Measuring the criticality of a Hopf bifurcation
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ORIGINAL PAPER
Measuring the criticality of a Hopf bifurcation Alexei Uteshev
· Tamás Kalmár-Nagy
Received: 9 March 2020 / Accepted: 23 August 2020 © The Author(s) 2020
Abstract This work is based on the observation that the first Poincaré–Lyapunov constant is a quadratic function of the coefficients of the two-dimensional vector field at a Hopf bifurcation. From a given parameter point, we find the distance to the “Hopf quadric.” This distance provides a measure of the criticality of the Hopf bifurcation. The viability of the approach is demonstrated through numerical examples. Keywords Hopf bifurcation · Distance to bifurcation · Discriminant
1 Introduction An important task in nonlinear science is to determine the nature of the bifurcations the system under study exhibits. A similarly important but less frequently studied problem is the robustness of the system. Quite naturally, the Euclidean distance between the actual parameter values and those of the closest bifurcation can be used to characterize the robustness of the system. A. Uteshev Faculty of Applied Mathematics, St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, Russia 199034 T. Kalmár-Nagy (B) Department of Fluid Mechanics, Faculty of Mechanical Engineering, Budapest University of Technology and Economics, 1111 Bertalan Lajos u. 4-6, Budapest, Hungary e-mail: [email protected]
Dobson [1] considers the general dynamical system
x˙ = f (x, u0 )
(1)
depending on parameters contained in the vector u0 . The distance-to-bifurcation problem seeks the minimal parameter variation γ = u0 − u for which Eq. (1) has a phase portrait qualitatively different from that of the original system. Dobson [1,2] proposed an iterative and a direct method of finding the closest bifurcation in parameter space. These initial approaches used numerical methods to find the minimum of γ , but were computationally demanding and only guaranteed locally optimal solutions. A global search was proposed by Kremer [3]. These methods use the normal vectors at a bifurcation curve in a parameter plane and are suitable for bifurcations of equilibria. Tamba and Lemmon [4] recast the minimum distance-to-bifurcation problem as a sum-of-squares relaxation for nonnegative dynamical systems that have kinetic realizations. Kitajima and Yoshinaga [5] extend Dobson’s method for periodic orbits. These methods and their extensions have been applied to hydraulic systems [3], gene systems [6], power systems [7–9] and bifurcations of arbitrary codimension [10]. Measuring the distance from a possible bifurcation in parameter space is also useful in making parametric changes to modify the nature of the bifurcation. Abed and Fu [11] proposed the use of nonlinear feedback control to invert the direction of the bifurcation. Yuen
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A. Uteshev
and Bau [12] demonstrated theoretically and experimentally that through the use of a nonlinear feedback controller, one can render a subcritical Hopf bifurcation supercritical and dramatically modify the nature of the flow in a the
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