Complex Dynamics of Systems Under Delta-Modulated Feedback

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, we cast the design of&lt;formula formulatype="inline"&gt;&lt;tex&gt;$\Delta$&lt;/tex&gt; &lt;/formula&gt;-modulated control of a high-order system into the study of control Lyapunov functions. We classify the complex dynamics of the closed-loop system in three cases. In the first case, we show how &lt;formula formulatype="inline"&gt; &lt;tex&gt;$\Delta$&lt;/tex&gt;&lt;/formula&gt;-modulated feedback introduces a finite set of globally attracting periodic points. We find the numbers and periods of all possible such periodic orbits. In addition, we characterize the attracting region for each of such periodic points. In the second case, we show that there is a maximal "stabilizable" region, and inside this region, there is a local attractor. In the last case, we show that all the states stabilizable by the &lt;formula formulatype="inline"&gt;&lt;tex&gt;$\Delta$&lt;/tex&gt;&lt;/formula&gt;-modulated feedback constitute a Cantor set. This Cantor set is a repeller, and the closed-loop system is chaotic on the Cantor set. </para>