Showing papers in "Journal of Applied Nonlinear Dynamics in 2017"
TL;DR: Fractional calculus represents the generalization of integration and differentiation to an arbitrary order and has been widely used in the control engineering field as discussed by the authors, where its use in control engineering has been gaining more and more popularity in both modeling and identification, as well as in controller tuning.
Abstract: Fractional calculus represents the generalization of integration and differentiation to an arbitrary order. Since the very first occurrence of fractional differentiation more than 300 years ago, fractional calculus and research related to its possible application have deserved ever-growing attention and interest. The research community has managed to bring forward ideas and concepts that justify the importance of fractional calculus for future engineering and science discoveries. What has begun as a means to describe abnormal behaviours in viscoelasticity or diffusion, power law phenomena, long range processes or fractal structures has spread to almost all engineering fields and applied sciences. Nowadays, its use in control engineering has been gaining more and more popularity in both modeling and identification, as well as in the controller tuning.
12 citations
12 citations
TL;DR: This work shows that a Krylov-Bogoliubov type analysis is a powerful method for analysing variants of the Mathieu equation and adopts an induction motor as a model system to show the details of the method.
Abstract: In this work we show that a Krylov-Bogoliubov type analysis is a powerful method for analysing variants of the Mathieu equation. We first demonstrate the technique by rederiving the results obtained by prior authors using different techniques and then apply it to a case where the system has a quasiperiodic drive (inhomogeneity) in addition to a quasiperiodic parametric term. A realistic system where such a forcing is present is an induction motor, so we adopt that as our model system to show the details of the method.
10 citations
8 citations
TL;DR: In this paper, the effects of controller parameters on the yaw rate of a mariner under a PDμ controller were analyzed by using the Nomoto model, and a region of safe behavior was identified and a strategy to reduce the YAW rate by an appropriated selection of controller parameter was discussed.
Abstract: This paper considers a mariner under PDμ controller and analyses the effects of controller parameters on the yaw rate by using the Nomoto model. The Nomoto model describing the time evolution of the yaw rate of the steering dynamics of a mariner is reduced to an asymmetric Duffing oscillator with fractional order derivative. Under the approximation of calm water, the steady behavior of the mariner shows an “imperfect” supercritical pitchfork bifurcation. Region of safe behavior is identified and strategy to reduce the yaw rate by an appropriated selection of controller parameters are discussed. The frequency analysis of the mariner shows the prominence of hysteresis is reduced for small order of the fractional derivative as well as the amplitude of the yaw rate. Evidence of chaotic response is illustrated using robust chaotic indicators such as the Lyapunov exponent and the fast Fourier transform.
7 citations
TL;DR: In this paper, the authors introduced the notion of identically distributed strictly non-Volterra cubic stochastic operator and showed that such an operator has a unique fixed point and has the property of being regular.
Abstract: We introduce the notion of identically distributed strictly non-Volterra cubic stochastic operator. We show that any identically distributed strictly non-Volterra cubic stochastic operator has a unique fixed point and that such operator has the property of being regular.
6 citations
TL;DR: In this article, a three dimensional Kolmogorov problem with extended forcing term for Navier-Stokes equations is considered and the Galerkin-Fourier method is applied and the symmetry preserving subset of solutions is considered.
Abstract: A three dimensional Kolmogorov problem with extended forcing term for Navier-Stokes equations is considered. The Galerkin-Fourier method is applied and the symmetry preserving subset of solutions is considered. The bifurcation patterns are revealed through the numerical analysis of eigenvalues of the linearized perturbed system from the analytical main stationary solution and through the analysis of phase space trajectories that the system generates. It was found that the initial stage of laminar-turbulent transition undergoes pitchfork bifurcation, through which the system can either go through the series of cycle cascades or through continuous tori bifurcations in accordance with the FShM scenario.
5 citations
TL;DR: In this paper, the authors initiated the rigorous analysis of controlled CTRW and their scaling limits, which paves the way to the real application of the research on CTRW, anomalous diffusion and related processes.
Abstract: In this paper we initiate the rigorous analysis of controlled Continuous Time Random Walks (CTRWs) and their scaling limits, which paves the way to the real application of the research on CTRWs, anomalous diffusion and related processes. For the first time the convergence is proved for payoff functions of controlled scaled CTRWs and their position dependent extensions to the solution of a new pseudodifferential equation which may be called the fractional Hamilton- Jacobi-Bellman equation.
