Groote Stefan, Beck Christian
Teoreetilise Füüsika Instituut, Tartu Ulikool, Tähe 4, 51010 Tartu, Estonia and Institut für Physik der Universität Mainz, Staudingerweg 7, 55099 Mainz, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 Oct;74(4 Pt 2):046216. doi: 10.1103/PhysRevE.74.046216. Epub 2006 Oct 30.
Coupled map lattices of nonhyperbolic local maps arise naturally in many physical situations described by discretized reaction diffusion equations or discretized scalar field theories. As a prototype for these types of lattice dynamical systems we study diffusively coupled Tchebyscheff maps of Nth order which exhibit strongest possible chaotic behavior for small coupling constants a. We prove that the expectations of arbitrary observables scale with sqrt of a in the low-coupling limit, contrasting the hyperbolic case which is known to scale with a . Moreover we prove that there are log-periodic oscillations of period ln N2 modulating the sqrt of a dependence of a given expectation value. We develop a general 1st order perturbation theory to analytically calculate the invariant one-point density, show that the density exhibits log-periodic oscillations in phase space, and obtain excellent agreement with numerical results.
非双曲局部映射的耦合映射格自然地出现在许多由离散反应扩散方程或离散标量场理论描述的物理情形中。作为这类格动力系统的一个原型，我们研究了N阶扩散耦合切比雪夫映射，对于小耦合常数a，它展现出最强可能的混沌行为。我们证明，在低耦合极限下，任意可观测量的期望值与√a成比例，这与已知的与a成比例的双曲情形形成对比。此外，我们证明存在周期为ln N²的对数周期振荡，调制给定期望值对√a的依赖关系。我们发展了一种通用的一阶微扰理论来解析计算不变单点密度，表明该密度在相空间中呈现对数周期振荡，并与数值结果取得了极好的一致性。
