Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 173along with the initial conditions F 0 = identity. The flow property generalizesthe situation where M = V is a linear space, X(x) = A x for a(bounded) linear operator A, and where F t (x) = e tA x – to the nonlinearcase. Therefore, the flow F t (m) can be defined as a formal exponentialF t (m) = exp(t X) = (I + t X + t2 2 X2 + ...) =∞∑ X k t k.k!recall that a time–dependent vector–field is a map X : M × R →T Msuch that X(m, t) ∈ T m M for each point m ∈ M and t ∈ R. An integralcurve of X is a curve γ(t) in M such thatk=0˙γ(t) = X (γ (t) , t) , for all t ∈ I ⊆ R.In this case, the flow is the one–parameter group of diffeomorphismsF t,s : M → M such that t ↦→ F t,s (m) is the integral curve γ(t) with initialcondition γ(s) = m at t = s. Again, the existence and uniqueness Theoremfrom ODE–theory applies here, and in particular, uniqueness gives thetime–dependent flow property, i.e., the Chapman–Kolmogorov lawF t,r = F t,s ◦ F s,r .If X happens to be time independent, the two notions of flows are relatedby F t,s = F t−s (see [Marsden and Ratiu (1999)]).3.6.1.7 Categories of ODEsOrdinary differential equations are naturally organized into their categories(see [Kock (1981)]). First order ODEs are organized into a category ODE 1 .A first–order ODE on a manifold–like object M is a vector–field X : M →T M, and a morphism of vector–fields (M 1 , X 1 ) → (M 2 , X 2 ) is a map f :M 1 → M 2 such that the following diagram commutesT M 1T f ✲ T M 2✻✻X 2X 1M 1✲ Mf2A global solution of the differential equation (M, X), or a flow line of avector–field X, is a morphism from ( R, ∂∂x)to (M, X).