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首页> 外文期刊>Memoirs of the American Mathematical Society >The Creation of Strange Non-Chaotic Attractors in Non-Smooth Saddle-Node Bifurcations
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The Creation of Strange Non-Chaotic Attractors in Non-Smooth Saddle-Node Bifurcations

机译：非光滑鞍结分叉中奇异非混沌吸引子的产生

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We propose a general mechanism by which strange non-chaotic attractors (SNA) are created during the collision of invariant curves in quasiperiodically forced systems. This mechanism, and its implementation in different models, is first discussed on an heuristic level and by means of simulations. In the considered examples, a stable and an unstable invariant circle undergo a saddle-node bifurcation, but instead of a neutral invariant curve there exists a strange non-chaotic attractor-repeller pair at the bifurcation point. This process is accompanied by a very characteristic behaviour of the invariant curves prior to their collision, which we call 'exponential evolution of peaks'. This observation is then used to give a rigorous description of non-smooth saddle-node bifurcations and to prove the existence of SNA in certain parameter families of quasiperiodically forced interval maps. The non-smoothness of the bifurcations and the occurrence of SNA is established via the existence of 'sinksource- orbits', meaning orbits with positive Lyapunov exponent both forwards and backwards in time. The important fact is that the presented approach allows for a certain amount of flexibility, which makes it possible to treat different models at the same time - even if the results presented here are still subject to a number of technical constraints. This is unlike previous proofs for the existence of SNA, which are all restricted to very specific classes and depend on very particular properties of the considered systems. In order to demonstrate this flexibility, we also discuss the application of the results to the Harper map, an example which is well-known from the study of discrete Schrodinger operators with quasiperiodic potentials. Further, we prove the existence of strange non-chaotic attractors with a certain inherent symmetry, as they occur in non-smooth pitchfork bifurcations.
机译：我们提出了一种一般机制，通过该机制，在准周期强迫系统中不变曲线的碰撞过程中会产生奇怪的非混沌吸引子（SNA）。该机制及其在不同模型中的实现首先在启发式级别上通过模拟进行讨论。在所考虑的示例中，稳定和不稳定的不变圆经历了鞍形节点分叉，但是在分叉点处存在一个奇怪的非混沌吸引子-排斥子对，而不是中性不变曲线。这个过程伴随着不变曲线在碰撞之前的非常特征性的行为，我们称其为“峰的指数演化”。然后，该观察结果用于对非光滑的鞍节点分叉进行严格描述，并证明准周期性间隔图的某些参数族中存在SNA。分叉的非光滑性和SNA的发生是通过“汇源轨道”的存在而建立的，“汇源轨道”的意思是时间前后都具有正Lyapunov指数的轨道。重要的事实是，所提出的方法具有一定程度的灵活性，这使得可以同时处理不同的模型，即使此处给出的结果仍然受到许多技术约束。这与SNA存在的先前证明不同，后者仅限于非常特定的类并且取决于所考虑系统的非常特殊的属性。为了证明这种灵活性，我们还讨论了将结果应用于Harper图的例子，该例子是研究具有准周期电位的离散Schrodinger算子的一个著名例子。此外，我们证明了奇异的非混沌吸引子的存在，它们具有一定的固有对称性，因为它们发生在非光滑的干草叉分叉中。

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