Abstract: The concept of differentiation and integration to non-integer order has its origins in the seventeen century. However, only in the second-half of the twenty century appeared the first applications related to the area of control theory. In this paper we consider the study of a heat diffusion system based on the application of the fractional calculus concepts. In this perspective, several control methodologies are investigated and compared. Simulations are presented assessing the performance of the proposed fractional-order algorithms.

Abstract: We introduce a method for tracking nonlinear oscillations and their bifurcations in nonlinear dynamical systems. Our method does not require a mathematical model of the dynamical system nor the ability to set its initial conditions. Instead it relies on feedback stabilizability, which makes the approach applicable in an experiment. This is demonstrated with a proof-of-concept computer experiment of the classical autonomous dry-friction oscillator, where we use a fixed time step simulation and include noise to mimic experimental limitations. For this system we track in one parameter a family of unstable nonlinear oscillations that forms the boundary between the basins of attraction of a stable equilibrium and a stable stick-slip oscillation. Furthermore, we track in two parameters the curves of Hopf bifurcation and grazing-sliding bifurcation that form the boundary of the bistability region.

Abstract: This paper is concerned with the problem of synchronization for stochastic discrete-time drive-response networks with time-varying delay. By employing the Lyapunov functional method combined with the stochastic analysis as well as the feedback control technique, several sufficient conditions are established that guarantee the exponentially mean-square synchronization of two identical delayed networks with stochastic disturbances. These sufficient conditions, which are expressed in terms of linear matrix inequalities (LMIs), can be solved efficiently by the LMI toolbox in Matlab. A particular feature of the LMI-based synchronization criteria is that they are dependent not only on the connection matrices in the drive networks and the feedback gains in the response networks, but also on the lower and upper bounds of the time-varying delay, and are therefore less conservative than the delay-independent ones. Two numerical examples are exploited to demonstrate the feasibility and applicability of the proposed synchronization approaches.

Abstract: A methodology for deployment/retrieval optimization of tethered satellite systems is presented. Previous research has focused on the case where the tether is modeled as an inelastic, straight rod for the determination of optimal system trajectories. However, the tether shape and string vibrations can often be very important, particularly when the deployment/retrieval speed changes rapidly, or when external forces such as aerodynamic drag or electrodynamic forces are present. An efficient mathematical model for flexible tethered systems is first derived, which treats the tether as composed of a system of lumped masses connected via inelastic links. A tension control law is presented based on a discretization of the tether length dynamics via Chebyshev polynomials. A scheme that minimizes the second derivative of length over the trajectory based on physically meaningful coefficients is presented. This is utilized in conjunction with evolutionary optimization methods to minimize the rigid body and flexible modes of the system during deployment/retrieval. It is shown that only a very small number of parameters are required to generate accurate trajectories. The results are compared to the case where the tether is modeled as a straight rod.

Abstract: This paper can be regarded as a continuation of our previous work [1, 2] on the hierarchy of the dynamical behavior of nonholonomic systems. We consider different mechanical systems with nonholonomic constraints; in particular, we examine the existence of tensor invariants (laws of conservation) and their connection with the behavior of a system. Considerable attention is given to the possibility of conformally Hamiltonian representation of the equations of motion, which is mainly used for the integration of the considered systems.

Abstract: Bifurcations and route to chaos of the Mathieu–Duffing oscillator are investigated by the incremental harmonic balance (IHB) procedure. A new scheme for selecting the initial value conditions is presented for predicting the higher order periodic solutions. A series of period-doubling bifurcation points and the threshold value of the control parameter at the onset of chaos can be calculated by the present procedure. A sequence of period-doubling bifurcation points of the oscillator are identified and found to obey the universal scale law approximately. The bifurcation diagram and phase portraits obtained by the IHB method are presented to confirm the period-doubling route-to-chaos qualitatively. It can also be noted that the phase portraits and bifurcation points agree well with those obtained by numerical time-integration.

