A computational approach to obtain normal forms for equilibrium points of three-dimensional autonomous systems, having a linear degeneracy corresponding to a triple-zero eigenvalue, is presented. Also, we provide the explicit expressions for the normal form coefficients, and analyze some additional simplifications that can be achieved.

A basic methodology to understand the dynamical behavior of a system relies on its decomposition into simple enough functional blocks. In this work, following that idea, we consider a family of piecewise-linear systems that can be written as a feedback structure. By using some results related to control systems theory, a simplifying procedure is given. In particular, we pay attention to obtain equivalent state equations containing both a minimum number of nonzero coefficients and a minimum number of nonlinear dynamical equations (canonical forms). Two new canonical forms are obtained, allowing to classify the members of the family in different classes. Some consequences derived from the above simplified equations are given. The state equations of different electronic oscillators with two or three state variables and two or three linear regions are studied, illustrating the proposed methodology.

For discontinuous switched linear systems, even when they are built by composing stable systems, examples of unstable systems are known. Here, three-dimensional homogeneous continuous piecewise linear systems composed by two linear systems sharing a boundary plane are considered. If the two linearization matrices are non-singular, then the only equilibrium point is at the origin, which is in the separation plane of the linear regions. For the important case where both matrices have complex eigenvalues and their whole spectra are in the open-left half of the complex plane, the possible counter-intuitive instability of the origin is analytically proved. Thus, in this paper, examples of unstable continuous switched linear system with just two stable components are shown. Copyright c 2005 IFAC.

The bifurcation of limit cycles in single-input single-output control systems with saturation is considered. Under some non-degeneracy conditions, a theorem characterizing such bifurcation is stated for the cases of dimension two and three. In terms of the deviation from the critical value of the bifurcation parameter, expressions in form of power series for the period, amplitude and the characteristic multipliers of the bifurcating limit cycle are obtained. These results are similar to the Hopf bifurcation theorem for differentiable systems, but they show some differences coming from the non-smooth character of the saturation characteristic.

Boundary equilibrium bifurcations in piecewise smooth discontinuous systems are characterized by the collision of an equilibrium point with the discontinuity surface. Generically, these bifurcations are of codimension one, but there are scenarios where the phenomenon can be of higher codimension. Here, the possible collision of a non-hyperbolic equilibrium with the boundary in a twoparameter framework and the nonlinear phenomena associated with such collision are considered.

This paper examines a typical structure of multi-stage converters present in direct current (dc) distribution systems. In such electric power distribution systems, point-of-load converters behave as a constant power load (CPL). Such loads may result in destabilizing nonlinear effects to their supply bus voltage. An analysis of the open loop dynamic behavior of the line regulating converter feeding CPLs is performed. This analysis gives the background to understand the proposed washout sliding mode controller to regulate the dc bus voltage under constant impedance load (CIL) and CPL changes. Also, the operating limits of the controlled system are determined in terms of the maximum power drained by CPL and CIL connected to the bus. The theoretical analysis is validated through simulation results.