Vol. 4, No. 1 (2015): Regular Issue

We investigate a phenomenon observed in systems of the form dx/dt = a1 (t)x + a2(t)y, dy/dt = a3(t)x + a4(t)y, where ai(t) = Pi + εQicos2t, where Pi, Qi and ε are given constants, and where it is assumed that when ε=0 this system exhibits a pair of linearly independent solutions of period 2π. Since the driver cos2t has period π, we have the ingredients for a 2:1 subharmonic resonance which typically results in a tongue of instability involving unbounded solutions when ε>0. We present conditions on the coefficients Pi, Qi such that the expected instability does not occur, i.e., the tongue of instability has disappeared.

In this paper, we study the numerical solution of fuzzy delay functional differential equations by using Euler method. Examples are presented to illustrate the computational aspects of the above method.

The nonlinear analysis of a typical electrical oscillator coupled nonlinearly to a mechanical one, as encountered in mechatronics applications for sensing, actuation and energy harvesting, is approached by using a state-space decomposition inspired by Volterra theory representation. The equation of motion of the mechanical subsystem includes an electromagnetic force directly proportional to the electric current squared. The nonlinear coupled dynamics is investigated systematically by partitioning the coupled system state vector in such a way as to fully exploit the mechanical low-pass and the electrical band-pass intrinsic features of free dynamics. In particular, by employing the Hilbert Transform, a low-pass equivalent system is derived and verified by using standard perturbation analysis. Then, a typical case is investigated thoroughly by means of numerical simulation of the original coupled low and band-pass, real-state-variable system and the low-pass-equivalent, complex-state-variable derived one. The nonlinear model equations considered here pave the way for a systematic investigation of nonlinear feedback control options designed to operate mechatronic transducers in energy harvesting, sensing or actuation modes.

In this paper, to solve the nonlinear convex programming problems with linear equality and bounded constraints, a new neural network model is constructed. It is proved that if the initial point lies in the linear equality region, the state of the proposed neural network is convergent to an exact optimal solution of the optimization problem. Compared with the existed neural networks, the proposed in this paper has a low model complexity and avoid estimating the penalty parameters in advance. In the end, several numerical simulations illustrate the effectiveness of the proposed neural network

This paper presents results of a numerical experiment, during which different numerical criteria of exact values setting in the accuracy assessment of fractional order derivatives / integrals numerical calculations are tested. Although traditional accuracy criteria in form of relative error expressed in % are applied, the values assumed as exact, necessary for comparison, are now: value of a function, classical 1st derivative, integral of the 1st order and Mittag-Leffler function’s values. For that purpose, fractional order differentiation and integration operators concatenation rules are applied. The methods allow to assess the accuracy of numerical calculations of fractional derivatives and integrals for each required function and not only for ones, for which mathematical formulas are available. The proposed measures are employed to determine proper operation and assess the accuracy of fractional order derivatives and integrals numerical algorithms.The algorithms utilize Riemann-Liouville and Grunwald-Letnikov frac-¨ tional order derivatives and integrals formulas.

The basic aim of the given paper is presentation of the basic principles of the new method defined as Non-orthogonal Amplitude Frequency Analysis of the Smoothed Signals (NAFASS). The new method is based on linear principle for the strongly-correlated variables and presentation of nonlinear signals. We demonstrate the possibilities of the NAFASS approach on analysis of real data related to dynamics and the fine structure of the Solar activity

The describing function(DF) is one method often used for the analysis of nonlinear systems and the prediction of limit-cycles. In this study, we explore the DF using frequency response methods in order to analyze the effectiveness of this technique in a fractional-order nonlinear system. Since it is common to find different types of nonlinearities in real systems, the DF method may reveal of great practical interest. In this perspective, we investigate the limit-cycle prediction and frequency response analysis of a fractional-orderplant model with hysteresis nonlinearity. The results presented may give some guidelines for the design of linear and nonlinear controllers of arbitrary order

In this work we study a generalization of the well known Fisher equation in cylindrical coordinates. We determine the subclasses of these equations which are nonlinear self-adjoint. By using a general theorem on conservation laws proved by Nail Ibragimov and the symmetry generators we find conservation laws for these partial differential equations without classical Lagrangians.