Abstract and Applied Analysis

Nonhyperbolic Periodic Orbits of Vector Fields in the Plane Revisited

Denis de Carvalho Braga, Luis Fernando Mello, and Antonio Carlos Zambroni de Souza

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The main goal of this paper is to present a theory of approximation of periodic orbits of vector fields in the plane. From the theory developed here, it is possible to obtain an approximation to the curve of nonhyperbolic periodic orbits in the bifurcation diagram of a family of differential equations that has a transversal Hopf point of codimension two. Applications of the developed theory are made in Liénard-type equations and in Bazykin’s predator-prey system.

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Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 212340, 19 pages.

First available in Project Euclid: 26 February 2014

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Braga, Denis de Carvalho; Mello, Luis Fernando; Zambroni de Souza, Antonio Carlos. Nonhyperbolic Periodic Orbits of Vector Fields in the Plane Revisited. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 212340, 19 pages. doi:10.1155/2013/212340. https://projecteuclid.org/euclid.aaa/1393449977

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  • Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, NY, USA, 2nd edition, 1998.
  • Yu. A. Kuznetsov, “Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODE's,” SIAM Journal on Numerical Analysis, vol. 36, no. 4, pp. 1104–1124, 1999.
  • D. C. Braga, L. F. Mello, C. Rocşoreanu, and M. Sterpu, “Con-trollable Hopf bifurcations of codimensions one and two in lin-ear control systems,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 21, no. 9, pp. 2665–2678, 2011.
  • W.-J. Beyn, A. Champneys, E. Doedel, W. Govaerts, Y. A. Kuznetsov, and B. Sandstede, “Numerical continuation, and computation of normal forms,” in Handbook of Dynamical Systems, Vol. 2, chapter 4, pp. 149–219, North-Holland, Amsterdam, The Netherlands, 2002.
  • A. Dhooge, W. Govaerts, and Yu. A. Kuznetsov, “MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs,” Association for Computing Machinery. Transactions on Mathematical Software, vol. 29, no. 2, pp. 141–164, 2003.
  • G. Iooss and D. Joseph, Elementary Stability and Bifurcation Theory, Springer, New York, NY, USA, 2nd edition, 2004.
  • Yu. A. Kuznetsov, H. G. E. Meijer, W. Govaerts, and B. Sautois, “Switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in ODEs,” Physica D, vol. 237, no. 23, pp. 3061–3068, 2008.
  • J. Sotomayor, L. F. Mello, and D. C. Braga, “Bifurcation analysis of the Watt governor system,” Computational & Applied Mathematics, vol. 26, no. 1, pp. 19–44, 2007.
  • C. Chicone, Ordinary Differential Equations with Applications, Springer, New York, NY, USA, 2006.
  • A. D. Bazykin and A. I. Khibnik, “On sharp excitation of self-oscillations in a Volterra-type model,” Biophysika, vol. 26, pp. 851–853, 1981 (Russian).