Limit Cycles and Hopf Bifurcation

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Limit Cycles and Hopf Bifurcation

• Chris Inabnit
• Brandon Turner
• Thomas Buck

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Direction Field
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Let the functions F and G have continuous first
partial derivatives in a domain D of the
xy-plane. A closed trajectory of the system
must necessarily enclose at least one
critical (equilibrium) point. If it encloses
only one critical point, the critical point
cannot be a saddle point.

• Theorem

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Graphical Interpretation
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Graphical Interpretation
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Specific Case of Theorem

• Find solutions for the following system
• Do both functions have continuous first order
  partial derivatives?

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Specific Case of Theorem

• Critical point of the system is (0,0)
• Eigenvalues are found by the corresponding linear
  system
• which turn out to be .

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What does this tell us?

• Origin is an unstable spiral point for both the
  linear system and the nonlinear system.
• Therefore, any solution that starts near the
  origin in the phase plane will spiral away from
  the origin.

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Trajectories of the System
Forming a system out of and
yields the trajectories shown.
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Using Polar Coordinates
Using x r cos(?) y r sin(?)
r 2 x 2 y 2
Goes to
Critical points ( r 0 , r 1 )

Thus, a circle is formed at r 1
and a point at r 0.
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Stability of Period Solutions
Orbital Stability Semi-stable Unstable
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Example of Stability
Given the Previous Equation
If r gt 1, Then, dr/dt lt 0 (meaning the
solution moves inward) If 0 lt r lt 1, Then, dr/dt
gt 0 (meaning the solutions movies outward)
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Bifurcation
Bifurcation occurs when the solution of an
equation reaches a critical point where it then
branches off into two simultaneous solutions.
y 0 y x
A simple example of bifurcation is the solution
of y2 x . When x lt 0 , y is identical to zero.
However, when x 0 , a second solution (y /-
x) emerges.
_
gt
Combining the two solutions, we see the
bifurcation point at x 0 . This type of
bifurcation is called pitchfork bifurcation.
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Hopf Bifurcation
Introducing the new parameter ( µ )
Converting to polar form as in previous slide
yields
r µ
Critical Points are now r 0 and
r µ
r 0
If you notice, these solutions are extremely
similar to those of the previous example y2 x
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Hopf Bifurcation
As the parameter µ increases through the value
zero, the previously asymptotically stable
critical point at the origin loses its stability,
and simultaneously a new asymptotically stable
solution (the limit cycle) emerges.
Thus, µ 0 is a bifurcation point. This type of
bifurcation is called Hopf bifurcation, in honor
of the Austrian mathematician Eberhard Hopf who
rigorously treated these types of problems in a
1942 paper.
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References

• Boyce, William, and DiPrima, Richard.
  Differential Equations. Hoboken John Wiley
  Sons, Inc.
• Bronson, Richard. Schaums Outlines Differential
  Equations. McGraw-Hill Companies, Inc., 1994
• Leduc, Steven. Cliffs Quick Review Differential
  Equations. Wiley Publishing, Inc., 1995.