Picone’s identity for -Laplace operator and its applications

UDC 517.9We prove a nonlinear analogue of Picone's identity for -Laplace operator. As an application, we give a Hardy type inequality and Sturmian comparison principle.We also show the strict monotonicity of the principle eigenvalue and degenerate elliptic system.

Invariant spectral foliations with applications to model order reduction and synthesis

The paper introduces a technique that decomposes the dynamics of a nonlinear system about an equilibrium into low-order components, which then can be used to reconstruct the full dynamics. This is a nonlinear analogue of linear modal analysis. The dynamics is decomposed using Invariant Spectral Foliation (ISF), which is defined as the smoothest invariant foliation about an equilibrium and hence unique under general conditions. The conjugate dynamics of an ISF can be used as a reduced order model. An ISF can be fitted to vibration data without carrying out a model identification first. The theory is illustrated on a analytic example and on free-vibration data of a clamped-clamped beam.

Nonlinear analogue of the May−Wigner instability transition

We study a system of N≫1 degrees of freedom coupled via a smooth homogeneous Gaussian vector field with both gradient and divergence-free components. In the absence of coupling, the system is exponentially relaxing to an equilibrium with rate μ. We show that, while increasing the ratio of the coupling strength to the relaxation rate, the system experiences an abrupt transition from a topologically trivial phase portrait with a single equilibrium into a topologically nontrivial regime characterized by an exponential number of equilibria, the vast majority of which are expected to be unstable. It is suggested that this picture provides a global view on the nature of the May−Wigner instability transition originally discovered by local linear stability analysis.

Diagnosis of Soft Spot Short Defects in Analog Circuits Considering the Thermal Behaviour of the Chip

The paper deals with fault diagnosis of nonlinear analogue integrated circuits. Soft spot short defects are analysed taking into account variations of the circuit parameters due to physical imperfections as well as self-heating of the chip. A method enabling to detect, locate and estimate the value of a spot defect has been developed. For this purpose an appropriate objective function was minimized using an optimization procedure based on the Fibonacci method. The proposed approach exploits DC measurements in the test phase, performed at a limited number of accessible points. For illustration three numerical examples are given.