14 1. Differential Equations and Their Solutions condition p. The interval on which σp is defined will be denoted by (α(p),ω(p)), where of course α(p) is either −∞ or a negative real number and ω(p) is either or a positive real number. As we have seen, a maximal solution σp need not be defined on all of R, and it is important to know just how the solution “blows up” as t approaches a finite endpoint of its interval of definition. A priori it might seem that the solution could remain in some bounded region, but it is an important fact that this is impossible—if ω(p) is finite, then the reason the solution cannot be continued past ω(p) is simply that it escapes to infinity as t approaches ω(p). 1.1.10. No Bounded Escape Theorem. If ω(p) ∞, then lim t→ω(p) σp(t) = ∞, and similarly, if α(p) −∞, then lim t→α(p) σp(t) = ∞. Exercise 1–9. Prove the No Bounded Escape Theorem. (Hint: If limt→ω(p) σ(p) = ∞, then by Bolzano-Weierstrass there would be a sequence tk converging to ω(p) from below, such that σp(tk) q. Then use the local existence theorem around q to show that you could extend the solution beyond ω(p). Here is where we get to use the fact that there is a neighborhood O of q such that a solution exists with any initial condition q in O and defined on the whole interval (−, ). For k sufficiently large, we will have both σp(tk) in O and tk ω , which quickly leads to a contradiction.) Here is another special property of autonomous systems. Exercise 1–10. Show that the images of the σp partition Rn into disjoint smooth curves (the “streamlines” of smoke particles). These curves are referred to as the orbits of the ODE. (Hint: If x(t) and ξ(t) are two solutions of the same autonomous ODE and if x(t0) = ξ(t1), then show that x(t0 + s) = ξ(t1 + s).)
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