Transcript Periodic Solutions and Limit Cycles

Limit Cycles and Hopf Bifurcation
Chris Inabnit
Brandon Turner
Thomas Buck
Direction Field
dy
 2 y
dx
Theorem
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
dx
 F ( x, y )
dt
dy
 G ( x, y )
dt
must necessarily enclose at least one critical
(equilibrium) point. If it encloses only one
critical point, the critical point cannot be a
saddle point.
Graphical Interpretation
dx
y
dt

dy
x
dt
Graphical Interpretation


 x  y
    
 y  x
Specific Case of Theorem
Find solutions for the following system

 x   x  y  x( x 2  y 2 ) 

   
2
2 
y

x

y

y
(
x

y
)
  
 Do both functions have continuous
first order partial derivatives?
Specific Case of Theorem
 Critical point of the system is (0,0)
 Eigenvalues are found by the corresponding
linear system

 x
 1 1 x 
   
 
 y
  1 1 y 
which turn out to be 1  i .
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.

dx
2
2
 x yx x  y
dt


dy
2
2
 x  y  y x  y
dt

Trajectories of the System
Forming a system out of
dx
dt
and
dy
dt
yields the trajectories shown.
Using Polar Coordinates
Using x = r cos()
y = r sin()

dx
 x  y  x x2  y2
dt
Goes to:

dr
 r 1 r2
dt

Critical points ( r = 0 , r = 1 )
Thus, a circle is formed at r = 1
and a point at r = 0.

r ^2 = x ^2 + y ^2

dy
 x  y  y x 2  y 2
dt

Stability of Period Solutions
Orbital Stability
Semi-stable
Unstable
Example of Stability
Given the Previous Equation:

dr
 r 1 r2
dt

If r > 1,
Then, dr/dt < 0
(meaning the solution moves inward)
If 0 < r < 1,
Then, dr/dt > 0
(meaning the solutions movies outward)
Bifurcation
Bifurcation occurs when the solution of an equation reaches a critical
point where it then branches off into two simultaneous solutions.
y=0
A simple example of bifurcation is
the solution of y2 = x .
When x < 0 , y is identical to zero.
_ 0 , a second
However, when x >
solution (y = +/- x) emerges.
Combining the two solutions, we see the
bifurcation point at x = 0 . This type of
bifurcation is called pitchfork bifurcation.
y= x
Hopf Bifurcation
Introducing the new parameter ( μ )

dx
 x  y  x x 2  y 2
dt


dy
  x  y  y x 2  y 2
dt
Converting to polar form as in previous slide yields:

dr
 r  r2
dt
Critical Points are now:
r=0
and
r=

r=
r=0
μ
If you notice, these solutions are
extremely similar to those of the
previous example y2 = x
μ

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.
References
 Boyce, William, and DiPrima, Richard.
Differential Equations. Hoboken: John Wiley &
Sons, Inc.
 Bronson, Richard. Schaum’s Outlines
Differential Equations. McGraw-Hill
Companies, Inc., 1994
 Leduc, Steven. Cliff’s Quick Review
Differential Equations. Wiley Publishing, Inc.,
1995.