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Control of the Planar Takens–Bogdanov Bifurcation with Applications

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Abstract

It is well-known that on a versal deformation of the Takens–Bogdanov bifurcation is possible to find dynamical systems that undergo saddle-node, Hopf, and homoclinic bifurcations. In this document a nonlinear control system in the plane is considered, whose nominal vector field has a double-zero eigenvalue, and then the idea is to find under which conditions there exists a scalar control law such that be possible establish a priori, that the closed-loop system undergoes any of the three bifurcations: saddle-node, Hopf or homoclinic. We will say then that such system undergoes the controllable Takens–Bogdanov bifurcation. Applications of this result to the averaged forced van der Pol oscillator, a population dynamics, and adaptive control systems are discussed.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Abed, E.H., Fu, J.H.: Local feedback stabilization and bifurcation control II. Stationary bifurcation. Syst. Control Lett. 8, 467–473 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bazykin, A., Kuznetsov, Y., Khibnik, A.: Bifurcation diagrams of planar dynamical systems. Research Computing Center, USSR Academy of Sciences, Pushkino (1985) (in Russian)

  4. Bogdanov, R.I.: Versal deformations of a singular point on the plane in the case of zero eigenvalues. Funct. Anal. Appl. 9, 144–145 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  5. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences, vol. 42. Springer, New York (1993)

    Google Scholar 

  6. Hamzi, B., Kang, W., Barbot, J.P.: Analysis and control of Hopf bifurcations. SIAM J. Control Optim. 42(6), 2200–2220 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Kang, W.: Bifurcation control via state feedback for systems with a single uncontrollable mode. SIAM J. Control Optim. 38, 1428–1452 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Salam, F.M.A., Bai, S.: Complicated dynamics of a prototype continuous-time adaptive control system. IEEE Trans. Circuits Syst. 35(7) (1988)

  9. Takens, F.: Forced oscillations and bifurcations. Applications of global analysis I. Commun. Math. Inst. Rijksuniversitat Utrecht 3, 1–59 (1974)

    MathSciNet  Google Scholar 

  10. Verduzco, F.: Control of codimension one stationary bifurcations. Int. J. Bifurc. Chaos 17(2), 575–582 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Verduzco, F., Alvarez, J.: Hopf bifurcation control: A new approach. Syst. Control Lett. 55, 437–451 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. Texts in Applied Mathematics, vol. 2. Springer, New York (2003)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, University of Sonora, Rosales and Transversal, Hermosillo, Sonora, Mexico

    Francisco A. Carrillo & Fernando Verduzco

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  1. Francisco A. Carrillo
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  2. Fernando Verduzco
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Correspondence to Fernando Verduzco.

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Carrillo, F.A., Verduzco, F. Control of the Planar Takens–Bogdanov Bifurcation with Applications. Acta Appl Math 105, 199–225 (2009). https://doi.org/10.1007/s10440-008-9272-9

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