Abstract: The improved F-expansion method with a computerized symbolic computation is used to construct the exact traveling wave solutions of four nonlinear evolution equations in physics. As a result, many exact traveling wave solutions are obtained which include new soliton-like solutions, trigonometric function solutions, and rational solutions. The method is straightforward and concise, and it holds promise for many applications.

Abstract: This research studies the effects of axial preload on nonlinear dynamic characteristics of a flexible rotor supported by angular contact ball bearings. A dynamic model of ball bearings is improved for modeling a five-degree-of-freedom rotor bearing system. The predicted results are in good agreement with prior experimental data, thus validating the proposed model. With or without considering unbalanced forces, the Floquet theory is employed to investigate the bifurcation and stability of system periodic solution. With the aid of Poincare maps and frequency response, the unstable motion of system is analyzed in detail. Results show that the effects of axial preload applied to ball bearings on system dynamic characteristics are significant. The unstable periodic solution of a balanced rotor bearing system can be avoided when the applied axial preload is sufficient. The bifurcation margins of an unbalanced rotor bearing system enhance markedly as the axial preload increases and relates to system resonance speed.

Abstract: This paper presents the nonlinear optimal feedback control for the deployment process of a tethered subsatellite model, which involves not only the usually addressed in-plane motion, but also the out-of-plane motion. The model also takes the uncertainties in the mass parameter, the perturbations in initial states, and the external disturbance forces into consideration from an engineering point of view. The proposed controller is on the basis of a shrinking horizon and online grid adaptation scheme. Even though the proposed feedback law is not analytically explicit, it is easy to determine it by using a rapid recomputation of the open-loop optimal control, which generates the initial guesses for controls by interpolating the results from the previous computation. The case studies in the paper well demonstrate the effectiveness, robustness, and dominant real-time merits of the proposed controller.

Abstract: For usage of noncontact atomic force microscopy (NC-AFM) in a liquid environment, we propose a “van der Pol”-type self-excited cantilever probe whose steady state amplitude can be controlled to be sufficiently small so as not to damage the surface of the observation object. The self-exciting technique for the micro cantilever probe has become a powerful tool for obtaining atomically resolved images in environments where the cantilever has a very low quality factor Q. It is important to maintain the steady state amplitude of the cantilever to prevent damage to the surface, to reduce the contact force with the sample surface, and to avoid destruction of a soft material having an irregular surface such as biomolecules. In contrast to external excitation, the response amplitude of a self-excited oscillator is generally determined by the amount of nonlinearity of the system. The greater the nonlinearity, the smaller the steady state amplitude. We apply nonlinear feedback control in addition to linear feedback control to increase inherent nonlinearity in the system. Thereby, the cantilever probe has nonlinear characteristics that are equivalent to a van der Pol oscillator. It is shown analytically via an asymptotic perturbation approach that the self-excited oscillation has small steady state amplitude. Experiments are performed using a micro cantilever that is fabricated from a Pt/Ti/PZT/Pt/Ti/SiO2/SOI multi-layered structure. The validity and advantage of the proposed control method are confirmed to realize the stable self-excited oscillation with as small an amplitude as possible for the NC-AFM for a liquid environment.

Abstract: In this paper, the Exp-function method is used to obtain generalized solitonary solutions and periodic solutions of a KdV equation with five arbitrary functions The results show that the Exp-function method with the help of symbolic computation provides a very effective and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics

Abstract: This study investigates a low degree-of-freedom (DoF) mechanical model of shimmying wheels. The model is studied using bifurcation theory and numerical continuation. Self-excited vibrations, that is, stable and unstable periodic motions of the wheel, are detected with the help of Hopf bifurcation calculations. These oscillations are then followed over a large parameter range for different damping values by means of the software package AUTO97. For certain parameter regions, the branches representing large-amplitude stable and unstable periodic motions become isolated following an isola birth. These regions are extremely dangerous from an engineering point of view if they are not identified and avoided at the design stage.