5 citations
TL;DR: In this article, an experimental study of the mass transfer kinetics for methanol in mesoporous silica is presented, and the experimental data shows that there is no good correspondence between them and corresponding solutions found according to the second Fick's law for various pores geometries of the silica.
Abstract: Experimental study of the mass transfer kinetics for methanol in mesoporous silica is presented. Analysis of the experimental data shows that there is no good correspondence between them and corresponding solutions found according to the second Fick’s law for various pores geometries of the silica. Contrary, we show a good fit of the experimental data by a solution of the time-fractional diffusion equation with proper boundary conditions that correspond to experiment. Our results support that mass transfer in silica, which is a geometrically restricted media, may exhibit anomalous features, due to the geometrical constraints associated with randomly porous structure of a solid.
5 citations
TL;DR: In this paper, the positive and minimal realisation problem for fractional continuous-time linear single-input and single-output (SISO) systems is formulated and an algorithm based on the one-dimensional digraph for finding a positive and minimization of a given proper transfer function is proposed.
Abstract: The positive and minimal realisation problem for fractional continuous-time linear single-input and single-output (SISO) systems is formulated. Method based on the one-dimensional digraph for finding a positive and minimal realisation of a given proper transfer function is proposed. Two special cases of the digraph structure are given. Sufficient conditions for the existence of a positive minimal realisation of a given proper transfer function systems are established. The algorithm for computation of a positive minimal realisation is proposed and illustrated with a numerical example. The algorithm is based on a parallel computing method to gain needed speed and computational power for such a solution.
4 citations
TL;DR: This paper shows that introducing non-integer (fractional) differentiation to edge detectors (Fractional Canny, Fractional LoG, Fractionsal Derivative operators) can impr ove automatic edge detection results.
Abstract: To perform a robot-assisted surgery of a prosthesis implantation on a patient’s femur, we may need to get the bone’s exact perpendicular direction for the applicati on. We can extract that information from Computed Tomography scans, using image processing. In image processing, edge detection often makes use of integer-order differentiation operators (e.g. Canny and L oG operators). This paper shows that introducing non-integer (fractional) differentiation to edge detectors (Fraction al Canny, Fractional LoG, Fractional Derivative operators) can impr ove automatic edge detection results.
TL;DR: The fractional Rucklidge system is a new kind of chaotic models which hold the feature of memory effects and can depict the long history interactions as mentioned in this paper, and a numerical formula is proposed by use of fast Adomian polynomials.
Abstract: The fractional Rucklidge system is a new kind of chaotic models which hold the feature of memory effects and can depict the long history interactions. A numerical formula is proposed by use of the fast Adomian polynomials. Chaotic behavior are discussed and the Poincare sections are given for various fractional cases. It’s also applied in chaos synchronization of the fractional system.
TL;DR: In this paper, the effect of throughflow and temperature modulation on a rotating porous medium is investigated, which is computed numerically in terms of the Nusselt number, governed by a non-autonomous complex Ginzburg-Landau equation.
Abstract: The effect of throughflow and temperature modulation on a rotating porous medium is investigated. The generalized Darcy model is used for the momentum equation. Heat transfer analysis is based on weakly nonlinear thermal instability. It is computed numerically in terms of the Nusselt number, which is governed by a non-autonomous complex Ginzburg-Landau equation. Both concepts, rotation and throughflow are used as an external mechanism to regulate heat transfer. The effect of amplitude and frequency of modulation on heat transport is discussed and presented graphically. The effect of throughflow has duel by nature on heat transfer, the outflow enhances and inflow diminishes the heat transfer. It is found that, high rotational rates promotes heat transfer than low rotational rates. Further, the effect of modulation on mean Nusselt number depends on both the phase difference and frequency rather than on only the choice of the frequency of small amplitude modulation.
TL;DR: In this article, the existence and blow-up of solutions of reaction diffusion system with p(x)− growth conditions were investigated and the existence of weak solution was proved by using the Galerkin method.
Abstract: This paper is concerned with the existence and blow-up of solutions of reaction diffusion system with p(x)− growth conditions. The existence of weak solution is proved by using the Galerkin method. The blow-up of solutions is established by applying the method of comparison with suitable blow-up of self-similar subsolutions. Finally the theoretical results are illustrated by numerical examples.
TL;DR: In this article, the Lagrangian of nonlinear dynamics for the whole system is formulated, and the natural frequencies and global mode shapes of the system are determined, and orthogonality relations of the global mode shape are established.
