In this paper the stability of the Korteweg-de Vries (KdV) equation is investigated. It is shown analytically and numerically that small perturbations of solutions of the KdV-equation introduce effects of dispersion, hence the perturbation propagates with a different velocity then the unperturbed solution. This effect is investigated analytically by formulating a differential equation for pertu...

New perturbation theorems are proved for simultaneous bases of singular subspaces of real matrices. These results improve the absolute bounds previously obtained in [6] for general (complex) matrices. Unlike previous results, which are valid only for the Frobenius norm, the new bounds, as well as those in [6] for complex matrices, are extended to any unitarily invariant matrix norm. The bounds ...

Folded saddle-nodes occur generically in one parameter families of singularly perturbed systems with two slow variables. We show that these folded singularities are the organizing centers for two main delay phenomena in singular perturbation problems: canards and delayed Hopf bifurcations. We combine techniques from geometric singular perturbation theory – the blow-up technique – and from delay...

By virtue of an operator-theoretical approach, we deal with hyperbolic singular perturbation problems for integrodifferential equations. New convergence theorems for such singular perturbation problems are obtained, which generalize some previous results by This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, an...

In this paper, we study the sensitivity and second order perturbation expansions of simple nonzero finite generalized singular values of a complex matrix pair, which is analytically dependent on several parameters. Our results generalize the perturbation analysis given by Sun [J. Comput. Math., 6 (1988), pp. 258–266] for simple nonzero singular values.

Modern Lightweight robots are constructed to be collaborative, which often results in a low structural stiffness compared conventional rigid robots. Therefore, the controller must able handle dynamic oscillatory effect mainly due intrinsic joint elasticity. Singular perturbation theory makes it possible decompose flexible dynamics into fast and slow subsystems. This model separation provides ad...

A novel composite control scheme for a manipulator with flexible links and joints is presented that uses the singular perturbation technique (SPT) to divide the manipulator dynamics into reduced order slow and fast subsystems. A neural network controller is then applied for the slow subsystem and a state-feedback H∞ controller for the fast subsystem. Results are presented that demonstrate impro...

Reaction-diffusion systems produce a variety of patterns such as spots, labyrinths, and rotating spirals. Circular spots may be stationary or unstable to oscillating motion. The oscillations are sometimes steady but may lead to collapsing or infinitely expanding spots. Using a singular perturbation technique we derive a set of ordinary differential equations for the dynamics of circular spots. ...

Educational issues of Proportional, Integral (PI) and Proportional, Integral, and Derivative (PID) type controller design in undergraduate course on control, graduate course on control, advanced control topics are discussed. The main attention is devoted to multi-time-scale control systems where PI PID type controllers designed for nonlinear systems in the presence of plant parameter variations...