For example if a jacobian matrix has a pair of

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Unformatted text preview: a pair of complex conjugate imaginary eigenvalues and all of the other eigenvalues are nonzero, it is not singular. Example Fishery model with constant harvesting. One way that we have written the equation of motion is . Let be an equilibrium point. . When and , the equilibrium point is degenerate; corresponds to the saddle-node bifurcation. For the equilibrium point disappears. ■ 5 Ch8Lecs If an equilibrium point is nondegenerate, it cannot be removed by sufficiently small perturbations (such as a small change in the value of Theorem 8.1 (Implicit Function). Let be an open set in . Suppose there is a point such that nonsingular matrix. Then there are open sets function for which and and and and . with is a and a unique The proof of this famous theorem probably appears in your favorite analysis book. To gain a rough understanding of why the condition on the Jacobian is necessary , expand about : . Neglect the higher-order terms and solve for to obtain . This can be done for arbitrary only if is nonsingular. When discussing the system , the application of the implicit function theorem to preservation of equilibria corresponds to and . Corollary 8.2 (Preservation of a Nondegenerate Equilibrium) Suppose the vector field is in both and and that is a nondegenerate equilibrium point for parameter value Then there exists a unique curve of equilibria passing through at . Proof. The matrix governs the stability of the equilibrium, and since of its eigenvalues nonzero, is nonsingular. Then theorem 8.1 implies that there is a neighborhood of for which there is a curve of equilibria .□ 8.3 Unfolding Vector Fields Change of Variables and Topological Conjugacy Recall topological conjugacy between flows corresponds to the diagram: , . has all 6 Ch8Lecs where is a homeomorphism. The diagram implies . Bifurcation theory studies systems that depend on parameters. Let be the flow of and let be the flow of . If these flows are conjugate for each value of then there is the diagram: . where is a homeomorphism. The diagram implies . Example. Consider the system . Change variables using to obtain . Note that is a homeomorphism (in fact, a diffeomorphism). The corresponding flows are topologically conjugate. Meiss extends this terminology to the vector fields; he also calls and conjugate. ■ For a fixed value of , family of vector fields. gives a vector field. For all possible values of , A family of vector fields is induced by a family such that . If the family the same dynamics as or is simpler than . Example. The family map values of if there is a continuous map is induced by the family , then is induced by the family . The family gives a has using the has the same dynamics as . (In f...
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