1.7. Invariance Properties of Flows 33 Now let us consider the case of a real ODE, dx dt = V (x), but assume that the vector field V : Rn → Rn is analytic. This means simply that each component Vi(x1,... , xn) is given by a convergent power series. Then these same power series extend the definition of V to an analytic map of Cn to itself, and we are back to the situation above. (In fact, this is just the special case of what we considered above when the coeﬃcients of the power series are all real.) Of course, if we consider only the solutions of this “complexified” ODE whose initial conditions z0 are real and also restrict the time parameter τ to real values, then we get the solutions of the original real equation. So what we learn from this excursion to Cn and back is that when the right-hand side of the ODE dx dt = V (x) is an analytic function of x, then the solutions are also analytic functions of the time and the initial conditions. This complexification trick is already useful in the simple case that the vector field V is linear, i.e., when Vi(x) = ∑n i Aijxj for some n × n real matrix A. The reason is that the characteristic polynomial of A, P (λ) = det(A − λI), always factors into linear factors over C, but not necessarily over R. In particular, if P has distinct roots, then it is diagonalizable over C and it is trivial to write down the solutions of the IVP in an eigenbasis. We will explore this in detail in Chapter 2 on linear ODEs. 1.7. Invariance Properties of Flows In this section we suppose that V is some complete vector field on Rn and that φt is the flow on Rn that it generates. For many purposes it is important to know what things are “preserved” (i.e., left fixed or “invariant”) under a flow. For example, the function F : Rn → R is said to be invariant under the flow (or to be a “constant of the motion”) if F ◦ φt = F for all t. Note that this just means that each solution curve σp lies on the level surface F = F (p) of the function F . (In particular, in case n = 2, where the level “surfaces” are level curves, the solution curves will in general be entire connected components of these curves.)

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