Zaslavsky G M, Stanislavsky A A, Edelman M
Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA.
Chaos. 2006 Mar;16(1):013102. doi: 10.1063/1.2126806.
We consider a nonlinear oscillator of the Duffing type with fractional derivative of the order 1<alpha<2. In this system replacement of the regular derivative by the fractional one leads to decaying solutions. The main feature of the system is that decay is asymptotically the powerwise situation that appears in different applications. Perturbed by a periodic force, the system exhibits chaotic motion called fractional chaotic attractor (FCA). The FCA is compared to the "regular" chaotic attractor that exists in the periodically forced Duffing oscillator. The properties of the FCA are discussed and the "pseudochaotic" case is demonstrated numerically for the case of the "dying attractor." We call "pseudochaos" the case when the randomness exists with zero Lyapunov exponent, i.e., the dispersion of initially close trajectories is subexponential.
我们考虑一个具有1<α<2阶分数导数的杜芬型非线性振荡器。在这个系统中，用分数导数代替常规导数会导致解的衰减。该系统的主要特征是，衰减在渐近上是不同应用中出现的幂律情形。在周期力的扰动下，该系统表现出称为分数混沌吸引子（FCA）的混沌运动。将FCA与周期性强迫杜芬振荡器中存在的“常规”混沌吸引子进行比较。讨论了FCA的性质，并针对“衰减吸引子”的情况进行了数值演示“伪混沌”情形。当随机性以零李雅普诺夫指数存在时，即初始接近轨迹的离散是次指数的，我们称这种情况为“伪混沌”。
