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On anticontrol of Hopf bifurcations
Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187), 2002The control of the amplitude of the oscillations in an underactuated mechanical system is treated. Two different procedures for the design of the controller are proposed. One is based on Hopf bifurcation theory, and the other is derived from considerations about the system's energy.
Diego M. Alonso +2 more
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On the control of Hopf bifurcations
Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187), 2002Linear and quadratic normal forms of nonlinear systems with a pair of imaginary uncontrollable modes are derived. Based on the normal form, formulae of feedbacks are found to control the bifurcation of the system. The Hopf bifurcation cannot be removed from the closed-loop system, because the imaginary eigenvalues are uncontrollable.
Boumediene Hamzi +2 more
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On the analysis of hopf bifurcations
International Journal of Engineering Science, 1983Abstract The oscillatory instability and the family of limit cycles associated with a general autonomous dynamical system described by n nonlinear first order differential equations and an independently assignable scalar parameter are examined via an intrinsic method of harmonic analysis.
Huseyin, K., Atadan, A. S.
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Hopf bifurcation and the Hopf fibration
Nonlinearity, 1994Summary: We present techniques for studying the local dynamics generated by an equivariant Hopf bifurcation. We show that under general hypothesis we can expect the formation of a branch of attracting invariant spheres with capture all the local dynamics.
Field, Mike, Swift, James W.
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Turing Instabilities at Hopf Bifurcation
Journal of Nonlinear Science, 2009A simple procedure for deriving a uniform asymptotic expansion for the limit cycle in the vicinity of the Hopf bifurcation point for a two dimensional reaction system \[ u_{t} =D_{u}\Delta u+f\left( u,v;a\right) , \] \[ v_{t} =D_{v}\Delta v+g\left( u,v;a\right) \tag{b} \] is suggested. First, an algorithm allowing reduction of the system (ref {b}) to a
Ricard, M.R., Mischler, Stéphane
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Analysis and Control of Hopf Bifurcations
SIAM Journal on Control and Optimization, 2004Summary: In this paper, control systems with two uncontrollable modes on the imaginary axis are studied. The main contributions include the local orientation control of periodic solutions and center manifolds, the quadratic normal form of systems with two imaginary uncontrollable modes, the stabilization of the Hopf bifurcation by state feedback, and ...
Boumediene Hamzi +2 more
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1985
The term Hopf bifurcation refers to a phenomenon in which a steady state of an evolution equation evolves into a periodic orbit as a bifurcation parameter is varied. The Hopf bifurcation theorem (Theorem 3.2) provides sufficient conditions for determining when this behavior occurs.
Martin Golubitsky, David G. Schaeffer
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The term Hopf bifurcation refers to a phenomenon in which a steady state of an evolution equation evolves into a periodic orbit as a bifurcation parameter is varied. The Hopf bifurcation theorem (Theorem 3.2) provides sufficient conditions for determining when this behavior occurs.
Martin Golubitsky, David G. Schaeffer
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Hopf bifurcation and Hopf hopping in recurrent nets
IEEE International Conference on Neural Networks, 2002Some aspects of the learning dynamics of recurrent neural networks are discussed. It is shown that as a two-unit fully recurrent network is trained to oscillate, the learning process brings the network to a point where a small change in any one of the weights can push the network through a Hopf bifurcation to create stable oscillation.
Fu-Sheng Tsung, Garrison W. Cottrell
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Anti-control of Hopf bifurcations
IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 2001Summary: Bifurcation control generally means the design of a controller that is capable of modifying the bifurcation characteristics of a bifurcating nonlinear system, thereby achieving some desirable dynamical behaviors. A typical objective is to delay and/or stabilize an existing bifurcation. In this paper, we consider the problem of anti-controlling
Chen, Dong S. +2 more
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A Hopf bifurcation with spherical symmetry
ZAMP Zeitschrift f�r angewandte Mathematik und Physik, 1992The authors consider Hopf bifurcation on a ten dimensional center manifold in a \(O(3)\)-symmetric system. The main hypothesis is that the representation on the critical modes is given by the representation on \(V_ 5\oplus V_ 5\), the sum of two copies of the absolutely irreducible representations of \(O(3)\) on the spherical harmonics of order two ...
HAAF, H, ROBERTS, M, STEWART, I
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