Other solutions are asymptotically parallel to as and

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Unformatted text preview: lly parallel to as . Sketch [C] Saddle: the origin on . Sketch -plane, and , several trajectories. . Straight line solutions move away from the origin on and towards . Other solution are asymptotic to as and asymptotic to as -plane, and , several trajectories. Like Meiss Fig. 2.3 [D] Unstable focus: . Both the eigenvalues and eigenvectors are complex conjugate. (To see this just take the complex conjugate of A .). Let , where and are real vectors. The solution is still given by [4], where and must also be conjugate in order that the solution be real. Let . Substituting these expressions into the right hand side of [4] one obtains [5] 6 . We have written The solution is an exponentially expanding spiral. Example. The following phase portrait was constructed for the linear system with For this case , , and so that . 0.4 0.2 0 .2 0.1 0.1 0.2 0.2 0.4 0.6 ■ [E] Stable focus: . The analysis is similar to that for an unstable focus. The solution is an exponentially contracting spiral. There are also several cases corresponding to boundaries between the 5 regions on the -plane. The first two cases correspond to non-isolated equilibria, which occur when there is a zero eigenvalue giving . 7 [a] Unstable degenerate equilibrium: , . The eigenspace is a line of unstable equilibria. Solutions that originate off this line move away exponentially as along lines parallel to . [b] Stable degenerate equilibrium: , . The eigenspace is a line of stable equilibria. Solutions that originate off this line move away exponentially as along lines parallel to . [c] Center: are obtained for and . The analysis is given by [5] with . Trajectories are ellipses. This case is particularly important both because it occurs in bifurcations (chapter 8) and because Hamiltonian systems always have . To analyze the boundary states corresponding to the parabola we need to recall some concepts from linear algebra, beginning with two concepts of the multiplicity of an eigenvalue: An eigenvalue has algebraic multiplicity if the characteristic polynomial can be written where , i.e. is a -fold root of the characteristic polynomial. An eigenvalue has geometric multiplicity if it has linearly independent eigenvectors , i.e. and . A theorem of linear algebra is that the geometric multiplicity is at most the algebraic multiplicity. When the geometric multiplicity is less than the algebraic multiplicity, the matrix is defec...
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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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