Lie-Poisson Deformation of the Poincar\'e Algebra

We find a one parameter family of quadratic Poisson structures on ${\bf R}^4\times SL(2,C)$ which satisfies the property {\it a)} that it is preserved under the Lie-Poisson action of the Lorentz group, as well as {\it b)} that it reduces to the standard Poincar\'e algebra for a particular limiting value of the parameter. (The Lie-Poisson transformations reduce to canonical ones in that limit, which we therefore refer to as the `canonical limit'.) Like with the Poincar\'e algebra, our deformed Poincar\'e algebra has two Casimir functions which we associate with `mass' and `spin'. We parametrize the symplectic leaves of ${\bf R}^4\times SL(2,C)$ with space-time coordinates, momenta and spin, thereby obtaining realizations of the deformed algebra for the cases of a spinless and a spinning particle. The formalism can be applied for finding a one parameter family of canonically inequivalent descriptions of the photon.Comment: Latex file, 26 page

Bifurcations and Chaos in Time Delayed Piecewise Linear Dynamical Systems

We reinvestigate the dynamical behavior of a first order scalar nonlinear delay differential equation with piecewise linearity and identify several interesting features in the nature of bifurcations and chaos associated with it as a function of the delay time and external forcing parameters. In particular, we point out that the fixed point solution exhibits a stability island in the two parameter space of time delay and strength of nonlinearity. Significant role played by transients in attaining steady state solutions is pointed out. Various routes to chaos and existence of hyperchaos even for low values of time delay which is evidenced by multiple positive Lyapunov exponents are brought out. The study is extended to the case of two coupled systems, one with delay and the other one without delay.Comment: 34 Pages, 14 Figure