Dynamical analysis of fractional-order Mathieu equation

The dynamical characteristics of Mathieu equation with fractional-order derivative is analytically studied by the Lindstedt-Poincare method and the multiple-scale method. The stability boundaries and the corresponding periodic solutions on these boundaries for the constant stiffness δ0=n2 (n = 0, 1, 2, …), are analytically obtained. The effects of the fractional-order parameters on the stability boundaries and the corresponding periodic solutions, including the fractional coefficient and the fractional order, are characterized by the equivalent linear damping coefficient (ELDC) and the equivalent linear stiffness coefficient (ELSC). The comparisons between the transition curves on the boundaries obtained by the approximate analytical solution and the numerical method verify the correctness and satisfactory precision of the analytical solution. The following analysis is focused on the effects of the fractional parameters on the stability boundaries located in the δ-ε plane. It is found that the increase of the fractional order p could make the ELDC larger and ELSC smaller, which could result into the rightwards and upwards moving of the stability boundaries simultaneously. It could also be concluded the increase of the fractional coefficient K1 would make the ELDC and ELSC larger, which could move the transition curves to the left and upwards at the same time. These results are very helpful to design, analyze or control this kind of system, and could present beneficial reference to the similar fractional-order system

Stability analysis of visco-elastically damped structure through Bagley Torvik Equation

The stability of fractional-order visco-elastically damped linear system Bagley Torvik equation is analyzed in this paper. The fundamental novelty of this paper is the application of Caputo derivative. Prevailing sufficient spectral conditions are considered to guarantee the stability of linear models. Laplace transform, and Mittag-Leffler functions are utilized to develop the results. Furthermore, asymptotical stability of linear fractional-order models are also achieved through spectral values of the characteristic polynomials. Numerical examples are given to display the effectiveness of suggested method

Parametric Resonance For Complex Fields

Recently, there have been studies of parametric resonance decay of oscillating real homogeneous cosmological scalar fields, in both the narrow-band and broad-band case, primarily within the context of inflaton decay and (p)reheating. However, many realistic models of particle cosmology, such as supersymmetric ones, inherently involve complex scalar fields. In the oscillations of complex scalars, a relative phase between the oscillations in the real and imaginary components may prevent the violations of adiabaticity that have been argued to underly broad-band parametric resonance. In this paper, we give a treatment of parametric resonance for the decay of homogeneous complex scalar fields, analyzing properties of the resonance in the presence of out of phase oscillations of the real and imaginary components. For phase-invariant coupling of the driving parameter field to the decay field, and Mathieu type resonance, we give an explicit mapping from the complex resonance case to an equivalent real case with shifted resonance parameters. In addition, we consider the consequences of the complex field case as they apply to ``instant preheating,'' the explosive decay of non-convex potentials, and resonance in an expanding FRW universe. Applications of our considerations to supersymmetric cosmological models will be presented elsewhere.Comment: 20 pages, 2 figure

Multi-objective Active Control Policy Design for Commensurate and Incommensurate Fractional Order Chaotic Financial Systems

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.In this paper, an active control policy design for a fractional order (FO) financial system is attempted, considering multiple conflicting objectives. An active control template as a nonlinear state feedback mechanism is developed and the controller gains are chosen within a multi-objective optimization (MOO) framework to satisfy the conditions of asymptotic stability, derived analytically. The MOO gives a set of solutions on the Pareto optimal front for the multiple conflicting objectives that are considered. It is shown that there is a trade-off between the multiple design objectives and a better performance in one objective can only be obtained at the cost of performance deterioration in the other objectives. The multi-objective controller design has been compared using three different MOO techniques viz. Non Dominated Sorting Genetic Algorithm-II (NSGA-II), epsilon variable Multi-Objective Genetic Algorithm (ev-MOGA), and Multi Objective Evolutionary Algorithm with Decomposition (MOEA/D). The robustness of the same control policy designed with the nominal system settings have been investigated also for gradual decrease in the commensurate and incommensurate fractional orders of the financial system

Effect of fractional orders in the velocity control of a servo system

AbstractThe application of fractional-order PID controllers is now an active field of research. This article investigates the effect of fractional (derivative and integral) orders upon system’s performance in the velocity control of a servo system. The servo system consists of a digital servomechanism and an open-architecture software environment for real-time control experiments using MATLAB/Simulink tools. Experimental responses are presented and analyzed, showing the effectiveness of fractional controllers. Comparison with classical PID controllers is also investigated

LQR based improved discrete PID controller design via optimum selection of weighting matrices using fractional order integral performance index

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.The continuous and discrete time Linear Quadratic Regulator (LQR) theory has been used in this paper for the design of optimal analog and discrete PID controllers respectively. The PID controller gains are formulated as the optimal state-feedback gains, corresponding to the standard quadratic cost function involving the state variables and the controller effort. A real coded Genetic Algorithm (GA) has been used next to optimally find out the weighting matrices, associated with the respective optimal state-feedback regulator design while minimizing another time domain integral performance index, comprising of a weighted sum of Integral of Time multiplied Squared Error (ITSE) and the controller effort. The proposed methodology is extended for a new kind of fractional order (FO) integral performance indices. The impact of fractional order (as any arbitrary real order) cost function on the LQR tuned PID control loops is highlighted in the present work, along with the achievable cost of control. Guidelines for the choice of integral order of the performance index are given depending on the characteristics of the process, to be controlled.This work has been supported by the Dept. of Science & Technology (DST), Govt. of India under PURSE programme

Delayed Reheating and the Breakdown of Coherent Oscillations

We analyze the evolution of the perturbations in the inflaton field and metric following the end of inflation. We present accurate analytic approximations for the perturbations, showing that the coherent oscillations of the post-inflationary condensate necessarily break down long before any current phenomenological constraints require the universe to become radiation dominated. Further, the breakdown occurs on length-scales equivalent to the comoving post-inflationary horizon size. This work has implications for both the inflationary "matching" problem, and the possible generation of a stochastic gravitational wave background in the post-inflationary universe.Comment: 18 pages, 5 figures, v2: references added, extended discussion in section