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By assumption there is an such if were finite we would

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Unformatted text preview: we would have . This contradicts the theorem on escape from compact sets. So cannot be finite. Similarly, cannot be finite. Therefore the solution defines a flow by lemma 4.2. □ Remark. If a physical system were exactly described by an initial value problem of form [1], its vector field would presumably be bounded. ■ Example. The overdamped pendulum, Theorem 4.4’ (Reparameterization of Time). If , , defines a flow by theorem 4.3’. ■ then is equivalent to [3] 4 Chapter 4A upon reparameterization of time. The vector field of bounded. defines a flow since it is Proof. The solution of [1] has a maximum interval of existence using Since and . Define is increasing and the transformation is one-to-one. By the chain rule . Therefore . Clearly is bounded. It remains to show that the partial derivatives . Let are . View , as a function of and We need to show , fixing the other components of . Then . We are given . Note . and *3’+ is except at values where Suppose Observe that . Then . Then . It is sufficient to show that all of except possibly at those values. 5 Chapter 4A Further, if , From the last expression in *3’+, Thus is continuous at remains bounded as . Therefore .□ and Example. The undamped pendulum , defines a complete flow on upon reparameterization of time. ■ Example. The reparameterization of time [3] also converts some vector fields on domains with boundaries into flows defined for all times. The Lotka-Volterra system for the competitive interaction of two species is , . The domain is . The right hand of this system is in the interior of . Although the domain has boundaries, trajectories do not escape the domain by approaching the boundaries. Sketch the domain and the flow on the boundaries. However, for certain initial conditions they may escape to in finite time. (Suppose and are positive, , and . The trajectory will escape to in finite negative time.) Upon reparameterization of time by [3] this system defines a flow for all time. ■ There are also cases where where is a proper open subset of . The following theorem and its proof are found in Perko (3rd edition, theorem 2 in section 3.2). Theorem (Global Existence on Open Domains with Boundaries). Suppose of and . Then there is a function such that is an open subset is equivalent to [4] 6 Chapter 4A upon reparameterization of time. The vector field Idea of Proof. and let where defines a complete flow. is given by [3]. Define the closed set . is a distance from to the boundary of : . The effect of is to slow trajectories that approach any boundary of . The maximum interval of existence for the reparameterized system is .□ Sketch bo ndary and tra e tory approaching boundary Example. The initial value problem on flow. ■ and , which we analyzed in chapter 3. is . Upon reparameterization of t...
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