Substitute this into to obtain consequently the

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Unformatted text preview: imensional case. It is easy to see that . Since is , so is , and differentiation of [2] with respect to gives . Since , this implies that . This relationship helps compute the required derivatives of : . Thus the needed hypotheses for Theorem 8.3 are satisfied and there exists an extremal value such that when crosses zero the number of equilibria changes from zero to two. The stability of the equilibria follows by considering the stability of equilibria in the one dimensional case in conjunction with the nonhyperbolic Hartman-Grobman theorem from our earlier studies of the center manifold. □ Now we can see why this is called a saddle-node bifurcation. Consider theorem 8.6 for sketch in 1D case; two equilibria for equilibria for . sketch for for sketch for node; for , degenerate equilibrium for : for a stable node and saddle; for a converging flow without equilibrium , no a degenerate node; : for a saddle and an unstable node; for a diverging flow without equilibrium a degenerate Tranversality The following theorem gives a condition that guarantees that parameter is varied. . changes sign as a 14 Ch8Lecs Corollary 8.7 (transversality) Assume that any single parameter such that satisfies the hypotheses of theorem 8.6. If , is (transversality) then a saddle-node bifurcation takes place when Proof. Show that . Use to denote the critical point of as a function of . Then The first derivatives crosses zero. and and both vanish by hypothesis, then the transversality .□ assumption gives Example. Let’s apply the systematic procedure suggested in the statement of theorem 8.6 to the system [3] , which has an equilibrium at the origin. First, rewrite this in matrix form . The Jacobian matrix and has eigenvalues and . Corresponding eigenvectors are . The matrix equation has the general form , where . Let the transformation matrix . Then . Set and . or . We have 15 Ch8Lecs . This has the form , , given in the statement of theorem 8.6 (see [1]). The system satisfies the singularity conditions stated in the theorem and the nondegeneracy condition . Furthermore, , so the transversality condition is satisfied. Since , has a minimum and since , the minimum decreases through as increases. Going back to the original system [3], we can easily solve for the equilibria to get , confirming our result. ■ and 8.5 Normal Forms In chapter 2 we transformed linear systems in order to put them in a simple form (diagonalizing the matrices in the semisimple case). In chapter 4 we learned that these changes of coordinates were diffeomorphisms. In this section we continue the program of applying diffeomorphisms to transform to simpler forms, only now we seek to simplify the no...
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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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