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Stability and Stabilization of Nonlinear Systems pp 75–93Cite as

Control of dynamic bifurcations

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Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 246))

Abstract

We consider differential equations xẋ = f (x, λ) where the parameter λ = εt moves slowly through a bifurcation point of f. Such a dynamic bifurcation is often accompanied by a potentially dangerous jump transition. We construct smooth scalar feedback controls which avoid these jumps. For transcritical and pitchfork bifurcations, a small constant additive control is usually sufficient. For Hopf bifurcations, we have to construct a more elaborate control creating a suitable bifurcation with double zero eigenvalue.

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References

  1. E.H. Abed, Bifurcation-theoretic issues in the control of voltage collapse, Dynamical systems approaches to nonlinear problems in systems and circuits, Proc. Conf. Qual. Methods Anal. Nonlinear Dyn., Henniker/UK 1986 (1988)

    Google Scholar 

  2. E.H. Abed, Local bifurcation control, in J.H. Chow (Ed.), Systems and control theory for power systems (Springer, New York, 1995).

    Google Scholar 

  3. E.H. Abed, J.-H. Fu, Local feedback stabilization and bifurcation control. I. Hopf bifurcation, Systems Control Lett. 7:11–17 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  4. D. Aeyels, Stabilization of a class of nonlinear systems by a smooth feedback control, Systems Control Lett. 5:289–294 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  5. S.M. Baer, T. Erneux, J. Rinzel, The slow passage through a Hopf bifurcation: delay, memory effects, and resonance, SIAM J. Appl. Math. 49:55–71 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  6. E. Benoît (Ed.), Dynamic Bifurcations, Proceedings, Luminy 1990 (Springer-Verlag, Lecture Notes in Mathematics 1493, Berlin, 1991).

    Google Scholar 

  7. N. Berglund, Adiabatic Dynamical Systems and Hysteresis, Thesis EPFL no 1800 (1998). Available at http://dpwww.epfl.ch/instituts/ipt/berglund/these.html

  8. N. Berglund, On the Reduction of Adiabatic Dynamical Systems near Equilibrium Curves, Proceedings of the International Workshop “Celestial Mechanics, Separatrix Splitting, Diffusion”, Aussois, France, June 21–27, 1998.

    Google Scholar 

  9. N. Berglund, Control of Dynamic Hopf Bifurcations, preprint WIAS-479, mp-arc/99-89 (1999). To be published.

    Google Scholar 

  10. N. Berglund, H. Kunz, Chaotic Hysteresis in an Adiabatically Oscillating Double Well, Phys. Rev. Letters 78:1692–1694 (1997).

    Article  Google Scholar 

  11. N. Berglund, H. Kunz, Memory Effects and Scaling Laws in Slowly Driven Systems, J. Phys. A 32:15–39 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  12. J. Carr, Applications of Centre Manifold Theory (Springer-Verlag, New York, 1981).

    MATH  Google Scholar 

  13. F. Colonius, G. Häckl, W. Kliemann, Controllability near a Hopf bifurcation, Proc. 31st IEEE Conf. Dec. Contr., Tucson, Arizona (1992).

    Google Scholar 

  14. F. Colonius, W. Kliemann, Controllability and stabilization of one-dimensional systems near bifurcation points, Syst. Control Lett. 24:87–95 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  15. T. Erneux, P. Mandel, Imperfect bifurcation with a slowly-varying control parameter, SIAM J. Appl. Math. 46:1–15 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  16. N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eq. 31:53–98 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  17. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983).

    MATH  Google Scholar 

  18. R. Haberman, Slowly varying jump and transition phenomena associated with algebraic bifurcation problems, SIAM J. Appl. Math. 37:69–106 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  19. G. Häckl, K.R. Schneider, Controllability near Takens-Bogdanov points, J. Dynam. Control Systems 2:583–598 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  20. L. Holden, T. Erneux, Slow passage through a Hopf bifurcation: From oscillatory to steady state solutions, SIAM J. Appl. Math. 53:1045–1058 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  21. A.I. Khibnik, B. Krauskopf, C. Rousseau, Global study of a family of cubic Liénard equations, Nonlinearity 11:1505–1519 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  22. N.R. Lebovitz, R.J. Schaar, Exchange of Stabilities in Autonomous Systems, Stud. in Appl. Math. 54:229–260 (1975).

