Lecture4A - 1 Chapter 4A 4.1 Definitions A dynamical system...

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1 Chapter 4A 4.1 Definitions A dynamical system is an evolution rule that defines trajectories in phase space: . is a differentiable map that is parameterized by time. is the solution of the initial value problem discussed in chapter 3. However, this chapter emphasizes a geometrical point of view and first discusses without reference to the initial value problem. It then comes back to the initial value problem through the properties of . The orbit or trajectory of a point is . Sketch , The forward orbit is . The preorbit or backward orbit is . Example . If is an equilibrium point then . Example . If , a periodic orbit with period , then and . We think of the map as operating on sets as well as points. Sketch Set is invariant under rule if for all . This means that, for each for all . is forward invariant if for . is backward invariant if for . 4.2 Flows Recall that if is on an open set then the initial value problem has a solution that is differentiable with respect to both and . A complete flow is a differentiable mapping such that (a) (b) for all The symbol denotes composition: . Property (b) is the group property , and implies

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2 Chapter 4A The mapping is invertible: . Any time may be viewed as the initial time: . Trajectories cannot cross. sketch , trajectories starting on and and then crossing If , then apply to both sides to obtain . That is, and are on the same trajectory. A flow may refer to a complete flow, but may instead refer to a mapping for which property (b) does not hold for all because trajectories are not defined for all times. A vector field is a function that defines a vector at each point . The vector field associated with a given flow is . Lemma 4.1 If is a flow, then it is a solution of the initial value problem , Proof. In text. Don’t give. If the flow is complete, the maximum interval of existence is . Example . Show that the function defined by is a complete flow on . Find the vector field associated with the flow. First note that is defined for all and . satisfies property (a) since . To verify that satisfies property (b), simplify the composition: . The vector field associated with the flow is
3 Chapter 4A . Lemma 4.2. Let be an open subset of , and a vector field such that the initial value problem , has a unique solution that exists for all and all . Then is a complete flow. Proof. In text. Don’t give. 4.3 Global Existence of Solutions The solutions of initial value problems for ODEs sometimes have finite maximum intervals of existence. How can we make use of the elegant framework of complete flow? Consider , [1] . [2] Theorem 4.3’ (Bounded Global Existence). If and bounded, then the solution of [1] defines a complete flow. Proof. Denote the maximum interval of existence . By assumption, there is an such that . For , [2] gives .

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