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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