Generic Hopf Bifurcation From Lines of Equilibria Without Parameters Iii: Binary Oscillations

Fiedler, Bernold; Liebscher, Stefan; Alexander, Robert R.

doi:10.1142/s0218127400001018

Cited by 38 publications

(36 citation statements)

References 6 publications

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“…The limit controller (7) (when it exists) is not guaranteed to yield a stabilized limit system (8). Surprisingly, failure of achieving a stabilized limit system can be quite dramatic, as is the case in Example 1.…”

Section: Preliminaries and Motivationmentioning

confidence: 97%

“…We already determined the single-wedge, semi-centre and semi-saddle bifurcations for the closed-loop dynamics in the 1 + 1 case and the spherical-spiral and conical-spiral bifurcations in the 2 + 1 case; see also [16,20]. Secondly, we want to make connections with recent developments of so-called bifurcations without parameters, again by locally classifying the possible closed-loop dynamics; see [5,6,7,8]. Finally, we visualize the dynamics using DsTool [3] and a specially designed extension module [18].…”

Section: Preliminaries and Motivationmentioning

confidence: 99%

“…2 (d). This is called the hyperbolic case in [5,6,7,8]. Similar to the saddle-type transcritical bifurcation, the hyperbolic Hopf bifurcation can only occur in the context of Rokni-Lamooki, Townley & Osinga Bifurcations and limit dynamics 25 adaptive control if another critical point exist on the positive K-axis such that the equilibria to the right of this other critical point are again stable and attract the initial conditions that escape a neighbourhood of the Hopf point.…”

Section: Hopf Bifurcation Without Parametersmentioning

confidence: 99%

“…2 (c). This is called the elliptic case in [5,6,7,8]. Note that the cubic polynomial (ω 2 − 1 + ν 2 1 ) (ω + 1) has only one real root at ω = −1 < 0 if ν 1 > 1.…”

Section: Hopf Bifurcation Without Parametersmentioning

confidence: 99%

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Bifurcations and Limit Dynamics in Adaptive Control Systems

Lamooki

Townley

Osinga

2005

Int. J. Bifurcation Chaos

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Adaptive controllers are used in systems where one or more parameters are unknown. Such controllers are designed to stabilize the system using an estimate for the unknown parameters that is adapted automatically as part of the stabilization. One drawback in adaptive control design is the possibility that the closed-loop limit system is not stable. The worst situation is the existence of a destabilized limit system attracting a large open subset of initial conditions. These situations lie behind bad behaviour of the closed-loop adaptive control system. The main issue in this paper is to identify and characterize the occurrence of such bad behaviour in the adaptive stabilization of first-and second-order systems with one unknown parameter. We develop normal forms for all possible cases and find the conditions that lead to bad behaviour. In this context we discuss a number of bifurcation-like phenomena.1 Rokni-Lamooki, Townley & Osinga Bifurcations and limit dynamics 1 Preliminaries and motivationAdaptive control is an established approach to the control of (parametrically) uncertain systems. Loosely speaking, there are two approaches to adaptive control: in one approach the adaptive controller tries to 'learn' the unknown parameters, the learning is guaranteed under suitable assumptions and the adaptive controller converges to a non-adaptive limit controller which is then itself guaranteed to be stabilizing; see for example [10,11]. In a second so-called direct approach there is no attempt to learn the unknown parameters but instead the aim is simply to stabilize the system's dynamics. Again under suitable assumptions the adaptive controller converges to a limiting closed-loop system. However, there is no guarantee that the limit controller is stabilizing. In fact, the possibility that such destabilizing limit controllers can be the limit for a large open set of initial conditions, i.e. the phenomenon is generic, is of great concern. Such situations give rise to undesirable closed-loop behaviour of the adaptive control system. In this paper we aim to understand and characterize the occurence of such undesirable behaviour. From the viewpoint of dynamical systems theory, we are classifying normal forms for two-and three-dimensional systems. However, there is an added complication because the adaptive part of the controller causes the generic case to have one zero eigenvalue and there are also further degeneracies. This leads us to consider a number of different concepts of bifurcations.This paper focusses on the class of adaptive back-stepping control for systems in strict feedback form. Let us, therefore, begin by explaining these notions. As a simple example, we consider a chain of two integrators given by ẋ = f 1 (x, y), y = f 2 (x, y, u),where f 1 (0, 0) = 0 and f 2 (0, 0, 0) = 0. The goal is to find a control u as a function of x and y such that the origin becomes asymptotically stable. The idea of back-stepping is to stabilize this system using feedback via two steps: in the first step we consider the ...

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Section: Preliminaries and Motivationmentioning

confidence: 97%

Section: Preliminaries and Motivationmentioning

confidence: 99%

Section: Hopf Bifurcation Without Parametersmentioning

confidence: 99%

“…2 (c). This is called the elliptic case in [5,6,7,8]. Note that the cubic polynomial (ω 2 − 1 + ν 2 1 ) (ω + 1) has only one real root at ω = −1 < 0 if ν 1 > 1.…”

Section: Hopf Bifurcation Without Parametersmentioning

confidence: 99%

See 2 more Smart Citations

Bifurcations and Limit Dynamics in Adaptive Control Systems

Lamooki

Townley

Osinga

2005

Int. J. Bifurcation Chaos

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“…We give three examples next. For further details we refer to section 12 below, as well as to [AA86], [AF89], [Far84], [Lie97], [Lie00], [FLA00a], [FL00], [FLA00b].…”

Section: Introduction and Examplesmentioning

confidence: 99%

Global Analysis of Dynamical Systems

Broer¹,

Krauskopf²,

Vegter³

2001

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Generic Hopf Bifurcation from Lines of Equilibria without Parameters

Fiedler¹,

Liebscher²,

Alexander³

2000

Journal of Differential Equations

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Motivated by decoupling effects in coupled oscillators, by viscous shock profiles in systems of nonlinear hyperbolic balance laws, and by binary oscillation effects in discretizations of systems of hyperbolic balance laws, we consider vector fields with a one-dimensional line of equilibria, even in the absence of any parameters. Besides a trivial eigenvalue zero we assume that the linearization at these equilibria possesses a simple pair of nonzero eigenvalues which cross the imaginary axis transversely as we move along the equilibrium line.In normal form and under a suitable nondegeneracy condition, we distinguish two cases of this Hopf-type loss of stability, hyperbolic and elliptic. Going beyond normal forms we present a rigorous analysis of both cases. In particular, :-and |-limit sets of nearby trajectories consist entirely of equilibria on the line.

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Generic Hopf Bifurcation From Lines of Equilibria Without Parameters Iii: Binary Oscillations

Cited by 38 publications

References 6 publications

Bifurcations and Limit Dynamics in Adaptive Control Systems

Bifurcations and Limit Dynamics in Adaptive Control Systems

Global Analysis of Dynamical Systems

Generic Hopf Bifurcation from Lines of Equilibria without Parameters

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