Synchronization of a periodically forced Duffing oscillator with a periodically excited pendulum

In this paper, the analytical conditions for a periodically forced Duffing oscillator synchronized with a chaotic pendulum are developed through the theory of discontinuous dynamical systems. From the analytical conditions, the synchronization invariant domains are developed. For a better understanding of synchronization of two different dynamical systems, the partial and full synchronizations of the Duffing oscillator with the chaotic pendulum are presented for illustrations. The control parameter map is developed from the analytical conditions. Under special parameters, the two systems can be fully and partially synchronized. Since the forced pendulum has librational and rotational chaotic motions, the periodically forced Duffing oscillator can be synchronized only with the librational chaotic motions of the pendulum. It is impossible for the forced Duffing oscillator to be synchronized with the rotational chaotic motions.     

Chaos of a parametrically excited undamped pendulum

The shooting method is applied to prove that a pendulum with oscillatory forcing makes chaotic motions for certain parameters. The method is more intuitive than using the Poincare’ map and provides more information about when the chaos occurs. It proves that more chaotic solutions exit.     

Chaotic motions of a parametrically excited pendulum

The shooting method is applied to prove that a pendulum with oscillatory forcing makes chaotic motions for certain parameters. The method is more intuitive than an using the Poincare’ map and provides more information about when the chaos occurs. It proves that more chaotic solutions exit.     

A generalized perturbed pendulum

The simple pendulum is a paradigm in the study of oscillations and other phenomena in physics and nonlinear dynamics. This explains why it has deserved much attention, from many viewpoints, for a long time. Here, we attempt to describe what we call a generalized perturbed pendulum, which comprises, in a single model, many known situations related to pendula, including different forcing and nonlinear damping terms. Melnikov analysis is applied to this model, with the result of general formulae for the appearance of chaotic motions that incorporate several particular cases. In this sense, we give a unified view of the pendulum.     

The chaotic synchronization of a controlled pendulum with a periodically forced,damped Duffing oscillator

In this paper, the chaotic synchronization of the Duffing oscillator and controlled pendulum is investigated. From the analytical conditions developed in [1], the partial and full synchronizations of the controlled pendulum with chaotic motions in the Duffing oscillator are discussed. Compared with the periodic synchronization, in the chaotic synchronization, switching points for appearance and vanishing of the partial synchronization are chaotic. The control parameter map for the synchronization is developed from the analytical conditions, and the partial and full synchronizations are illustrated to show the analytical conditions. This synchronization is different from the controlled Duffing oscillator synchronizing with chaotic motion in the periodically excited pendulum. For a better understanding of synchronization characteristics between two different dynamical systems, effects with other parameters will be discussed later.     

Vibrations of a Pendulum with Oscillating Support and Extra Torque

The motion of a driven planar pendulum with vertically periodically oscillating point of suspension and under the action of an additional constant torque is investigated. We study the influence of the torque strength on the transition to chaotic motions of the pendulum using Melnikov's analysis. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear Dynamics of the 3D Pendulum

A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force. 3D pendulum dynamics have been much studied in integrable cases that arise when certain physical symmetry assumptions are made. This paper treats the non-integrable case of the 3D pendulum dynamics when the rigid body is asymmetric and the center of mass is distinct from the pivot location. 3D pendulum full and reduced models are introduced and used to study important features of the nonlinear dynamics: conserved quantities, equilibria, relative equilibria, invariant manifolds, local dynamics, and presence of chaotic motions. The paper provides a unified treatment of the 3D pendulum dynamics that includes prior results and new results expressed in the framework of geometric mechanics. These results demonstrate the rich and complex dynamics of the 3D pendulum.     

Predictability portraits for chaotic motions

The reliability of forecasts for chaotic motions varies with the state of the dynamical system. We define quantities that measure predictability and investigate their dependence on the initial state for different forecasting times. Two model systems are investigated, a driven damped pendulum and the Lorenz system. We use two different numerical methods to analyse the effect of finite resolution in determining the initial conditions on the reliability of forecasts. For the pendulum we also compare numerical forecasts with experimental data.     

Periodic and chaotic synchronizations of two distinct dynamical systems under sinusoidal constraints

In this paper, periodic and chaotic synchronizations between two distinct dynamical systems under specific constraints are investigated from the theory of discontinuous dynamical systems. The analytical conditions for the sinusoidal synchronization of the pendulum and Duffing oscillator were obtained, and the invariant domain of sinusoidal synchronization is achieved. From analytical conditions, the control parameter map is developed. Numerical illustrations for partial and full sinusoidal synchronizations of chaotic and periodic motions of the controlled pendulum with the Duffing oscillator are carried out. This paper presents how to apply the theory of discontinuous dynamical systems to dynamical system synchronization with specific constraints. The function synchronization of two distinct dynamical systems with specific constraints should be identified only by G-functions. The significance of function synchronization of distinct dynamical systems is to make the synchronicity behaviors hidden, which is very useful for telecommunication synchronization and network security.     

The dynamics of a spherical pendulum with a vibrating suspension

The motion of a spherical pendulum whose point of suspension performs high-frequency vertical harmonic oscillations of small amplitude is investigated. It is shown that two types of motion of the pendulum exist when it performs high-frequency oscillations close to conical motions, for which the pendulum makes a constant angle with the vertical and rotates around it with constant angular velocity. For the motions of the first and second types the centre of gravity of the pendulum is situated below and above the point of suspension, respectively. A bifurcation curve is obtained, which divides the plane of the parameters of the problem into two regions. In one of these only the first type of motion can exist, while in the other, in addition to the first type of motion, there are two motions of the second type. The problem of the stability of these motion of the pendulum, close to conical, is solved. It is shown that the first type of motion is stable, while of the second type of motion, only the motion with the higher position of the centre of gravity is stable.     