Abstract: The global modal approach is employed to obtain a set of ordinary differential equations of motion describing the nonlinear dynamics of an L-shaped beam structure in this paper. Firstly, the Lagrangian of nonlinear dynamics for the whole system is formulated. The linear partial differential equations of transverse motion are derived for each beam, along with their boundary and matching conditions. Consequently, the characteristic equation is formulated for the whole system. The natural frequencies and global mode shapes of the system are determined, and orthogonality relations of the global mode shapes are established. Then, the Lagrange's procedure is employed to obtain the nonlinear ODEs of motion for the structure with multiple- DoF. A comparison between the natural frequencies obtained by the proposed method and those from finite element method is given to illustrate the validity of our approach. Through the nonlinear ODEs presented in this article, a study on the variation of dynamic responses for the systems with different number of global modes is performed to give a suggestion of how many modes should be taken for vibration analysis of the structure.
TL;DR: In this article, an analytical formula is derived for the inverse Laplace transform of fractional order integrator, 1/sα where α∈R and 0<α<1 using Stirling's formula and Gamma function.
Abstract: There is considerable interest in the study of fractional order derivative integrator but obtaining analytical impulse and step responses is a difficult problem. Therefore all methods reported on to date use approximations for the fractional derivative/integrator both for analytical based computations and more relevantly in simulation studies. In this paper, an analytical formula is first derived for the inverse Laplace transform of fractional order integrator, 1/sα where α∈R and 0<α<1 using Stirling’s formula and Gamma function. Then, the analytical step response of fractional integrator has been computed from the derived impulse response of 1/sα. The obtained analytical formulas for impulse and step responses of fractional order integrator are exact results except the very small error due to the neglected terms of Stirling’s series. The results are compared with some well known integer order approximation methods and Grunwald-Letnikov (GL) approximation technique. It has been shown via numerical examples that the presented method is very successful according to other methods.
TL;DR: In this article, the nonlinear dynamics of Colpitts oscillator under control of MEMS varactor in feedback connectivity has been analyzed with objectives for generation and control of high frequency chaotic signals.
Abstract: The nonlinear dynamics of Colpitts oscillator under control of MEMS varactor in feedback connectivity has been analyzed with objectives for generation and control of high frequency chaotic signals. The feedback signal derived from the capacitive divider in the standard Colpitts oscillator is modified by the MEMS varactor response mirrored by a voltage-controlled current multiplier. The latter implements MEMS capacitance multiplication and serves as a control parameter. The effects of voltage nonlinearity of the MEMS capacitance and the capacitance multiplication factor (α) have been analyzed by employing Lyapunov exponent, bifurcation diagram, phase portrait and Fourier transform methods. The modified feedback network facilitates high frequency chaos generation due to frequency doubling and high pass filtering effects of the MEMS capacitance. The latter emphasizes high frequency generation and attenuates lower frequencies. The variation of capacitance multiplication factor allows systematic changes in the qualitative nature of oscillator dynamics from a stable low frequency noisy state to Hopf bifurcation to period doubling/ tripling to chaos generation. The analysis suggests new MEMS based tuning and control of chaotic Colpitts oscillations.
TL;DR: In this article, the femoral head-neck orientation is extracted from computed tomography scans using image processing and non-integer (fractional) differentiation operators (e.g., fractional Canny and LoG operators) are used for edge detection.
Abstract: To perform a robot-assisted surgery of a prosthesis implantation on a patient’s femur, we may need to get the femoral head-neck orientation for the application. We can extract that information from Computed Tomography scans, using image processing. In image processing, edge detection often makes use of integer-order differentiation operators (e.g. Canny and LoG operators). This paper shows that introducing non-integer (fractional) differentiation to edge detectors (Fractional Canny, Fractional LoG, Fractional Derivative operators) can improve automatic edge detection results
TL;DR: In this paper, the sampling rate and discretization methods are used for parameter identification of a NARX model for a system with hysteresis, where the improved Euler and fourth order Runge-Kutta methods are applied in a Bouc-Wen model.
Abstract: Hysteresis is a nonlinear behaviour, which has been considered very hard to model. It is commonly found in actuators and sensors, involving quasi-static memory effects between input and output variables. Usually, continuous time models are used to model this feature. However, polynomial NARX model has come up as an alternative to model this behaviour. Since NARX models are discrete-time models, it is important to verify how the sampling rate interfere in obtaining the mathematical model. Further, frequently continuous-time models are used as a bench test, to generate data for identification of several nonlinear behaviour, including hysteresis. This paper investigates how the sampling rate and discretization methods affects the parameter identification of a NARX model for a system with hysteresis. Improved Euler and fourth order Runge-Kutta methods are applied in a Bouc-Wen model for a magneto-rheological damper, which is used as a system to be identified by a NARX model, considering the above mentioned scenario. Least-square based technique is used in this work to estimate model parameters.
