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

$$ \dot x = v(x). $$