Abstract: Chaos synchronization of Rikitake system applying the passive control method is investigated in this paper. Based on the passive technique, the passive controllers are designed. The nonlinear controller for the synchronization of two identical Rikitake systems or two different chaotic systems is simple and convenient to realize. Both theoretical analysis and numerical results show the effectiveness of the proposed method.

Abstract: This paper deals with the global exponential stability analysis problem for a general class of uncertain stochastic neural networks with mixed time delays and Markovian switching. The mixed time delays under consideration comprise both the discrete time-varying delays and the distributed time-delays. The main purpose of this paper is to establish easily verifiable conditions under which the delayed stochastic neural network is robustly exponentially stable in the mean square in the presence of parameters uncertainties, mixed time delays, and Markovian switching. By employing new Lyapunov–Krasovskii functionals and conducting stochastic analysis, a linear matrix inequality (LMI) approach is developed to derive the criteria for the robust exponential stability, which can be readily checked by using some standard numerical packages such as the Matlab LMI Toolbox. The criteria derived are dependent on both the discrete time delay and distributed time delay, and, are therefore, less conservative. A simple example is provided to demonstrate the effectiveness and applicability of the proposed testing criteria.

Abstract: The nonlinear vibration of shallow cables, equipped with a semiactive control device is considered in this paper. The control device is represented by a tuned mass damper with a variable out-of-plane inclination. A suitable control algorithm is designed in order to regulate the inclination of the device and to dampen the spatial cable vibrations. Numerical simulations are conducted under free spatial oscillations through a nonlinear finite element model, solved in two different computational environments. A harmonic analysis, in the region of the primary resonance, is also performed through a control-oriented nonlinear Galerkin model, including detuning effects due to the cable slackening.

Abstract: Time-delay feedback control of container cranes is robustly stable and insensitive to initial conditions for most of the linearly stable region. To better understand this robustness and any limitations of the technique, we undertake a nonlinear analysis of the system. To this end, we develop a nonlinear model of the crane system by modeling the crane-hoist-payload assembly as a double pendulum. Then, we derive a linear approximation specific to this model. Finally, we derive a cubic model of the dynamics for nonlinear analysis. Using linear analysis, we determine the gain and time delay factors for stabilizing controllers. Also, we show that the controller undergoes a Hopf bifurcation at the linear stability boundary. Using the method of multiple scales on the cubic model, we determine the normal form of the Hopf bifurcation. We then show that for practical operating ranges, the controller undergoes a supercritical bifurcation that helps explain the robustness of the controller.

Abstract: An electromechanical system with flexible arm is considered. The mechanical part is a linear flexible beam and the electrical part is a nonlinear self-sustained oscillator. Oscillatory solutions are obtained using an averaging method. Chaotic behavior is studied via the Lyapunov exponent. The synchronization of regular and chaotic states of two such devices is discussed and the stability boundaries for the synchronization process are derived using the Floquet theory. We compare the results obtained from a finite difference simulation to those from the classical modal approach.

Abstract: The frequency-locking area of 2:1 and 1:1 resonances in a fast harmonically excited van der Pol–Mathieu–Duffing oscillator is studied An averaging technique over the fast excitation is used to derive an equation governing the slow dynamic of the oscillator A perturbation technique is then performed on the slow dynamic near the 2:1 and 1:1 resonances, respectively, to obtain reduced autonomous slow flow equations governing the modulation of amplitude and phase of the corresponding slow dynamics These equations are used to determine the steady state responses, bifurcations and frequency-response curves Analysis of quasi-periodic vibrations is carried out by performing multiple scales expansion for each of the dependent variables of the slow flows Results show that in the vicinity of both considered resonances, fast harmonic excitation can change the nonlinear characteristic spring behavior from softening to hardening and causes the entrainment regions to shift It was also shown that entrained vibrations with moderate amplitude can be obtained in a small region near the 1:1 resonance Numerical simulations are performed to confirm the analytical results