Synchronization dynamics of two different dynamical systems

In this paper, synchronization dynamics of two different dynamical systems is investigated through the theory of discontinuous dynamical systems. The necessary and sufficient conditions for the synchronization, de-synchronization and instantaneous synchronization (penetration or grazing) are presented. Using such a synchronization theory, the synchronization of a controlled pendulum with the Duffing oscillator is systematically discussed as a sampled problem, and the corresponding analytical conditions for the synchronization are presented. The synchronization parameter study is carried out for a better understanding of synchronization characteristics of the controlled pendulum and the Duffing oscillator. Finally, the partial and full synchronizations of the controlled pendulum with periodic and chaotic motions are presented to illustrate the analytical conditions. The synchronization of the Duffing oscillator and pendulum are investigated in order to show the usefulness and efficiency of the methodology in this paper. The synchronization invariant domain is obtained. The technique presented in this paper should have a wide spectrum of applications in engineering. For example, this technique can be applied to the maneuvering target tracking, and the others.     

Chaotic motion in a micro-electro–mechanical system with non-linearity from capacitors

This investigation is to provide a possible prediction for design, manufacturing, testing and industrial applications of a simplified micro-electro–mechanical system (MEMS). The chaotic motion in a certain frequency band of such a MEMS device is investigated, and the corresponding equilibrium, natural frequency and responses are determined. Under alternating current (AC) voltage, the resonant condition for such a system is obtained. It is observed that the lower-order resonant motions can be easily converted to the mechanical force and sensed to the electrical signal. The chaotic motions in the vicinity of a specified resonant separatrix are investigated analytically and numerically. For given voltages, the AC frequency bands are obtained for chaotic motion in the specific resonant layers and resonant motions, and such chaotic motions can be very easily sensed by the output transducer in MEMS.     

Nonlinear dynamics and chaos in a fractional-order financial system

This study examines the two most attractive characteristics, memory and chaos, in simulations of financial systems. A fractional-order financial system is proposed in this study. It is a generalization of a dynamic financial model recently reported in the literature. The fractional-order financial system displays many interesting dynamic behaviors, such as fixed points, periodic motions, and chaotic motions. It has been found that chaos exists in fractional-order financial systems with orders less than 3. In this study, the lowest order at which this system yielded chaos was 2.35. Period doubling and intermittency routes to chaos in the fractional-order financial system were found.     

An inverted pendulum on a fixed and a moving base

The plane motions of a controlled single-link pendulum with a fixed suspension point and a pendulum with its suspension point located at the centre of a wheel which rolls without sliding along a flat horizontal surface are considered. The control torque, applied to the pendulum at the suspension point, is bounded in absolute magnitude. A controllability domain is constructed in the linear approximation for the one and the other pendulum, from all points of which the pendulum can be brought into the upper unstable equilibrium position without oscillations about the lower equilibrium. It is shown that the domain of controllability is greater for a pendulum mounted on a wheel, as a result it is more easily stabilizable. Control laws are constructed, under which the domain of attraction is identical to the controllability domain and is thereby the largest possible domain.     

Chaos in a four-dimensional system consisting of fundamental lag elements and the relation to the system eigenvalues

A simple system consisting of a second-order lag element (a damped linear pendulum) and two first-order lag elements with piecewise-linear static feedback that has been derived from a power system model is presented. It exhibits chaotic behavior for a wide range of parameter values. The analysis of the bifurcations and the chaotic behavior are presented with qualitative estimation of the parameter values for which the chaotic behavior is observed. Several characteristics like scalability of the attractor and globality of the attractor-basin are also discussed.     

保守双摆的不可积性和混沌

本文用Birkhoff级数正则变换方法求出保守双摆运动方程的近似积分,并把近似积分的等值曲线与数值仿真结果作了比较.由此清楚地看出.当能级提高时,系统由近可积的成为不可积的,即其运动情况由规则的转变为混沌的.本文还介绍了演示上述性态的一个保守双摆模型.     

The influence of dissipative and constant forces on the form and stability of steady motions of mechanical systems with cyclic coordinates

Mechanical systems with cyclic coordinates subject to dissipative forces with complete dissipation and constant forces applied only to the cyclic variables are considered. Problems of the existence of steady motions in such systems and the conditions for their stability are discussed. It is shown, in particular, that if the Rayleigh function is proportional to the kinetic energy, the stability conditions for the steady motions of the system are the same as or (under certain assumptions) similar to such conditions for steady motions of a corresponding conservative system. The example of a physical pendulum is used to show that such conclusions are generally false: dissipative and constant forces may cause destabilization of stable motions of the system.     

Construction and quatitative characterization of a chaotic saddle from a pendulum experiment

We experimentally study the behaviour of a parametrically driven damped pendulum in a parameter region where a transient chaotic motion is observed. We reconstruct the chaotic saddle and a chaotic attractor near an interior crisis in a stroboscopic phase representation and give an estimation of the corresponding f() spectra.     

Towards a computer-assisted proof for chaos in a forced damped pendulum equation

We report on the first steps made towards the computational proof of the chaotic behaviour of the forced damped pendulum. Although, chaos for this pendulum was being conjectured for long, and it has been plausible on the basis of numerical simulations, there is no rigorous proof for it. In the present paper we provide computational details on a fitting model and on a verified method of solution. We also give guaranteed reliability solutions showing some trajectory properties necessary for complicate chaotic behaviour.     

Chaotic behavior in differential equations driven by a Brownian motion

In this paper, we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed point under an unbounded random forcing driven by a Brownian motion. We prove that, for almost all sample paths of the Brownian motion in the classical Wiener space, the forced equation admits a topological horseshoe of infinitely many branches. This result is then applied to the randomly forced Duffing equation and the pendulum equation.