Vol. 6, No. 1 (2017): Regular Issue

In this work, we consider a nonlinear Itô type stochastic integrodifferential equation and study the Freidlin-Wentzell type large deviation principle for its solution processes. The weak convergence approach is employed to establish the Laplace principle which in turn is equivalent to the large deviation principle. The compactness criterion is verified by means of sequential compactness of solutions of the associated controlled equation. The weak convergence result is asserted via solutions of the controlled equation with stochastic perturbation. Finally, examples are included to illustrate the theory.

In this paper, we consider a class of nonlinear oscillators whose equations of motion are in the form of that of a cubic Duffing oscillator extended by a term which is a quadratic monomial in the velocity and whose coefficient is a rational function of the position. We apply a combination of harmonic balance and Newton method to seek analytical approximations to the periodic solutions to the equation. The analysis can be applied directly to the equation in its "natural" rational form or after reducing it to the same denominator and considering only the numerator. The advantages and drawback of these two usages of the method are also discussed.

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.

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.

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.

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.

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.

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.

We present our investigation on the effect of parametric force on the response amplitude in the single Duffing oscillator and unidirectionally coupled n Duffing oscillators. In the single oscillator parametric perturbation is of the form f xsinωt. Parametric perturbation induced oscillatory motion is found for values of f above a critical value. In the oscillatory motion the dominant frequency is found to be ω/2. A(ω/2), the amplitude of oscillation at the frequency ω/2, is found to vary linearly with ω. We consider unidirectionally coupled n oscillators with first oscillator alone driven by a parametric force and the other oscillators are nonlinearly or linearly coupled but one-way only. Depending upon the values of the coupling strength δ the oscillators, after first several oscillators, exhibit damped or undamped signal propagation. In the nonlinearly coupled oscillators the dominant frequency of oscillation is ω/2. In the linearly coupled system the frequency ω/2 is absent. The oscillators other than the first oscillator exhibiting oscillatory motions have frequencies ω or 2ω or both depending upon the values of the coupling strength.