Abstract: The radial diffusion in a sphere of radius R is described using time-fractional diffusion equation. The Caputo fractional derivative of the order 0<α<2 is used. The Laplace and finite sin-Fourier transforms are employed. The solution is written in terms of the Mittag–Leffler functions. For the first and second time-derivative terms, the obtained solutions reduce to the solutions of the ordinary diffusion and wave equations. Several examples of signaling, source and Cauchy problems are presented. Numerical results are illustrated graphically.

Abstract: In this effort, a six-degree-of-freedom (DOF) model is presented for the study of a machine-tool spindle-bearing system. The dynamics of machine-tool spindle system supported by ball bearings can be described by a set of second order nonlinear differential equations with piecewise stiffness and damping due to the bearing clearance. To investigate the effect of bearing clearance, bifurcations and routes to chaos of this nonsmooth system, numerical simulation is carried out. Numerical results show when the inner race touches the bearing ball with a low speed, grazing bifurcation occurs. The solutions of this system evolve from quasi-periodic to chaotic orbit, from period doubled orbit to periodic orbit, and from periodic orbit to quasi-periodic orbit through grazing bifurcations. In addition, the tori doubling process to chaos which usually occurs in the impact system is also observed in this spindle-bearing system.

Abstract: A nonlinear controller is presented for the stabilization of the underactuated inverted pendulum mounted on a cart. The fact that this system can be expressed as a chain of integrators, with an additionally nonlinear perturbation, allows us to use a nested saturation control technique to bring the pendulum to the top position, with zero displacement of the cart. The obtained closed-loop system is semiglobal, asymptotically stable, and locally exponentially stable, under the assumption that the position of the angle is initialized above the upper half plane.

Abstract: In this paper, we discuss a new application of the variational iteration method considering Adomian’s polynomials on nonlinear physical equations. Two models of interest in physics are considered and solved by means of the variational iteration method. The behavior of the variational iteration method and the effects of different values of t are investigated. Comparisons are made among the standard Adomian decomposition method, exact solutions, and the proposed method. He’s variational iteration method is introduced to overcome the difficulty arising in calculating the Adomian polynomial in Adomian decomposition method. The results reveal that the proposed method is very effective and simple and can be applied to other nonlinear problems.

Abstract: The nonlinear dynamic behavior of a rigid rotor supported by a spherical gas journal bearing sys- tem is analyzed using a hybrid method combining the differential transformation method (DTM) and the fi- nite difference method (FDM). The analytical results reveal that the bearing system has a complex dynamic behavior comprising periodic, subharmonic, and quasi- periodic responses of the rotor center. The evolution in the dynamic behavior of the bearing system is system- atically examined as the rotor mass and bearing number are increased. The analytical results are found to be in good agreement with those of other numerical meth- ods. Hence, the validity of the proposed hybrid method as a means of gaining insights into the nonlinear dy- namics of spherical gas film rotor-bearing systems is confirmed.

Abstract: Atomic force microscopes (AFM) are used to estimate material and surface properties. When using contact-mode AFM, the sample or the probe is excited near a natural frequency of the system to estimate the linear coefficient of the contact stiffness. Because higher modes offer lower thermal noise, higher quality factors, and higher sensitivity to stiff samples, their use in this procedure is more desirable. However, these modes are candidates for internal resonances, where the energy being fed into one mode may be channeled to another mode. Ignoring such interactions could distort or affect the accuracy of measurements. The method of multiple scales is used to derive an approximate analytical expression to the probe response in the presence of two-to-one autoparametric resonance between the second and third modes. We examine characteristics of this solution in relation to a single-mode response and consider its implications in AFM measurements. We find that the influence of this interaction extends over a considerable range of the tip-sample contact stiffness.