TL;DR: In this article, the effects of controller parameters on the yaw rate of a mariner under a PDμ controller were analyzed by using the Nomoto model, and a region of safe behavior was identified and a strategy to reduce the YAW rate by an appropriated selection of controller parameter was discussed.
Abstract: This paper considers a mariner under PDμ controller and analyses the effects of controller parameters on the yaw rate by using the Nomoto model. The Nomoto model describing the time evolution of the yaw rate of the steering dynamics of a mariner is reduced to an asymmetric Duffing oscillator with fractional order derivative. Under the approximation of calm water, the steady behavior of the mariner shows an “imperfect” supercritical pitchfork bifurcation. Region of safe behavior is identified and strategy to reduce the yaw rate by an appropriated selection of controller parameters are discussed. The frequency analysis of the mariner shows the prominence of hysteresis is reduced for small order of the fractional derivative as well as the amplitude of the yaw rate. Evidence of chaotic response is illustrated using robust chaotic indicators such as the Lyapunov exponent and the fast Fourier transform.
TL;DR: In this paper, the authors established sufficient conditions for the controllability of neutral fractional integrodifferential systems with infinite delay and infinite NF systems with implicit derivative, where fixed point approaches are employed for achieving the required results.
Abstract: In this paper, we establish sufficient conditions for the controllability of neutral fractional integrodifferential systems with infinite delay and infinite neutral fractional systems with implicit derivative. Fixed point approaches are employed for achieving the required results. Examples are provided to illustrate the efficiency of the results.
TL;DR: In this paper, a nonlinear sliding mode control (SMC) and nonlinear SLM observer (SMO) for a class of linear time-invariant (LTI) singularly perturbed systems (SPS) subject to impulsive effects is presented.
Abstract: This paper addresses the problems of designing a nonlinear sliding mode control (SMC) and nonlinear sliding mode observer (SMO) for a class of linear time-invariant (LTI) singularly perturbed systems (SPS) subject to impulsive effects. The continuous states are viewed as an interconnected (or composite) system with two-time scale (slow and fast) subsystems. The main goal is to design a SMC law through the slow reduced order subsystems to achieve closed-loop stability of the full order system. This approach in turn results in lessening some unnecessary sufficient conditions on the fast subsystem. Then, assuming that partial output measurement of the slow subsystem is available, a similar control design is adopted to estimate the states of full order SPS, where a sliding mode modification of a Luenberger observer is used.
TL;DR: In this article, the existence of solutions for a class of semilinear fractional differential equations with nonlocal conditions and involving abstract Volterra operators was studied. But the existence was not proved for the class of fractional control problems involving Caputo fractional derivatives.
Abstract: In this paper we study the existence of solutions for a class of semilinear fractional differential equations with nonlocal conditions and involving abstract Volterra operators. The existence of an optimal solution for a class of fractional control problem involving Caputo fractional derivatives is obtained. An example is presented to illustrate our main result.
TL;DR: In this article, Chaotic properties of ACT system are analyzed through a detailed analysis of its bifurcation diagram, attractor formation, bi-parametric Lyapunov plots, and stability analysis.
Abstract: Chaotic properties of a new nonlinear dynamical system, namely ACT system, are analyzed through a detailed analysis of its bifurcation diagram, attractor formation, bi-parametric Lyapunov plots. Due to the presence of many parameters in the system it shows a very rich structure in all respects. Details of stability analysis and its relation to the corresponding center manifold reduction are also studied.
TL;DR: This paper presents a simple steering control algorithm for a rigid body model, which is a famous example of non-holonomic control systems with drift, and proposes a back-stepping-based adaptive controller design under the strict-feedback form.
Abstract: This paper presents a simple steering control algorithm for a rigid body model, which is a famous example of non-holonomic control systems with drift. The controllability Lie Algebra of a rigid body model contains Lie brackets of depth two. We propose a back-stepping-based adaptive controller design under the strict-feedback form. We analyze two cases for continuous steering. In the first case, the parameters of the model are assumed to be known while in the second case these are estimated by considering them unknown. This approach does not necessitate the conversion of the system model into a “chained form”, and thus does not rely on any special transformation techniques. The practical effectiveness of the controller is illustrated by numerical simulations and graceful stabilization.