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bifurcation-analysis.nb
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bifurcation-analysis.nb
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(* Content-type: application/vnd.wolfram.mathematica *)
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Cell[CellGroupData[{
Cell["Bifurcation Analysis", "Title",
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"Below is a sequence of examples of four common types of bifurcation: \
saddle-node, transcritical, pitchfork, and Hopf. For each example we provide \
a specific autonomous ODE system which displays that type of bifurcation. In \
all examples, ",
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"As a review, a bifurcation occurs when changing the value of the parameter ",
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" causes a qualitative change to the equilibria of the system. The specific \
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it describes a change in the number or the stability of the equilibria. We \
can attempt to look for bifurcations by solving for the equilibria ",
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"A bifurcation also occurs when the stability of an equilibrium changes. In \
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of the Jacobian matrix. Given an ODE system ",
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"If all eigenvalues have negative real parts, then ",
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saddle point, or orbiting in a particular direction) can also be made. As \
with the equilibria, themselves, the eigenvalues of the Jacobian are \
functions of ",
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"Note that, in the case of a one-dimensional ODE system ",
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"Each section below describes a different type of bifurcation. First an ODE \
system will be presented, then a bifurcation analysis will be conducted by \
hand, and finally a Manipulate environment will be defined to show a \
visualization of the bifurcation in action. In all cases there is a slider to \
control the value of ",
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". Equilibria will be shown in red on the stream plot, either as a line (for \
the 1D case) or a point (in the 2D case). For the 1D cases, to the right of \
the stream plot will be a plot of the equilibrium values as a function of ",
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equilibria, while dotted lines and hollow dots correspond to unstable \
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A saddle-node bifurcation occurs when two equilibria collide and annihilate \
each other (or, if moving in the other direction, when two equilibria are \
spontaneously generated out of nothing). Specifically this can occur when \
changing the parameters causes a pair of equilibria to simultaneously become \
real or imaginary. The example below concerns the 1D system:\
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"We can conduct stability analysis by evaluating ",
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