Abstract: Computation of focus (or focal) values for nonlinear dynamical systems is not only important in theoretical study, but also useful in applications. In this paper, we compare three typical methods for computing focus values, and give a comparison among these methods. Then, we apply these methods to study two practical problems and Hilbert's 16th problem. We show that these different methods have the same computational complexity. Finally, we discuss the “minimal singular point value” problem.

Abstract: In this paper, the problem of the motion of a gyrostat fixed at one point under the action of a gyrostatic moment vector \(\vec{\ell}\) whose components are li (i=1,2,3) about the axes of rotation, similar to a Lagrange gyroscope is investigated. We assume that the center of mass G of this gyrostat is displaced by a small quantity relative to the axis of symmetry, and that quantity is used to obtain the small parameter e (Elfimov in PMM, 42(2):251–258, [1978]). The equations of motion will be studied under certain initial conditions of motion. The Poincare small parameter method (Malkin in USAEC, Technical Information Service, ABC. Tr-3766, [1959]; Nayfeh in Perturbation methods, Wiley-Interscience, New York, [1973]) is applied to obtain the periodic solutions of motion. The periodic solutions for the case of irrational frequencies ratio are given. The periodic solutions are analyzed geometrically using Euler’s angles to describe the orientation of the body at any instant t of time. These solutions are performed by our computer programs to get their graphical representations.

Abstract: By introducing impulsive biological control strategy, the dynamic behaviors of the two-prey one-predator model with defensive ability and Holling type-II functional response are investigated. By using Floquet’s Theorem and the small amplitude perturbation method, we prove that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical minimum value, and permanence conditions (that is, the impulsive period is greater than some critical maximum value) are established via the method of comparison involving multiple Liapunov functions. It is shown that our impulsive control strategy is more effective than the classical one. Furthermore, the effect of impulsive perturbations on the unforced continuous system is studied. From simulations, we find that the system has more complex dynamic behaviors and is dominated by periodic, quasi-periodic, and chaotic solutions.

Abstract: This paper reports a method for the stability analysis of the steady curving of vehicles based on equations of motion that are obtained using multibody dynamics. The use of multibody dynamics techniques allows the systematic accurate analysis of vehicle dynamics in complex scenarios. However, stability analyses of vehicles are much more complicated than the use of conventional vehicle dynamics methods. The use of global coordinates and rotational parameters for the bodies involved implies the description of steady motions of the vehicle as periodic orbits rather than equilibrium points in the coordinate space. As a result, stability analyses must rely on Floquet’s theory instead of simple eigenvalue analyses of linearized equations. In practice, applying Floquet’s theory to large multibody systems involves very high computational costs. This paper reports an alternative stability analysis method based on two coordinate projections and a special eigenvalue analysis of differential algebraic equations. With this method, steady circular motions can be described in terms of equilibrium points rather than periodic motions. Stability analyses are thus made much more simple and computationally efficient. By way of example, the method was applied to a simple wheeled mechanism. The numerical results thus obtained were consistent with those of analytical and classical theories, which testifies to the accuracy of the proposed method.

Abstract: Nonlinear dynamics and stability of the rotor–bearing–seal system are investigated both theoretically and experimentally. An experimental rotor–bearing–seal device is designed and corresponding tests are carried out. The experimental rotor system is simplified as the Jeffcott rotor. The nonlinear oil–film forces are obtained under the short bearing theory and Muszynska nonlinear seal force model is used. Numerical method is utilized to solve the nonlinear governing equations. Bifurcation diagrams, waterfall plots, Poincare maps, spectrum plots and rotor orbits are drawn to analyze various nonlinear phenomena and system unstable processes. Theoretical results from numerical analysis are in good agreement with results from experiments. Conclusions are drawn and prove that this study will contribute to the further understanding of nonlinear dynamics and stability of the rotor system with the fluid-induced forces from oil–film bearings and the seals.