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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
matrix. Then there are open sets
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
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
Corollary 8.2 (Preservation of a Nondegenerate Equilibrium) Suppose the vector field
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
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
. If the family
the same dynamics as or is simpler than .
Example. The family
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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- Fall '14