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Ch8Lecs
8.1 Bifurcations of Equilibria
Bifurcation theory
studies qualitative changes in solutions as a parameter varies.
In general,
one could study the bifurcation theory of ODEs, PDEs, integrodifferential equations, discrete
mappings etc.
Of course, we are concerned with ODEs.
Local bifurcations
refer to qualitative changes occurring in a neighborhood of an equilibrium
point of a differential equation or a fixed point of an associated Poincaré map.
These can be
studied by expanding the equations of motion in power series about the point.
Consider an ODE depending on a parameter
:
,
,
.
Assume this system has an equilibrium point
that is a sink for
where
is a
critical value of the parameter.
Graph the evls as a function of
.
Real and imaginary
axes, region of evls with negative real parts for
, arrows showing
that the real parts of some evls are increasing with
A real evl or a cc pair may traverse the imaginary axis as
increases through
.
The sink
becomes a source or a saddle. Or the equilibrium point may simply disappear when it has a
zero evl!
Example
.
Fishery model with constant harvesting.
Recall the logistic model of population growth with an additional constant term.
Interpret as a model of a fishery with
the number of fish,
time,
the fish growth rate,
the
carrying capacity, and
a constant rate of harvesting.
The model may be simplified by
measuring the quantity of fish
and the harvest rate in units of the carrying capacity:
and
.
Further simplify by introducing a dimensionless time variable:
.
Then obtain
.
[1]
Equilibria correspond to
.
Analyze graphically
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Ch8Lecs
sketch
,
,
,
The direction of the arrows on the
axis now depends on the sign of
.
When
there are two equilibrium points, when
there is one, and when
there are none.
Alternately, we can study the equilibria algebraically.
Let
.
Then equilibrium points
of [1] correspond to
or
.
The equilibria are
,
These exist only for
.
add
, arrows to graph
We can write [1] in the form
,
[2]
where the dot now refers to the derivative with respect to dimensionless time.
The
eigenvalues of the equilibrium points are
,
which depends on
through
.
Since
, its eigenvalue is negative and
is a stable
equilibrium point.
Since
, its eigenvalue is positive and
is an unstable equilibrium
point.
As
increases toward
the equilibria converge and their eigenvalues approach 0.
At
the equilibria merge as a single nonhyperbolic equilibrium point and for
they
disappear.
Draw a bifurcation diagram
for the logistic model with constant harvesting by plotting the
equilibria as a function of
.
sketch
,
, the two branches of the saddlenode
A saddlenode bifurcation
occurs at a critical value of the parameter,
.
Simplify the logistic model with constant harvesting by centering the bifurcation at the origin.
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 Fall '14
 MarcEvans
 Linear Algebra, Equilibrium point, Bifurcation theory, Saddlenode bifurcation, Transcritical Bifurcation, Homological Operator

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