Ben Zion Yossi, Horwitz Lawrence
Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel.
Phys Rev E Stat Nonlin Soft Matter Phys. 2010 Apr;81(4 Pt 2):046217. doi: 10.1103/PhysRevE.81.046217. Epub 2010 Apr 29.
An effective characterization of chaotic conservative Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor derived from the structure of the Hamiltonian has been extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce the Hamilton equations of the original potential model through an inverse map in the tangent space. The second covariant derivative of the geodesic deviation in this space generates a dynamical curvature, resulting in (energy-dependent) criteria for unstable behavior different from the usual Lyapunov criteria. We show here that this criterion can be constructively used to modify locally the potential of a chaotic Hamiltonian model in such a way that stable motion is achieved. Since our criterion for instability is local in coordinate space, these results provide a minimal method for achieving control of a chaotic system.
根据与从哈密顿量结构导出的黎曼度量张量相关的曲率，对混沌保守哈密顿系统进行的有效表征，已通过共形度量的定义扩展到了一大类标准形式的势模型。测地线方程通过切空间中的逆映射再现了原始势模型的哈密顿方程。该空间中测地线偏差的二阶协变导数产生一个动力学曲率，从而得到与通常的李雅普诺夫准则不同的（与能量相关的）不稳定行为判据。我们在此表明，该判据可被建设性地用于局部修改混沌哈密顿模型的势，从而实现稳定运动。由于我们的不稳定性判据在坐标空间中是局部的，这些结果提供了一种实现对混沌系统控制的最小方法。
