurrence of chaotic motion was investigated by means of direct numerical integration and application of the classical stability theory with the aid of Hill's type equation. It was discovered that for the parameters studied, chaotic motion with only take place under the action of a train of extremely large waves. Such motion may develop either through period-doubling bifurcations or take place suddenly depending on the excitation amplitude.

Non-linear behavior and chaotic motions of an sdof system with piecewise-non-linear stiffness

Forced vibrations of a single-degree-of-freedom system with unsymmetric, piecewisenon-lincar restoring forces are investigated using direct numerical integration and approximate analytical methods. A simple and efficient algorithm was developed to analyse the piecewisenon-linear system by the harmonic balance method and fast Fourier transformation technique. The highly non-linear characteristics, including subharmonics, instabilities, bifurcations, Lyapunov exponents and domain of attraction, were examined. Possible occurrence of chaotic motion was investigated by means of direct numerical integration and the classical stability theory with the aid of Hill's type equation. It is found that the threshold of the onset of chaotic motion is very sensitive to any changes in system parameters, and chaotic motion will occur as the excitation amplitude exceeds certain limits.

On the dynamic response of the hysteretic system

In this study, the jump phenomena and the chaotic motion are investigated. Firstly, it is found that the jump phenomenon occurs under the harmonic excitation by means of computer simulation, and that it occurs under the narrow band random excitation by the averaging method, too. And the conditions for the occurrence of jump phenomena, the probability density function of dynamic response and the resonance curves are shown, respectively. Furthermore, the phenomena are considered from the viewpoint of mean square value, where the statistical linearization method and Liapunov matrix equation are used. Consequently, it seems that the method of equivalent linearization for smooth hysteretic systems under random excitation is versatile and has a great potential in the analysis of hysteretic system, but that it is impertinent to the research of jump phenomena. Secondly, the occurrence of chaotic motion is disscussed by means of the method of mapping and the averaging method. As the results, it is found that the chaotic motion occurs under the various parameters.