Systems & Control Letters 55 (2006) 437 – 451 www.elsevier.com/locate/sysconle Hopf bifurcation control: A new approach Fernando Verduzco a , ∗ , Joaquin Alvarez b a Sonora University (UNISON), Mexico b Scientific Research and Advanced Studies Center of Ensenada (CICESE), Mexico Received 16 January 2004; received in revised form 10 September 2005; accepted 20 September 2005 Available online 21 November 2005 Abstract In this paper we find conditions to control a Hopf bifurcation in a class of nonlinear systems whose linear approximation has two eigenvalues on the imaginary axis, without assuming that the system is controllable. We use the center manifold theorem to project the dynamics on a two dimensional manifold, and design a controller that permits us to decide the stability and direction of the emerging periodic solution. © 2005 Elsevier B.V.All rights reserved. Keywords: Hopf bifurcation; Center manifold; Bifurcation control 1. Introduction In the last years there has been a great interest to analyze control systems displaying complex dynamics. An emerging research field that has become very stimulating is the bifurcation control which, among other objectives, aims to modify the dynamical behavior of a system around bifurcation points, generates a particular bifurcation at a given parameter value [3], delays the onset of an inherent bifurcation [10], or stabilizes a bifurcated solution [1,2]. An overview of this field can be found in [5]. In [1,2,11,12] the bifurcation control problem is analyzed using state feedback. Kang and others [4,7,8] investigated this problem using normal forms and invariants. For the particular case of Hopf bifurcation, several procedures have been proposed to design the controller. In [1] the stability of the emerging periodic orbit is determined by its characteristic exponents. The center manifold theorem is used in [7] to project the system dynamics on a two dimensional manifold. They decompose the original system in two subsystems, a two-dimensional, non-hyperbolic system and a (n -2)-dimensional, hyperbolic one; then they represent the hyperbolic part in the Brunovski form, assuming controllability of this subsystem. Verduzco and others [11,12] analyzed the control of Hopf bifurcation in a similar way as Hamzi and others [7], removing the controllability assumption by using a Jordan form representation for the hyperbolic subsystem. In this document we follow the last method and complement the results obtained in [11]. 2. Preliminary results Two essential theorems are included in this section. The center manifold theorem reduces the analysis of the system to dimension two, while the Hopf bifurcation theorem establishes conditions to ensure the direction and stability of the limit cycle. Theorem 1 (The local Center Manifold Theorem, Perko [9]). Let f ∈ C r (U), where U is an open subset of R n containing the origin and r  1. Suppose that f(0) = 0 and that Df (0) has c eigenvalues with zero real parts and s eigenvalues with negative real ∗ Corresponding author. Tel.: +52 662 2592155; fax: +52 662 2592219. E-mail address: verduzco@gauss.mat.uson.mx (F. Verduzco). 0167-6911/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2005.09.007