1.1. First-Order ODE: Existence and Uniqueness 15 1.1.11. Definition. A C1 vector field V : Rn → Rn (and also the autonomous differential equation dx dt = V (x)) is called complete if α(p) = −∞ and ω(p) = ∞ for all p in Rn. In this case, for each t ∈ R we define a map φt : Rn → Rn by φt(p) = σp(t). The mapping t → φt is called the flow generated by the differential equation dx dt = V (x). 1.1.12. Remark. Using our smoke particle metaphor, the mean- ing of φt can be explained as follows: if a puff of smoke occupies a region U at a given time, then t units of time later it will occupy the region φt(U). Note that φ0 is clearly the identity mapping of Rn. Exercise 1–11. Show that the φt satisfy φt 1 +t2 = φt 1 φt 2 , so that in particular φ−t = φ−1. t In other words, the flow generated by a complete, autonomous vector field is a homomorphism of the additive group of real numbers into the group of bijective self-mappings of Rn. In the next section we will see that (t, p) → φt(p) is jointly contin- uous, so that the φt are homeomorphisms of Rn. Later (in Appendix F) we will also see that if the vector field V is Cr, then (t, p) → φt(p) is also Cr, so that the flow generated by a complete, autonomous, Cr differential equation dx dt = V (x) is a homomorphism of R into the group of Cr diffeomorphisms of Rn. The branch of mathematics that studies the properties of flows is called dynamical systems theory. • Example 1–1. Constant Vector Fields. The simplest exam- ples of autonomous vector fields in Rn are the constant vector fields V (x) = v, where v is some fixed vector in Rn. The maximal so- lution curve with initial condition p of dx dt = v is clearly the linearly parametrized straight line σp : R → Rn given by σp(t) = p+tv, and it follows that these vector fields are complete. The corresponding flow φt is given by φt(p) = p + tv, so for obvious reasons these are called constant velocity flows. In words, φt is translation by the vector tv, and indeed these flows are precisely the one-parameter subgroups of the group of translations of Rn. • Example 1–2. Exponential Growth. An important complete vector field in R is the linear map V (x) = kx. The maximal solution curves of dx dt = kx are again easy to write down explicitly, namely

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