And which we analyzed in chapter 3 is upon

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Unformatted text preview: ime by [4], this system defines a complete 4.4 Linearization Let , where and is an dimensional manifold. Consider . [1] Suppose has an equilibrium point about : . Let and expand sing Taylor’s theorem . Use to find . These expressions use as notation (“little-oh” notation). In general: if there is a neighborhood . There is also notation (“big-oh” notation): of such that 7 Chapter 4A as . Example. If if there is a neighborhood of and a such that .■ then Taylor’s theorem wo ld give The linearization of [1] at is [2] where . Here and . Recall the decomposition of into unstable, center and stable eigenspaces of : , where is the unstable eigenspace, and so forth. An equilibrium point is hyperbolic if none of the eigenvalues of have zero real part. Then is empty. Hyperbolic equilibrium points are important because, by the Hartman-Grobman theorem (section 4.8), the flow of [1] in some neighborhood of a hyperbolic equilibrium point is topologically equivalent to the flow of [2]. Distinguish three classes of hyperbolic equilibrium points : is a sink if all of the eigenvalues of have negative real parts. Then . is a source if all of the eigenvalues of have positive real parts. Then . is a saddle if some of the eigenvalues of have negative real parts and some have positive real parts. Then . Often very useful information about the behavior of can be obtained by finding the equilibrium points and analyzing the associated linearized systems using the methods of chapter 2. Example. The Lotka-Volterra Competition Model. , . The equilibrium points are sketch , (3,0), (0,2) and (1,1). -, Linearized systems have the form -, eq. pts. , where . 8 Chapter 4A The Jacobean matrix is Linearize about . Obtain . Notice that linearizing about the origin is equivalent to dropping the nonlinear terms in the Lotka-Volterra equations. See that , . This is a source. Add to sketch. and Linearize about Obtain . See , . See . This is a sink. Add to sketch. Linearize about Obtain . See , and find . . This is also a sink. Add to sketch. See Linearize about Obtain . Find . Find the corresponding eigenvectors and the eigenvalues and . This is a saddle. add to sketch. ■ Example. The Lorenz model with . , . Linearize about the equilibrium point at terms. Obtain . This is equivalent to dropping the nonlinear . This has block diagonal form. The block has eigenvalues . For all of the eigenvalues of the Lorenz model are negative and the origin is a sink. For one eigenvalue is positive and the origin is a saddle. The origin loses stability as increases through 1. A bifurcation occurs at .■ 9 Chapter 4A 4.5 Stability An equilibrium of a flow sketch , If , is Lyapunov stable or stable if . , , trajectory based at stays in such that is not stable then it is unstable. An equilibrium is asymptotically stable if it is Lyapuno...
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This document was uploaded on 02/24/2014 for the course MATH 512 at Washington State University .

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