    MathSciNet  MATH  Google Scholar 

  23. N.R. Lebovitz, R.J. Schaar, Exchange of Stabilities in Autonomous Systems II, Stud. in Appl. Math. 56:1–50 (1977).

    MathSciNet  Google Scholar 

  24. E.F. Mishchenko, N.Kh. Rozov, Differential Equations with Small Parameters and Relaxations Oscillations (Plenum, New York, 1980).

    Google Scholar 

  25. W. Müller, K.R. Schneider, Feedback Stabilization of Nonlinear Discrete-Time Systems, J. Differ. Equations Appl. 4:579–596 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  26. N.N. Nefedov, K.R. Schneider, Delayed exchange of stabilities in singularly perturbed systems, Zeitschr. Angewandte Mathematik und Mechanik (ZAMM) 78:199–202 (1998).

    Google Scholar 

  27. N.N. Nefedov, K.R. Schneider, Immediate exchange of stabilities in singularly perturbed systems, Diff. Integr. Equations, 1999 (to appear).

    Google Scholar 

  28. A.I. Neishtadt, Persistence of stability loss for dynamical bifurcations I, Diff. Equ. 23:1385–1391 (1987).

    MathSciNet  Google Scholar 

  29. A.I. Neishtadt, Persistence of stability loss for dynamical bifurcations II, Diff. Equ. 24:171–176 (1988).

    MathSciNet  Google Scholar 

  30. A.I. Neishtadt, On Calculation of Stability Loss Delay Time for Dynamical Bifurcations in D. Jacobnitzer Ed., XI th International Congress of Mathematical Physics (International Press, Boston, 1995).

    Google Scholar 

  31. L.S. Pontryagin, L.V. Rodygin, Approximate solution of a system of ordinary differential equations involving a small parameter in the derivatives, Dokl. Akad. Nauk SSSR 131:237–240 (1960).

    MathSciNet  Google Scholar 

  32. M.A. Shishkova, Examination of system of differential equations with a small parameter in highest derivatives, Dokl. Akad. Nauk SSSR 209:576–579 (1973). [English transl.: Soviet Math. Dokl. 14:384–387 (1973)].

    MathSciNet  Google Scholar 

  33. F. Takens, Forced oscillations and bifurcations, Comm. Math. Inst., Rijksuniversiteit Utrecht 3:1–59 (1974).

    MathSciNet  Google Scholar 

  34. A.B. Vasil’eva, V.F. Butusov, L.V. Kalachev, The Boundary Function Method for Singular Perturbation Problems (SIAM, Philadelphia, 1995).

    Google Scholar 

  35. V. Venkatasubramanian, H. Schaettler, J. Zaborszky, On the dynamics of differential-algebraic systems such as the balanced large electric power system, in J.H. Chow (Ed.), Systems and control theory for power systems (Springer, New York, 1995).

    Google Scholar 

  36. E.V. Volokitin, S.A. Treskov, Parametric portrait of the Fitz-Hugh differential system, Math. Modelling 6:65–78 (1994). (in Russian)

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, D-10117, Berlin, Germany

    Nils Berglund & Klaus R. Schneider

Authors
  1. Nils Berglund
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  2. Klaus R. Schneider
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Editor information

Editors and Affiliations

  1. Dept. of Systems Dynamics, Universiteit Gent, Technologiepark-Zwijnaarde 9, 9052, Gent, Belgium

    Dirk Aeyels (Professor) (Professor)

  2. Laboratoire des Signaux et Systèmes, CNRS, SUPELEC, 91192, Gif-sur-Yvette, France

    Françoise Lamnabhi-Lagarrigue (Docteur d’état) (Docteur d’état)

  3. Dept. of Applied Mathematics, University of Twente, PO Box 217, 7500, Enschede, The Netherlands

    Arjan van der Schaft (Professor) (Professor)

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© 1999 Springer-Verlag London Limited

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Berglund, N., Schneider, K.R. (1999). Control of dynamic bifurcations. In: Aeyels, D., Lamnabhi-Lagarrigue, F., van der Schaft, A. (eds) Stability and Stabilization of Nonlinear Systems. Lecture Notes in Control and Information Sciences, vol 246. Springer, London. https://doi.org/10.1007/1-84628-577-1_4

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  • DOI: https://doi.org/10.1007/1-84628-577-1_4

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  • Publisher Name: Springer, London

  • Print ISBN: 978-1-85233-638-7

  • Online ISBN: 978-1-84628-577-6

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