Limit Cycles Appearing From Piecewise Smooth Perturbations to a Reversible Nonlinear Center

This paper deals with the limit cycle bifurcation from a reversible differential center of degree [Formula: see text] due to small piecewise smooth homogeneous polynomial perturbations. By using the averaging theory for discontinuous systems and the complex method based on the Argument Principle, we obtain lower and upper bounds for the maximum number of limit cycles bifurcating from the period annulus around the center of the unperturbed system.

An Adaptive Regressor-Free Fourier Series-Based Tracking Controller for Robot Manipulators: Theory and Experimental Evaluation

SUMMARY In this article, we propose a nonlinear Proportional+Derivative (PD) tracking controller with adaptive Fourier series compensation. The proposed controller uses a regressor-free adaptive scheme that relies on a trigonometric polynomial with varying coefficients to solve the control problem. Asymptotic convergence of the position and velocity errors is proven via a formal stability analysis based on Lyapunov and LaSalle theory for discontinuous systems. The proposed controller is validated on a 2-degrees of freedom robot manipulator. The experimental results validate the theoretically obtained results and reflect the effect of certain parameters in the transient behavior of the error dynamics. Certain robustness properties are also observed.

On Periodic Solutions of a Time-Delayed, Discontinuous System with a Hyperbola Switching Control Law

In this paper, the existence of periodic motions of a discontinuous delayed system with a hyperbolic switching boundary is investigated. From the delay-related [Formula: see text]-function, the crossing, sliding and grazing conditions of a flow to the switching boundary are first developed. For this time-delayed discontinuous dynamical system, there are 17 classes of generic mappings in phase plane and 66 types of local mappings in a delay duration. The generic mappings are determined by subsystems in three domains and two switching boundaries. Periodic motions in such a delay discontinuous system are constructed and predicted analytically from specific mapping structures. Three examples are given for the illustration of periodic motions with or without sliding motion on the switching boundary. This paper shows how to develop switchability conditions of motions at the switching boundary in the time-delayed discontinuous systems and how to construct the specific periodic solutions for the time-delayed discontinuous systems. This study can help us understand complex dynamics in time-delayed discontinuous dynamical systems, and one can use such analysis to control the time-delayed discontinuous dynamical systems.