Attitude dynamics of a spacecraft with variable structure at presence of harmonic perturbations

Attitude dynamics of a spacecraft (SC) with variable structure (inertia–mass parameters variation) is examined. Equations of the motion of the SC are obtained on the base of Hamiltonian formalism in Serret–Andoyer variables. These equations can be used for analysis and synthesis of conditions of the SC attitude motion on active legs of orbital trajectories. Analytical and numerical modeling of the SC motion is realized. Existence of the SC chaotic modes of motion is demonstrated with the help of Melnikov method and Poincaré sections. Also attitude motion of a dual-spin spacecraft (DSSC) is considered at presence of small internal harmonic torque between DSSC coaxial bodies.     

Heteroclinic dynamics and attitude motion chaotization of coaxial bodies and dual-spin spacecraft

Heteroclinic dynamics of a free coaxial bodies system and dual-spin spacecraft is examined. New analytical solutions for heteroclinic orbits, corresponded to the polhodes-separatrices in the space of the angular moment components, are obtained. On the base of these analytical heteroclinic solutions analysis of possibility of the system motion chaotization with the help of Melnikov method is conducted. The analysis shows the polhode-separatrix-orbit splitting at presence of small harmonical perturbation torques between the coaxial bodies. The separatrix splitting generates the chaotic layer near the unperturbed separatrix region. This fact proves possibility of realization of non-regular dynamics and chaotic tilting motion of the dual-spin spacecraft.     

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)     

Steady-state,self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations

This paper analyzes a controlled servomechanism with feedback and a cubic nonlinearity by means of the Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations, from which steady-state, self-oscillating and chaotic behaviors will be investigated using the center manifold theorem. The system controller is formed by a Proportional plus Integral plus Derivative action (PID) that allows to stabilize and drive to a prescribed set point a body connected to the shaft of a DC motor. The Bogdanov–Takens bifurcation is analyzed through the second Lyapunov stability method and the harmonic-balance method, whereas the first Lyapunov value is used for the Andronov–Poincaré–Hopf bifurcation. On the basis of the results deduced from the bifurcation analysis, we show a procedure to select the parameters of the PID controller so that an arbitrary steady-state position of the servomechanism can be reached even in presence of noise. We also show how chaotic behavior can be obtained by applying a harmonical external torque to the device in self-oscillating regime. The advantage of achieving chaotic behavior is that it can be used so that the system reaches a set point inside a strange attractor with a small control effort. The analytical calculations have been verified through detailed numerical simulations.     

An alternative approach for chaos: A plane pendulum with oscillating torque

The shooting method is applied to obtain chaotic motions for a pendulum with a oscillatory torque excitation on its support. It shows that if the pendulum is placed at certain spots, the corresponding motion will become chaotic. It proves the coexistence of uncountably many non-periodic motions and countably many periodic motions of the pendulum.     

Local instability and localization of attractors. From stochastic generator to Chua's systems

This paper discusses the connection between various instability definitions (namely, Lyapunov instability, Poincaré or orbital instability, Zhukovskij instability) and chaotic movements. It is demonstrated that the notion of Zhukovskij instability is the most adequate for describing chaotic movements. In order to investigate this instability, a new type of linearization is offered and the connection between that and the theorems of Borg, Hartman-Olech, and Leonov is established. By means of new linearization, analytical conditions of the existence of strange attractors for impulse stochastic generators are obtained. The assumption is expressed that an analogous analytical tool may be elaborated for continuous dynamical systems describing Chua's circuits. The paper makes a first step in this direction and establishes a frequency criterion of the existence of positive invariant sets with positive Lebesgue measure for piecewise linear systems, which are unstable in every region of phase space where they are linear.     

Optimal Trajectory Tracking of Underwater Vehicle-Manipulator Systems Through the Clifford Algebras and of the Davies Method

The manipulation in singular regions generates an instantaneous reduction in the mechanism mobility which can result in some disturbances in the trajectory tracking. In proximity of the singularities, small velocities in the end-effectors generates high speeds in the joints due to the gradual reduction of the mobility. The phenomenon of kinematic singularity generates a instantaneous instability in torque profile of the redundant robotic systems by the transformation of secondary joints to primary joints. The disturbances of the underwater environment intensifies the effects of the kinematic singularities because the hydrodynamic strongly oppose to torque variations. This work presents a methodology for using dual quaternions in the posture feedback of a Underwater Vehicle-Manipulator System (UVMS) using the Davies method which avoids kinematic singularities and ensures the optimal torque profiles.     

Analysis and circuitry realization of a novel three-dimensional chaotic system

In this paper a new three-dimensional chaotic system is introduced. Some basic dynamical properties are analyzed to show chaotic behavior of the presented system. These properties are covered by dissipation of system, instability of equilibria, strange attractor, Lyapunov exponents, fractal dimension and sensitivity to initial conditions. Through altering one of the system parameters, various dynamical behaviors are observed which included chaos, periodic and convergence to an equilibrium point. Eventually, an analog circuit is designed and implemented experimentally to realize the chaotic system.     

