General Theory of Global Differential Dynamics

Abstract

In the theory of local dynamical systems we study differential equations in an open subset of the real number space Rn; whereas in global dynamics the space is a general differentiable manifold Mn. We specify a global dynamical system as a tangent vector field v on Mn; and in any local chart (x1 ,…, xn) on Mn we denote the dynamical system v by its components vi(x1 , …, xn), say $$ \matrix{ {v) {{dx} \over {dt}} = {v^i}({x^1}, \ldots ,{x^n})} & i \cr } = 1,2, \ldots ,n $$ or $$ \dot x = v(x). $$