The Hopf Bifurcation
and the Brusselator
Connor Smith and Oliver Morfin
December 3, 2010
Outline
• Hopf Bifurcation
– Limit Cycle
– Conditions
• The Jacobian Matrix
• The T-D Plane
• The Brusselator
– Proof
– Phase portraits
• Conclusion
2
The Hopf Bifurcation
• Critical point behaves normally, until…
• Small amplitude oscillations (limit-cycles)
• Weird!
3
A Limit Cycle
• Closed trajectory in phase space
• Solution curves converge and become periodic selfsustained oscillations
• Example:
Van der Pol Oscillator 
Image: http://www.wolframalpha.com/entities/calculators/van_der_Pol_oscillator/ov/bm/om/
4
The Jacobian Matrix
• Non-linear system of differential equations:
dx
 f (x, y; )
dt
dy
 g(x, y; )
dt
• Linearize by finding the Jacobian at the critical point:

f x
J(xeq )  
gx
f y 

gy 
• Get the eigenvalues:

TJ  TJ 2  4DJ
 ( ) 
2
5
Using the Trace-Determinant Plane
• When α = αH, two conditions must be true:
1. The TJ = 0 & DJ > 0 and,
2. The real part of the
eigenvalues satisfy the
condition:
 d 
  Re  
d 


 
H
0
• Hopf bifurcation!

Image: http://www.math.sunysb.edu/~scott/Book331/Fixed_Point_Analysis.html
6
The Brusselator
• Autocatalytic chemical reaction sequence:
A X
2X  Y  3X
• Rate equations:
dx
B  X Y  C
X D
 1  (b 1)x  ax y
2
dt
dy
 bx  ax 2 y
dt
7
First We Computed
• Equilibrium point:
• Jacobian matrix:

 b 
x eq  1, 
 a 
b 1 a 
J b   

1, 
b
a


 a 
• Trace & Determinant:

TJ  b 1  a
DJ  a

8

Then…
• Hopf bifurcation  TJ = 0 & DJ > 0
• a > 0 and b >0
– Therefore DJ = a > 0, as required.
– TJ  b 1  a  0  b  1 a
 b 
1,

• Since the system only has one equilibrium point  a 

a Hopf bifurcation occurs when b H  1 a.
• To prove that we have a Hopf bifurcation, there are

two conditions…

9
Condition #1
J
• Eigenvalues of 1, b  are purely imaginary and non a 
H
zero at b  1 a.
– λ’s of J1, b  are imaginary T2 - 4D <0
 a 
 TJ
– We know:
 b 1  a
DJ  a

– Given TJ =0, and b H  1 a  T disappears

– Thus
T 2  4D  4a  0 as required.


10
Condition #2
• The rate of change of the real part is greater than
zero at b H  1 a.
 



1
1
T  T 2  4D  T  Im()
2
2
1
Re()  T
2
• For any eigenvalue λ,
 d 
 Re( )  0
db 
• Sincea is a fixed constant:
 
 d Re( )  1 d b 1  a  1  0
db 
2 db
2
Thus, by #1 and #2, a Hopf bifurcation occurs at b H  1 a

11
b < a +1
A spiral SINK!
12
bH =a +1
• Outside the limit-cycle:
- A spiral SINK!
13
bH =a +1
• Inside the limit-cycle:
- A CENTER!
14
b >a +1
• Outside the limit cycle:
- A spiral SINK!
• Inside the limit cycle:
- A spiral SOURCE!
15
Conclusion
• Characteristics of a Hopf bifurcation
• How to find and determine the properties of a Hopf
bifurcation
• Analysis of the Brusselator’s Hopf bifurcation
16
References
• Arrowsmith, D.K., and C.M. Place. Ordinary Differential Equations: A
Qualitative Approach with Applications. London: Chapman and Hall, 1982.
• Ault, Shaun; Holmgreen, Erik. “Dynamics of the Brusselator”
Academia.edu, 16 March 2003. 11/27/10
<http://fordham.academia.edu/ShaunAult/Papers/83373/Dynamics_of_the_
Brusselator>
• Franke, Reiner. "A Precise Statement of the Hopf Bifurcation Theorem and
Some Remarks." 1-6 pp.
• <http://www.bwl.uni-kiel.de/gwif/files/handouts/dmt/HopfPrecise.pdf>.
• Guckenheimer, John; Myers, Mark; and Sturmfels, Bernd. "Computing
Hopf Bifurcations I." Society for Industrial and Applied Mathematics
Journal on Numerical Analysis 34 (1997): 1-21 pp. 11/11/10
<http://www.jstor.org/stable/2952033>.
• Kuznetsov, Yuri A. "Andronov-Hopf Bifurcation." Scholarpedia, 2006.
• Pernarowski, Mark. "Hopf Bifurcations - an Introduction." Montana State
University, 2004. 1-2.
• Wiens, Elmer G. "Bifurcations and Two Dimensional Flows." Egwald
Mathematics.
• Graphs were created using PPlane (math.rice.edu/~dfield/dfpp.html)
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