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Unformatted text preview: stant harvesting by centering the bifurcation at the origin.
Let and . Then [2] becomes
. [3] 3
Ch8Lecs This is the normal form of the saddlenode bifurcation. For
there are two hyperbolic
equilibrium points, for
there is a single nonhyperbolic equilibrium, and for
there
are no equilibria. ■
The saddlenode bifurcation requires 3 conditions on the vector field: Singularity condition (the equilibrium point [3] is nonhyperbolic at
).
Nondegeneracy condition (the coefficient of
in [3] is nonzero).
Transversality condition (that guarantees that the parameter perturbs the
nonhyperbolic equilibrium point in a transverse way:
in [3]). Local Bifurcations of Limit Cycles
To further illustrate the meaning of “local bifurcation” let’s briefly describe local bifurcations of
limit cycles.
sketch limit cycle , section , intersection
Consider a Poincaré map . corresponds to a fixed point of : .
The eigenvalues of are the Floquet multipliers (not including the trivial unit multiplier). Real and imaginary axes, region of evls within the unit circle for
, arrows showing that
evls can traverse the circle though
,
, or a complex conjugate pair can pass out of
the unit circle off the real axis.
Example A saddlenode bifurcation of limit cycles.
Suppose that for there is a semistable limit cycle and an associated Poincaré map: sketch semistable limit cycle in 2D and nearby trajectories
sketch
,
, the line
, the Poincaré map is concave down and
tangent to the line at the origin, cobweb paths inside and outside the limit cycle
sketch pair of limit cycles. The outer one is stable and the inner one unstable.
Nearby trajectories. sketch
,
, the line
, the Poincaré map has shifted so there are
now two intersections with the line. Three cobweb paths. 4
Ch8Lecs sketch spiral flow without limit cycle no intersection between the Poincare map with
the line
.
The saddle node bifurcation corresponds to a single eigenvalue reaching the unit circle at
and then disappearing.■ Global Bifurcations of Limit Cycles
Global bifurcations cannot be described by a local analysis.
Example Homoclinic bifurcation in
sketch like Meiss Fig. 8.18.
the limit cycle for
■ .
and without the point and axis. Notice 8.2 Preservation of Equilibria
Consider
. An equilibrium point
satisfies
. When we linearize
about the equilibrium point, the Jacobian matrix is
. If the Jacobian matrix has a
zero eigenvalue then it is singular and we say the equilibrium point is degenerate. Otherwise
the Jacobian matrix is nonsingular and the equilibrium point is nondegenerate.
Contrast this with the definition of a nonhyperbolic equilibrium point, where only the real part
of an eigenvalue need be zero. For example, if a Jacobian matrix has...
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This document was uploaded on 02/24/2014 for the course MATH 512 at Washington State University .
 Fall '14
 MarcEvans

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