Vol. 2, No. 4 (2013): Regular Issue

We study a three degree of freedom autonomous system with damping, composed of two linear coupled oscillators with an essentially nonlinear lightweight attachment. In particular, we are interested in strongly nonlinear interactions between the linear oscillators and the essentially nonlinear attachment. First, we reduce our system to a non-autonomous second order nonlinear damped oscillator. Then, we introduce a slow-fast partition of the dynamics and average out the main frequency components in order to obtain a reduced system that is studied through the Slow Invariant Manifold (SIM) approach. Depending on the pa- rameters of the system we find different interesting nonlinear dynamical phenomena. With the help of the SIM approach we can study how the parameters of the original problem influence the asymptotic behavior of the orbits of the system. This is accomplished with the application of Tikhonov’s theorem. We classify the different cases of the dynamics according to the values of the parameters and the theoretically predicted asymptotic behavior of the orbits. Interesting phenomena are reported such as orbit capture, relaxation oscillations and complex structure of the basins of attraction.

This paper illustrates the advantages and disadvantages of using integer- and fractional-order control algorithms for an example of a system with S-shape dynamics. A laboratory setup of communicating vessels, where the level is regulated by means of a pump, is described. Both PI and PID control are designed, im- plemented and tested on the real setup. The results show that fractional order control may outperform classical PID control, under specific conditions.

n comparison with Traditional Galerkin’s Method (TGM), an Approx- imate Inertial Manifolds with Time Delay (AIMTDs) is presented for the model reduction of shallow arch with large deformation, a kind of nonlinear continuous dynamic system governed by partial differ- ential equation with second order in time, and the numerical simulation is also presented for the dynamic buckling analysis of shallow arch under impact. By this method, the nonlinear governing equations, which is a kind of infinite dimensional dynamic system, are studied in the phase space, and the solutions of the governing equations are projected onto the complete space spanned by the eigenfunctions of the linear operator of the governing equations. With the introduction of AIMTDs, it decomposes the infinite-dimensional phase space into two complementary subspaces, namely, a finite-dimensional one and its infinite-dimensional complement, and the relation between these two subspaces is the one with a time delay, that is, the evolution of the high modes is not only relevant to the instantaneous low modes, but also to the past high modes. Hence, the system can be projected onto the subspace with finite-dimension, in combination with the relation between the two subspaces, implying the model is reduced. Then, the nonlinear Galerkin’s procedure is adapted to approach the solution on AIMTDs in which the final equilibrium positions are included. Finally, the method is applied to the numerical simulation of dynamic buck- ling of the shallow arch under impact, and some comparisons between Traditional Galerkin’s procedure and Approximate Inertial Manifolds with Time Delay are given. It can be concluded that the method pre- sented give a guarantee for the truncation of buckling modes as mode expansion is used, and are generally feasible for the model reduction of the nonlinear continuous dynamic systems with second order in time, and it is an efficient numerical method requiring less computing time and high accuracy. More, the AIMTDs can be easily approached by finite element method.

This article presents a comparative study of the regulators of a hydro-electromechanical system, composed of two pumps, a uniform tank and a level sensor. Both controllers are fractional orders where the first one is the generalized PID controller and the second one is the CRONE (French acronym: Commande Robuste d’Ordre Non Entier) controller. In a first time, the transfer function of the plant is presented after identification (using the graybox method) and simplification processes. Then, the realization of both regulators is shown where the first controller (generalized PID) is obtained after imposing the regulator model whereas the second one is deduced using the open-loop constraints. At the end, a comparison between the behavior of both controllers is made in the frequency domain around some functional points of the plant as its behavior is nonlinear.

In this article, we discuss the pulse phenomena for a class of impulsive differential systems from the angle of discontinuity, consider the systems as a global discontinuous one consisting of two uniquelycontinuous subsystems and investigate its chatter dynamics. We introduce the method of flow theory and focus on the dynamical behaviors on the impulse surface. By deliberating analytical criteria for the transversality of a flow to the boundary surface, some sufficient conditions that guarantee the absence of pulse phenomena are obtained. We generalize several known results and some examples are given to apply the theory.

We investigate the vibrational resonance in the Duffing system with different kinds of delayed feedback. Our approach is to consider the delayed feedback as a generalized delayed feedback in a fractional-order differential version. For three special cases, the generalized delayed feedback corresponds to displacement delayed feedback, velocity delayed feedback, and acceleration delayed feedback respectively. At first, based on the vibrational mechanism, the approximate solution of the system is obtained. Then, we give conditions for all resonance patterns. The the oretical predictions are verified by numerical simulations. Furthermore, the theoretical results are in good agreement with the numerical simulations. Since the delayed feedback is in a generalized form, our results can be regarded as universal for the vibrational resonance in nonlinear systems with different kinds of delayed feedback.

In this article we introduce and analyze a mathematical model of commercial fishing as a dynamical system presented by fractional differential equations (FDE). In the model two different species of fish are interacting as predator and prey. We discuss the long-run properties of this system, including stability of equilibrium points, non-existence of limit cycle and periodic solution for the movement of fish stocks for both species. In deriving our mathematical model, instead of customary use of classical positive integer order of differential equations, we use fractional order derivatives. As we will discuss, non-local properties of FDE and its advantages in modeling problems with non-smooth domains will yield more accurate results in comparison with ordinary differential equation counterpart.