Metriplectic Representation of Finite Dimensional Dynamical Systems

Banbao Xu

ABSTRACT
About 25 years ago, P. Morrison and others started to study representation of dissipative systems in the Metriplectic (MP) form, which was called "Double Bracket" by A. Brockett and A. Bloch in the control theory. Later, R. Vilela Mendes and J. Tabrda Duarte proved that a vector field on a Riemannian manifold can be decomposed locally into one gradient and one Hamiltonian vector field. But the question of how to decompose a n-dimensional (n > 2) system into a gradient and a Hamiltonian part so that the important characteristic of a dynamical system would have a clear geometrical form (similar to one for Hamiltonian system) is not investigated.

In the first part of this work, we discuss MP representation of low dimensional dynamical systems. We show that many 2D systems are formally Hamiltonian (possibly with singularity) and find out the Poisson structure and the Hamilton function in formal series for 2D Lotka-Volterra system at the generic cases.

In the second and third parts of the work, some 2D chemical reaction systems, Fitzhug-Nagumo reaction systems, 3D Lorenz, Rabinovich, RTW, Field-Noyes systems and general n-dimensional Lotka-Volterra systems are decomposed into different forms of Metriplectic (MP) representation.

Thursday, June 3, 2004
DISSERTATION COMMITTEE
Serge Preston, Chairman
Steven Bleiler
Andrew M. Fraser
Peter Veerman
Peter Leung, Graduate Studies Rep.