具有结构内阻尼磁性刚体航天器姿态动力学稳定性分析

在地球引力场和磁场中,在考虑了航天器结构内阻尼及气体阻力的影响条件下,研究磁性刚体航天器在绕地球圆轨道运行时可能出现的混沌问题.根据动量矩定理建立动力学模型,应用Melnikov方法证明了动力系统在一定条件下会发生混沌行为,并且给出了解析判据.最后利用数值仿真分析了系统的动力学行为,理论结果与数值仿真结果相一致.     

The problem of the time-optimal control of spacecraft reorientation

The use of Pontryagin's maximum principle to solve spacecraft motion control problems is demonstrated. The problem of the optimal control of the spatial reorientation of a spacecraft (as a rigid body) from an arbitrary initial angular position to an assigned final angular position in the minimum rotation time is investigated in detail. The case in which velocity parameters of the motion are constrained is considered. An analytical solution of the problem is obtained in closed form using the method of quaternions, and mathematical expressions for synthesizing the optimal control programme are given. The kinematic problem of spacecraft reorientation is solved completely. A design scheme for solving the maximum principle boundary-value problem for arbitrary turning conditions and inertial characteristics of the spacecraft is given. A solution of the problem of the optimal control of spatial reorientation for a dynamically symmetrical spacecraft is presented in analytical form (to expressions in elementary functions). The results of mathematical modelling of the motion of a spacecraft under optimal control, which confirm the practical feasibility of the control algorithm developed, are given. Estimates have shown that the turn time of modern spacecraft with a constrained magnitude of the angular momentum can be reduced by 15–25% compared with conventional reorientation methods. The greatest effect is achieved for turns through large angles (90° or more) when the final rotation vector is equidistant from the longitudinal axis and the transverse plane of the spacecraft.     

Energy cycle of brushless DC motor chaotic system

The vector field of the brushless DC motor (BLDCM) chaotic system is regarded as the force field of a pure mechanical system via the transformation of Kolmogorov system. The BLDCM force field is decomposed into four types of torque: inertial, internal, dissipative, and generalized external torque. The forcing effect of each term in the force field is identified via the analogue of the electrical and mechanical system. The BLDCM energy transformation of four forms of energy—kinetic, potential, dissipative, and generalized external is investigated. The physical interpretation of force decomposition and energy exchange is given. The rate of change of the Casimir energy is equivalent to the power exchanged between the dissipative energy and the energy supplied to the motor, and it governs the different dynamic modes. A simple and optimal supremum bound for the chaotic attractor is proposed using the Casimir function and optimization.     

一种改进的混沌序列产生方法

根据灰度图像的二维直方图的特点,在已有的二维Arnold混沌系统的基础上,结合Bernstein形式的Bézier曲线的生成算法,给出了一种基于生成Bézier曲线的de Casteljau算法构造伪随机序列的方法,实验结果表明生成的二维序列不仅具有伪随机性,而且还具有在近似圆盘中随机分布的性质,这使得该伪随机序列更适合对灰度图像的二维灰度直方图进行基于混沌优化的图像分割.在此基础上,给出了一种基于混沌优化的二维最大熵的灰度图像分割算法,该算法对于含噪图像取得了良好的分割效果.     

Chaotic and bifurcation for a simply-supported thermo-mechanical coupling circular plate with variable thickness

This paper presents an approach to characterize the conditions that can possibly lead to chaotic motion for a simply supported large deflection circular plate of thermo-mechanical coupling by utilizing the criterion of the maximum Lyapunov exponent. The governing partial differential equation of the simply supported large deflection circular plate of thermo-mechanical coupling is first derived and simplified to a set of three ordinary differential equations by the Galerkin method. Several different features including time history, Power spectra, phase plot, Poincare map and bifurcation diagram are then numerically computed. These features are used to characterize the dynamic behavior of the plate subjected various geometric and excitation conditions. Numerical examples are presented to verify the validity of the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. The modeling results of numerical simulation indicate that the chaotic motion may occurs in the lateral loads , η₁=1.1, β=0.5, and _∞=0.0007. As the thermo-elastic damping is great than a critical value, the dynamic motion of the thermal-couple plate is periodic. As the thickness parameter β of the concave circular plate is great than a critical value, the motion of the plate is periodic. The modeling result thus obtained by using the method proposed in this paper can be employed to predict the instability induced by the dynamics of the thermo-mechanical coupling circular plate in large deflection.     

