# Invariance Properties of Continuous Piecewise-Linear Vector Fields

### R. Lum and Leon O. Chua

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/ERL M90/38

1990

In the application of continuous piecewise-linear vector fields to the modelling of systems, it is often desirable that the model preserve certain properties of the system that it is supposed to emulate. These properties may represent fluid incompressibility, conservation of energy and symmetries. In this paper necessary and sufficient conditions are stated for the identification and imposition of several such properties. The first part of the paper addresses continuous piecewise-linear vector fields that are divergence free, gradient systems, and Hamiltonian systems. The second half of the paper determines possible relationships between a lattice piecewise-linear vector field and a transformation matrix. This will facilitate the identification of symmetries of a lattice vector field and the classification of lattice vector fields possessing certain symmetry properties. With these results, the modelling process via piecewise-linear vector fields will have the capacity to preserve intrinsic structure of the modeled system.

BibTeX citation:

@techreport{Lum:M90/38, Author = {Lum, R. and Chua, Leon O.}, Title = {Invariance Properties of Continuous Piecewise-Linear Vector Fields}, Institution = {EECS Department, University of California, Berkeley}, Year = {1990}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1990/1479.html}, Number = {UCB/ERL M90/38}, Abstract = {In the application of continuous piecewise-linear vector fields to the modelling of systems, it is often desirable that the model preserve certain properties of the system that it is supposed to emulate. These properties may represent fluid incompressibility, conservation of energy and symmetries. In this paper necessary and sufficient conditions are stated for the identification and imposition of several such properties. The first part of the paper addresses continuous piecewise-linear vector fields that are divergence free, gradient systems, and Hamiltonian systems. The second half of the paper determines possible relationships between a lattice piecewise-linear vector field and a transformation matrix. This will facilitate the identification of symmetries of a lattice vector field and the classification of lattice vector fields possessing certain symmetry properties. With these results, the modelling process via piecewise-linear vector fields will have the capacity to preserve intrinsic structure of the modeled system.} }

EndNote citation:

%0 Report %A Lum, R. %A Chua, Leon O. %T Invariance Properties of Continuous Piecewise-Linear Vector Fields %I EECS Department, University of California, Berkeley %D 1990 %@ UCB/ERL M90/38 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1990/1479.html %F Lum:M90/38