[en] Uniform index is a conception that can describe the uniformity of a finite point set in a polyhedron, and is closely related to chaos. In order to study uniform index, the concept of contained uniform index is defined, which is similar to uniform index and has good mathematical properties. In this paper, we prove the convergence of the contained uniform index, and develop the base of proving the convergence of uniform index.

[en] Let $R_{r_0},R_{r_1}:{\mathbb{S}}^1⟶<!--\; ???\; -->{\mathbb{S}}^1$ be rotations on the unit circle ${\mathbb{S}}^1$ and define $f:{\mathrm{\Sigma <!--\; ??\; -->}}_2\times <!--\; ??\; -->{\mathbb{S}}^1⟶<!--\; ???\; -->{\mathrm{\Sigma <!--\; ??\; -->}}_2\times <!--\; ??\; -->{\mathbb{S}}^1$ as $$f(x,t)=(\mathrm{\sigma <!--\; ??\; -->}(x),R_{r_{x_1}}(t)),$$ for $x=x_1{x}_2\cdots <!--\; ???\; -->\in <!--\; ???\; -->{\mathrm{\Sigma <!--\; ??\; -->}}_2:=\{0,1{\}}^{\mathbb{N}}$, $t\in <!--\; ???\; -->{\mathbb{S}}^1$, where $\mathrm{\sigma <!--\; ??\; -->}:{\mathrm{\Sigma <!--\; ??\; -->}}_2⟶<!--\; ???\; -->{\mathrm{\Sigma <!--\; ??\; -->}}_2$ is the shift, and $r_0$ and $r_1$ are rotational angles. It is first proved that the system $({\mathrm{\Sigma <!--\; ??\; -->}}_2\times <!--\; ??\; -->{\mathbb{S}}^1,f)$ exhibits maximal distributional chaos for any $r_0,r_1\in <!--\; ???\; -->\mathbb{R}$ (no assumption of $r_0,r_1\in <!--\; ???\; -->\mathbb{R}\setminus <!--\; ???\; -->\mathbb{Q}$), generalizing Theorem 1 in Wu and Chen (Topol. Appl. 162:91–99, 2014). It is also obtained that $({\mathrm{\Sigma <!--\; ??\; -->}}_2\times <!--\; ??\; -->{\mathbb{S}}^1,f)$ is cofinitely sensitive and $({\stackrel{^<!--\; ^\; -->}{\mathcal{M}}}^1,{\stackrel{^<!--\; ^\; -->}{\mathcal{M}}}^1)$-sensitive and that $({\mathrm{\Sigma <!--\; ??\; -->}}_2\times <!--\; ??\; -->{\mathbb{S}}^1,f)$ is densely chaotic if and only if $r_1-<!--\; ???\; -->r_0\in <!--\; ???\; -->\mathbb{R}\setminus <!--\; ???\; -->\mathbb{Q}$.

[en] In this paper, we investigate the generalized Q-S synchronization between the generalized Lorenz canonical form and the Roessler system. Firstly, we transform an arbitrary generalized Lorenz system to the generalized Lorenz canonical form, and the relation between the parameter of the generalized Lorenz system and the parameter of the generalized Lorenz canonical form are shown. Secondly, we extend the scheme present by [Yan ZY. Chaos 2005;15:023902] to study the generalized Q-S synchronization between the generalized Lorenz canonical form and the Roessler system, the more general controller is obtained. By choosing different parameter in the generalized controller obtained here, without much extra effort, we can get the controller of synchronization between the Chen system and the Roessler system, the Lue system and the Roessler system, the classic Lorenz system and the Roessler system, the Hyperbolic Lorenz system and the Roessler system, respectively. Finally, numerical simulations are used to perform such synchronization and verify the effectiveness of the controller.

[en] We investigate the motion of the globally coupled maps (logistic map) driven by uniform disorder. It is shown that this disorder can produce multi-synchronization for the globally coupled chaotic maps studied by us. The disorder determines the synchronized dynamics, leading to the emergence of a wide range of new collective behaviour in which the individual units in isolation are incapable of producing in the absence of the disorder. Our results imply that the disorder can tame the collective motion of the coupled chaotic maps

[en] Based on the generalized Lorenz system, a conjugate Lorenz-type system is introduced, which contains three different chaotic attractors, i.e., the conjugate Lorenz attractor, the conjugate Chen attractor and the conjugate Lue attractor. These new attractors are conjugate, respectively, to the Lorenz attractor, the Chen attractor and the Lue attractor in an algebraic sense. The conjugate attractors may be helpful for finally revealing the geometric structure of the Lorenz attractor.

[en] We have observed anti-synchronization phenomena in different chaotic dynamical systems. Anti-synchronization can be characterized by the vanishing of the sum of relevant variables. Anti-synchronization problem for different chaotic dynamical systems with fully unknown parameters in response system is analyzed. This technique is applied to achieve anti-synchronization between Lorenz system, Lue system and Four-scroll system. Numerical simulations are provided to verify the effectiveness of the proposed methods.

[en] The grey prediction of Lorenz chaotic system will be discussed carefully in this paper. We are mainly using GM(1,1) model to predict data sequences, and the usual prediction precision has exceeded 90%. In the symbolic prediction of Lorenz chaotic dynamical system, the precision of grey prediction certainly will decrease as the length of symbolic sequence is increasing. But in this place we have found a generating rule that may realize chaotic synchronization at least in a short and medium term, and we can analysis and predict in this way.

[en] This paper proposes a novel particle swarm optimization (PSO) algorithm, chaotically encoded particle swarm optimization algorithm (CENPSOA), based on the notion of chaos numbers that have been recently proposed for a novel meaning to numbers. In this paper, various chaos arithmetic and evaluation measures that can be used in CENPSOA have been described. Furthermore, CENPSOA has been designed to be effectively utilized in data mining applications.

[en] We investigate a kind of chaos generating technique on a type of n-dimensional linear differential systems by adding feedback control items under a discontinuous state. This method is checked with some examples of numeric simulation. A constructive theorem is proposed for generalized synchronization related to the above chaotic system

[en] Chaotic systems have been broadly exploited through the last two decades to build encryption methods. Recently, two new image encryption schemes have been proposed, where the encryption process involves a permutation operation and an XOR-like transformation of the shuffled pixels, which are controlled by three chaotic systems. This paper discusses some defects of the schemes and how to break them with a chosen-plaintext attack.