Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps
Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps
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保守和耗散二维图中任意高阶的同宿切线
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
L. Shilnikov
中科院分区：
文献类型：
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作者：
S. Gonchenko;D. Turaev;L. Shilnikov
我们证明了具有任意高阶同宿切线的映射以及由此而产生的具有任意退化周期轨道的映射在实解析保面积二维映射空间和一般实解析二维映射空间中的纽豪斯区域中是稠密的（这个结果以前只在光滑非保守映射空间中才知道）。在此基础上，我们表明，一个通用的面积保持地图从纽豪斯地区是“普遍的”在这个意义上说，它的迭代近似的动态任何其他面积保持地图具有任意好的精度。事实上，我们表明，每一个动力学现象，一般发生在任何开放的一组辛同胚的二维光盘，或在任何开放的一组有限参数家庭这样的同胚，可以遇到在扰动的任何区域保持二维映射的同宿切线。
We show that maps with homoclinic tangencies of arbitrarily high orders and, as a consequence, with arbitrarily degenerate periodic orbits are dense in the Newhouse regions in spaces of real-analytic area-preserving two-dimensional maps and general real-analytic two-dimensional maps (the result was earlier known only for the space of smooth non-conservative maps). Based on this, we show that a generic area-preserving map from the Newhouse region is ‘universal’ in the sense that its iterations approximate the dynamics of any other area-preserving map with arbitrarily good accuracy. In fact, we show that every dynamical phenomenon which occurs generically in any open set of symplectic diffeomorphisms of a two-dimensional disc, or in any open set of finite-parameter families of such diffeomorphisms, can be encountered at a perturbation of any area-preserving two-dimensional map with a homoclinic tangency.