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Chaos is associated with stochasticity, complex, irregular motion, etc. It has some peculiar properties such as ergodicity, highly initial value sensitivity, non-periodicity and long-term unpredictability. These pseudo random features lead chaotic systems to enormous applications such as random number generator, image encryption and secure communication. In general, the concept of chaos is never associated with similarity. However, we found the chaotic systems belonging to one chaos family (OCF) have similar dynamic behavior, which is a novel characteristic of chaos. In this work, three classical chaotic system family are studied, which are Lorenz family, Chua family and hyperbolic sine family. These systems contain different derived chaotic systems (Lorenz system, Chen system and Lu system), different order chaotic systems (Chua family and hyperbolic sine family), and different kinds of chaotic systems (chaos and hyper-chaos). Their PSPs demonstrate that there exist strong correlation in OCF. Moreover, we found that high order/dimensional chaotic systems will inherit all dynamic behavior of lower ones, and the similarity will decrease as the order/dimensional goes higher, which is analogous to genetic process in biology. All of these features are quantitatively evaluated by PPMCC and SSIM.
A new method of virtual unknown parameter is proposed to synchronize two different systems with unknown parameters and disturbance in finite time. Virtual unknown parameters are introduced in order to avoid the unknown parameters from appearing in th
Adaptive controllers are designed to synchronize two different chaotic systems with uncertainties, including unknown parameters, internal and external perturbations. Lyapunov stability theory is applied to prove that under some conditions the drive-r
Homoclinic and unstable periodic orbits in chaotic systems play central roles in various semiclassical sum rules. The interferences between terms are governed by the action functions and Maslov indices. In this article, we identify geometric relation
The problem of separation of an observed sum of chaotic signals into the individual components in the presence of noise on the path to the observer is considered. A noise threshold is found above which high-quality separation is impossible. Below the
Two deterministic models for Brownian motion are investigated by means of numerical simulations and kinetic theory arguments. The first model consists of a heavy hard disk immersed in a rarefied gas of smaller and lighter hard disks acting as a therm