In studying general perturbations of a dissipative twist map depending on two parameters, a frequency \(\nu\) and a dissipation \(\eta\), the existence of a Cantor set \(\mathcal{C}\) of curves in the \((\nu,\eta)\) plane such that the corresponding equation possesses a Diophantine quasi-periodic invariant circle can be deduced, up to small values of the dissipation, as a direct consequence of a normal form theorem in the spirit of Rüssmann and the “elimination of parameters” technique. These circles are normally hyperbolic as soon as \(\eta\not=0\), which implies that the equation still possesses a circle of this kind for values of the parameters belonging to a neighborhood \(\mathcal{V}\) of this set of curves. Obviously, the dynamics on such invariant circles is no more controlled and may be generic, but the normal dynamics is controlled in the sense of their basins of attraction.

As expected, by the classical graph-transform method we are able to determine a first rough region where the normal hyperbolicity prevails and a circle persists, for a strong enough dissipation \(\eta\sim O(\sqrt{\varepsilon}),\) \(\varepsilon\) being the size of the perturbation. Then, through normal-form techniques, we shall enlarge such regions and determine such a (conic) neighborhood \(\mathcal{V}\), up to values of dissipation of the same order as the perturbation, by using the fact that the proximity of the set \(\mathcal{C}\) allows, thanks to Rüssmann’s translated curve theorem, an introduction of local coordinates of the type (dissipation, translation) similar to the ones introduced by Chenciner in [7].

在研究取决于两个参数（频率\(\nu\)和耗散\(\eta\)）的耗散扭曲图的一般扰动时，存在曲线的康托集\(\mathcal{C}\)在 \ ((\nu,\eta)\)平面上，使得相应的方程拥有丢番图准周期不变圆，可以在耗散值很小的情况下推导出来，作为精神中的范式定理的直接结果Rüssmann 和“参数消除”技术。这些圆通常在\(\eta\not=0\)时是双曲形的，这意味着对于属于邻域\(\mathcal{V}\)的参数值，方程仍然具有这种圆这组曲线。显然，这种不变圆上的动力学不再受控，并且可能是通用的，但正常动力学在其吸引盆的意义上是受控的。

正如预期的那样，通过经典的图变换方法，我们能够确定第一个粗糙区域，其中法向双曲性占主导地位并且圆持续存在，以获得足够强的耗散 \(\eta\sim O(\ sqrt{\varepsilon})， \) \(\varepsilon\)是扰动的大小。然后，通过范式技术，我们将扩大这些区域并确定这样一个（圆锥）邻域\(\mathcal{V}\)，直到与扰动具有相同数量级的耗散值，通过使用以下事实： 由于 Rüssmann 平移曲线定理，集合\ (\mathcal{C}\)的邻近性允许引入类似于 Chenciner 在 [7] 中引入的类型（耗散、平移）的局部坐标。