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The problem of existence and stability of equilibria of linear systems with constant power loads is addressed in this paper. First, we correct an unfortunate mistake in our recent paper refid="ref10"/ pertaining to the sufficiency of the condition for existence of equilibria in multiport systems given there. Second, we give two necessary conditions for existence of equilibria. The first one is a simple linear matrix inequality hence it can be easily verified with existing software. Third, we prove that the latter condition is also sufficient if a set defined by the problem data is convex, which is the case for single and two-port systems. Finally, sufficient conditions for stability and instability for a given equilibrium point are given. The results are illustrated with two benchmark examples. [ABSTRACT FROM PUBLISHER]

In this paper we combine partial stability and set invariance methods, which is a necessary development for applications such as synchronization. We use set invariance methods to ensure that the ‘auxiliary’ variables remain on a restricted domain, and then use this framework to develop new results for both local and global partial stability theory. We apply the methodology to identical synchronization of oscillator networks, which gives rigorous conditions for existing local methodology as well as novel global methodology. The work also allows a common framework for synchronization analysis of both oscillator networks and power systems. We show the applicability by finding novel synchronization conditions for an example oscillator network. [ABSTRACT FROM PUBLISHER]

The 555 timer IC is a well-known integrated circuit having been widely used as a pulse generator. We propose a circuit containing a single 555 IC, an LED, an inductor, two capacitors, and three resistors that exhibits chaos. The IC serves as a hysteretic switch causing the system to be alternately attracted to each of two stable equilibria. Ours appears to be the first chaotic circuit governed by piecewise-linear equations that have stable but no unstable equilibria. Also unique is the sensitive dependence on initial conditions resulting only from a square-root map. This arises from the grazing impact of two voltages. The circuit is also unusual in exhibiting periodic orbits with a Farey tree structure and a transition to chaos via a period-adding cascade. This and other bifurcations seen experimentally are in agreement with those of the governing three-dimensional ODEs with hysteretic conditions. Inclusion of three extra resistors in the circuit lowers the characteristic frequency so that the bifurcations can be detected via the LED without the need of an oscilloscope. The circuit is therefore also suitable for experimentation by electronics hobbyists with limited resources. [ABSTRACT FROM AUTHOR]