Surnrnary. It is u'cll knowrr that lincar SISO systcms t,hat can bc rcrrdcrcri passlve through constant output fccdback can be iiclaptivcly st,altiliscd throul;li a single gain adaptation lau,.. \\'c rcvisit, t,hc dynatnical itchaviotrr of such systems and cxhibit througli a bifurcation analysis a rich varicty of potcnt,ial asymptot,lc dynarnics, for which wc providc a control theorctir: intcrprctation. This in turn lcads us to question the actual adaptivc cont,rol qucstiorr arid solution

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... ch. Asyrnptotic Dynamics irr Adaptivc Gain Control This observation of lack of robustness lead to the so called sigma rnodification. This idea was first suggested by Ioannou and Kokotovic [8] in the contcxt of adaptive model refercnce controi. It was adopted for systems of the form (1) by Bar-Kana and co-workcrs, sce [11] for a detailed overview of this work. Another possible rnoclification to overcomc this lack of robustness problem whilst preserving the simplicity of the controllcr is the so called ,\-l,racking approach (first suggested in [f] ana introduccd by Ilchmann and Ryan [6], see also Ilchmann [5]). In lhis line of work a dead-zonc in the gain adaptation is invoked. The dead-zone reduces the performance of regulation in the disturbance free case in that thc output converges to a prcspccified, )neighbourhood of the rcfcrcnce signal. This is a small pricc to pay for the addcd robrrstncss with respect to small disturbances. Thc .\-tracking modification will not be considered in the present paper. Compared to the above simple intuitive picture, the sigma modification alters the phase portrait in a non trivial way. This is the topic studied in this papcr. Thc prirnc purposc of thc sigma modification is to obtain robustness with respect, to measurcmcnt errors. This is indeed achieved, but at a price, as the a^syrnptot,ic dynamics arc no Iongcr as desired. We invcst,igatc this phcnomcnon in some detail, suggesting some altcrnatives.