Takens-Bogdanov bifurcations without parameters and oscillatory shock profiles

Fiedler, Bernold; Liebscher, Stefan

doi:10.1887/0750308036/b1058c8

Cited by 8 publications

(22 citation statements)

References 8 publications

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“…Finally, all three eigenvalues could be zero, which corresponds to a Bogdanov-Takens bifurcation; this bifurcation is of codimension two and we only encounter it for a particular value of the unknown parameter θ * . The Bogdanov-Takens bifurcation would occur generically in the case 2 + 2, which is discussed in detail in [6]. In the context of adaptive back-stepping control for the case 2 + 1 we briefly discuss the Bogdanov-Takens bifurcation in Sec.…”

Section: Rokni-lamooki Townley and Osinga Bifurcations And Limit Dynammentioning

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%

“…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].…”

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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: Rokni-lamooki Townley and Osinga Bifurcations And Limit Dynammentioning

confidence: 99%

Section: Hopf Bifurcation Without Parametersmentioning

confidence: 99%

mentioning

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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“…In this section and Section 5, the plane of equilibria of the system (14), under the condition of Theorem 3.2, will be studied from bifurcation point of view. The relevant notion of bifurcation is bifurcation without parameters; see for example [4]. Suppose h 1 = h ′ (0) and µ 1 = µ ′ (0).…”

Section: Eigenvalue Analysismentioning

confidence: 99%

“…Therefore, the origin of the reduced system will not have a non empty interior basin of attraction; more details about BogdanovTakens bifurcation is discussed, for example, in [4] and [5]. It is easy to check that if functions ψ 1 and ψ 2 contains no constants, then the center manifold will take the form (47), then the reduction of the system (36) to the center manifold takes the following form.…”

Section: Limit System Analysismentioning

confidence: 99%

Adaptive Partial Stabilization, Limit Dynamics and Bifurcation Analysis

Lamooki¹

2012

Journal of the Korean Mathematical Society

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Two-stroke relaxation oscillators

Jelbart

Wechselberger

2020

Nonlinearity

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Two-stroke relaxation oscillations consist of two distinct phases per cycle -one slow and one fast -which distinguishes them from the well-known van der Pol-type 'four-stroke' relaxation oscillations. These type of oscillations can be found in singular perturbation problems in non-standard form where the slow-fast timescale splitting is not necessarily reflected in a slow-fast variable splitting. We provide a framework for the application of geometric singular perturbation theory to problems of this kind, and apply it to prove the existence, uniqueness and stability of the observed relaxation oscillations. The analysis of such two-stroke oscillations is motivated by applications which arise in the dynamics of nonlinear transistors, and models for mechanical oscillators with friction.

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Takens-Bogdanov bifurcations without parameters and oscillatory shock profiles

Cited by 8 publications

References 8 publications

Bifurcations and Limit Dynamics in Adaptive Control Systems

Bifurcations and Limit Dynamics in Adaptive Control Systems

Adaptive Partial Stabilization, Limit Dynamics and Bifurcation Analysis

Two-stroke relaxation oscillators

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