具有局部双曲吸引子的扩散介质中双曲性的破坏
Violation of hyperbolicity in a diffusive medium with local hyperbolic attractor
摘要：
从两个非自治振幅方程的系统出发，证明双曲型混沌动力学，我们构建了一个一维介质作为通过扩散引入空间耦合的这种局部元素的系综。当介质的长度较小时，所有的空间细胞同步振荡，再现局部双曲动力学。这种状态的特征在于一个单一的积极的李雅普诺夫指数。当系统的长度变大，第二个李雅普诺夫指数超过零时，双曲性仍然存在，并且振荡在空间上变得不均匀。然而，在第三个李雅普诺夫指数变为正的点处，由于沿着吸引子上的轨迹收缩和膨胀的切子空间的一维交叉的出现，发生了一些分叉，导致违反双曲性。进一步增长的长度的结果在二维交叉的扩大和收缩的子空间，我们归类为一个更强的类型的违反。除了双曲性损失的点，系统表现出广泛的时空混沌典型的扩展混沌系统：当系统的长度增加的Kaplan-Yorke维数，正的李雅普诺夫指数的数量，和Kolmogorov-Sinai熵的上限估计线性增长，而李雅普诺夫谱趋于一个极限曲线。
英文摘要：
Departing from a system of two non-autonomous amplitude equations, demonstrating hyperbolic chaotic dynamics, we construct a 1D medium as ensemble of such local elements introducing spatial coupling via diffusion. When the length of the medium is small, all spatial cells oscillate synchronously, reproducing the local hyperbolic dynamics. This regime is characterized by a single positive Lyapunov exponent. The hyperbolicity survives when the system gets larger in length so that the second Lyapunov exponent passes zero, and the oscillations become inhomogeneous in space. However, at a point where the third Lyapunov exponent becomes positive, some bifurcation occurs that results in violation of the hyperbolicity due to the emergence of one-dimensional intersections of contracting and expanding tangent subspaces along trajectories on the attractor. Further growth of the length results in two-dimensional intersections of expanding and contracting subspaces that we classify as a stronger type of the violation. Beyond of the point of the hyperbolicity loss, the system demonstrates an extensive spatiotemporal chaos typical for extended chaotic systems: when the length of the system increases the Kaplan-Yorke dimension, the number of positive Lyapunov exponents, and the upper estimate for Kolmogorov-Sinai entropy grow linearly, while the Lyapunov spectrum tends to a limiting curve.
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