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The Hopf Bifurcation and the Brusselator

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. The Hopf Bifurcation.

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The Hopf Bifurcation and the Brusselator

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  1. The Hopf Bifurcationand the Brusselator Connor Smith and Oliver Morfin December 3, 2010

  2. Outline • Hopf Bifurcation • Limit Cycle • Conditions • The Jacobian Matrix • The T-D Plane • The Brusselator • Proof • Phase portraits • Conclusion

  3. The Hopf Bifurcation • Critical point behaves normally, until… • Small amplitude oscillations (limit-cycles) • Weird!

  4. A Limit Cycle • Closed trajectory in phase space • Solution curves converge and become periodic self-sustained oscillations • Example: Van der Pol Oscillator  Image: http://www.wolframalpha.com/entities/calculators/van_der_Pol_oscillator/ov/bm/om/

  5. The Jacobian Matrix • Non-linear system of differential equations: • Linearize by finding the Jacobian at the critical point: • Get the eigenvalues:

  6. Using the Trace-Determinant Plane • When α = αH, two conditions must be true: • The TJ = 0 & DJ > 0 and, • The real part of the eigenvalues satisfy the condition: • Hopf bifurcation! Image: http://www.math.sunysb.edu/~scott/Book331/Fixed_Point_Analysis.html

  7. The Brusselator • Autocatalytic chemical reaction sequence: • Rate equations:

  8. First We Computed • Equilibrium point: • Jacobian matrix: • Trace & Determinant:

  9. Then… • Hopf bifurcation  TJ = 0 & DJ > 0 • a > 0 and b >0 • Therefore DJ = a > 0, as required. • Since the system only has one equilibrium point a Hopf bifurcation occurs when . • To prove that we have a Hopf bifurcation, there are two conditions…

  10. Condition #1 • Eigenvalues of are purely imaginary and non-zero at . • λ’s of are imaginary T2 - 4D <0 • We know: • Given TJ =0, and  T disappears • Thus as required.

  11. Condition #2 • The rate of change of the real part is greater than zero at . • For any eigenvalue λ, • Since a is a fixed constant: Thus, by #1 and #2, a Hopf bifurcation occurs at

  12. b < a +1 A spiral SINK!

  13. bH=a +1 • Outside the limit-cycle: • - A spiral SINK!

  14. bH=a +1 • Inside the limit-cycle: • - A CENTER!

  15. b >a +1 • Outside the limit cycle: • - A spiral SINK! • Inside the limit cycle: • - A spiral SOURCE!

  16. Conclusion • Characteristics of a Hopf bifurcation • How to find and determine the properties of a Hopf bifurcation • Analysis of the Brusselator’s Hopf bifurcation

  17. 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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