Controlling bifurcations and chaos in TCP–UDP–RED

We study the bifurcation and chaotic behavior of the Transmission Control Protocol (TCP) and User Datagram Protocol (UDP) network with Random Early Detection (RED) queue management. These bifurcation and chaotic behaviors may cause heavy oscillation of an average queue length and induce network instability. We propose an impulsive control method for controlling bifurcations and chaos in the internet congestion control system. The theoretical analysis and the simulation experiments show that this method can obtain the stable average queue length without sacrificing the other advantages of RED.     

Chaotic effects on disease spread in simple eco-epidemiological system

In this paper, an eco-epidemiological model where prey disease is structured as a susceptible-infected model is investigated. Thresholds that control disease spread and population persistence are obtained. Existence, stability and instability of the system are studied. Hopf bifurcation is shown to occur where a periodic solution bifurcates from the coexistence equilibrium. Simulations show that the system exhibits chaotic phenomena when the transmission rate is varied.     

Generation of multi-wing chaotic attractor in fractional order system

In this paper, a novel approach is proposed for generating multi-wing chaotic attractors from the fractional linear differential system via nonlinear state feedback controller equipped with a duality-symmetric multi-segment quadratic function. The main idea is to design a proper nonlinear state feedback controller by using four construction criterions from a fundamental fractional differential nominal linear system, so that the controlled fractional differential system can generate multi-wing chaotic attractors. It is the first time in the literature to report the multi-wing chaotic attractors from an uncoupled fractional differential system. Furthermore, some basic dynamical analysis and numerical simulations are also given, confirming the effectiveness of the proposed method.     

Spatiotemporal Dynamic Analysis in a Time-space Discrete Brusselator Mode

In this paper, we study the spatiotemporal patterns of a Brusselator model with discrete time-space by using the coupled mapping lattice (CML) model. The existence and stability conditions of the equilibrium point are obtained by using linear stability analysis. Then, applying the center manifold reduction theorem and the bifurcation theory, the parametric conditions of the flip and the Neimark-Sacker bifurcation are described respectively. Under space diffusion, the model admits the Turing instability at stable homogeneous solutions under some certain conditions. Two nonlinear mechanisms, including flip-Turing instability and Neimark-Sacker-Turing instability, are presented. Through numerical simulation, periodic windows, invariant circles, chaotic phenomenon and some interesting spatial patterns are found.     

动力系统实测数据的非线性混沌模型重构

动力系统实测非线性混沌数据的模型重构技术是相空间重构的重要内容。在判定了实测数据的非线性混沌特征,计算了实测数据的分维数,Lyapunov指数,并对其进行了本征值分解和噪声去除及确定其模型阶数以后,提出了一个动力系统实测数据的非线性混沌模型,给出了相应的模型参数辨识方法,并用其确立的混沌模型进行了预测工作,计算结果表明:模型参数辨识方法能迅速地将参数估计值带到多峰目标函数的全局最少值附近,然后再采用优化理论能较准确地求出模型的参数,用得到的混沌模型对系统进行预测工作其预测效果良好,且混沌时序不可能作长期预测。     

A new accurate numerical method of approximation of chaotic solutions of dynamical model equations with quadratic nonlinearities

In this article the authors describe the method of construction of approximate chaotic solutions of dynamical model equations with quadratic nonlinearities in a general form using a new accurate numerical method. Numerous systems of chaotic dynamical systems of this type are studied in modern literature. The authors find the region of convergence of the method and offer an algorithm of construction and several criteria to check the accuracy of the approximate chaotic solutions.     

Investigation on nonlinear dynamic characteristics of combustion instability in the lean-burn premixed natural gas engine

In this paper, the nonlinear dynamic characteristics of combustion instability in the natural gas engine were investigated. The experiments covered the whole excess air ratio (λ) in range from 1 to 1.6 and spark advance angle (SAA) in range from 10°CA to 50°CA before top dead center (TDC). And the real-time series of in-cylinder pressure in combustion process were acquired through a piezoelectric transducer. A couple of new coordinates were proposed for the 0–1 test method. Then the characteristics of the experimentally obtained real-time series of in-cylinder pressure in combustion process were analyzed by using the 0–1 test, the largest Lyapunov exponent (LLE) and the phase space reconstruction methods. The effects of SAA and λ on the complexity of combustion instability of the natural gas engine were tested qualitatively and quantitatively. The results show that all the average asymptotic growth rate K_c are approximately equal to 1 and all the LLE are positive. This indicates the combustion process involves some chaotic characteristics. All the attractors are limited to the finite range of phase space and all the attractors have twist and folded geometry structure. This indicates the combustion process has some irregular deterministic components. Both K_c and LLE have higher values, also the attractor is more complex and looser under higher λ and too large or too small SAA conditions. One can conclude that the chaotic behavior is stronger and the combustion process is more complex and sensitive to small variations of initial condition under higher λ and too large or too small SAA conditions.