This paper focuses on the stability testing of fractional-delay systems. It begins with a brief introduction of a recently reportedalgorithm, a detailed demonstration of a failure in applications of the algorithm and the key points behind the failure. Then,it presents a criterion via integration, in terms of the characteristic function of the fractional-delay system directly, for testingwhether the characteristic function has roots with negative real parts only or not. As two applications of the proposed criterion,an algorithm for calculating the rightmost characteristic root and an algorithm for determining the stability switches, are proposed.The illustrative examples show that the algorithms work effectively in the stability testing of fractional-delay systems.
It begins with the study of damping representation of a linear vibration system of single degree of freedom (SDOF),from the view point of fractional calculus. By using the idea of stability switch,it shows that the linear term involving the fractional-order derivative of an order between 0 and 2 always acts as a damping force,so that the unique equilibrium is asymp-totically stable. Further,based on the idea of stability switch again,the paper proposes a scheme for determining the stable gain region of a linear vibration system under a fractional-order control. It shows that unlike the classical velocity feedback which can adjust the damping force only,a fractional-order feedback can adjust not only the damping force,but also the elastic re-storing force,and in addition,a fractional-order PDα control can either enlarge the stable gain region or narrow the stable gain region. For the dynamic systems described by integer-order derivatives,the asymptotical stability of an equilibrium is guaranteed if all characteristic roots stay in the open left half-plane,while for the systems with fractional-order derivatives,the asymptotical stability of an equilibrium is guaranteed if all characteristic roots stay within a sector in the complex plane. Analysis shows that the proposed method,based on the idea of stability switch,works effectively in the stability analysis of dynamical systems with fractional-order derivatives.
WANG ZaiHua1,2 & HU HaiYan1 1 Institute of Vibration Engineering Research,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China
This paper presents an investigation on the phenomenon of delayed bifurcation in time-delayed slow-fast differential systems.Here the two delayed's have different meanings.The delayed bifurcation means that the bifurcation does not happen immediately at the bifurcation point as the bifurcation parameter passes through some bifurcation point,but at some other point which is above the bifurcation point by an obvious distance.In a time-delayed system,the evolution of the system depends not only on the present state but also on past states.In this paper,the time-delayed slow-fast system is firstly simplified to a slow-fast system without time delay by means of the center manifold reduction,and then the so-called entry-exit function is defined to characterize the delayed bifurcation on the basis of Neishtadt's theory.It shows that delayed Hopf bifurcation exists in time-delayed slow-fast systems,and the theoretical prediction on the exit-point is in good agreement with the numerical calculation,as illustrated in the two illustrative examples.
ZHENG YuanGuang1 & WANG ZaiHua1,2 1 Institute of Vibration Engineering Research,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China
Periodic solutions occur commonly in linear or nonlinear dynamical systems. In some cases, the stability of a periodic solution holds if the rightmost characteristic root has negative real part. Based on the Lambert W function, this paper presents a simple algorithm for locating the rightmost characteristic root of periodic solutions of some nonlinear oscillators with large time delay. As application, the proposed algorithm is used to study the primary resonance and 1/3 subharmonic resonance of the Duffing oscillator under harmonic excitation and delayed feedback, as well as the control problem of the van der Pol oscillator under harmonic excitation by using delayed feedback, with a number of case studies. The main advantage of this algorithm is that though very simple in implementation, it works effectively with high accuracy even if the delay is large.
