Some classical chaotic systems such as the Lorenz system and Chua system have finite numbers of chaotic attractors. This letter develops a simple, effective method for constructing lower-dimensional autonomous systems with infinitely many chaotic attractors. As an application, a Lorenz-type system and a Rössler-type system with infinitely many chaotic attractors are constructed with bifurcation analysis, and with an extension to the fractional-order setting.

Similarity measures are very effective and meaningful tool used for evaluating the closeness between any two attributes which are very important and valuable to manage awkward and complex information in real-life problems. Therefore, for better handing of fuzzy information in real life, Ullah et al. (Complex Intell Syst 6(1): 15–27, 2020) recently introduced the concept of complex Pythagorean fuzzy set (CPyFS) and also described valuable and dominant measures, called various types of distance measures (DisMs) based on CPyFSs. The theory of CPyFS is the essential modification of Pythagorean fuzzy set to handle awkward and complicated in real-life problems. Keeping the advantages of the CPyFS, in this paper, we first construct an example to illustrate that a DisM proposed by Ullah et al. does not satisfy the axiomatic definition of complex Pythagorean fuzzy DisM. Then, combining the 3D Hamming distance with the Hausdorff distance, we propose a new DisM for CPyFSs, which is proved to satisfy the axiomatic definition of complex Pythagorean fuzzy DisM. Moreover, similarly to some DisMs for intuitionistic fuzzy sets, we present some other new complex Pythagorean fuzzy DisMs. Finally, we apply our proposed DisMs to a building material recognition problem and a medical diagnosis problem to illustrate the effectiveness of our DisMs. Finally, we aim to compare the proposed work with some existing measures is to enhance the worth of the derived measures.

In this paper, some properties of a strictly $A$ -coupled-expanding map in compact subsets of a metric space are studied, where $A$ is a transition matrix. It is shown that this map has a compact invariant set on which it is topologically semi-conjugate to the subshift for $A$ . If the subshift for $A$ has positive topological entropy, then the map is chaotic in the sense of Li-Yorke. Moreover, in the one-dimensional case, the map is at most two-to-one conjugate to the subshift for $A$ and chaotic in the sense of Devaney.

In this paper, we first give an example to show that Theorem 1 in Hung and Yang (Inf Sci 178(6):1641–1650, 2008) does not hold, implying that the J-divergence introduced by Hung and Yang does not satisfy the axiomatic definition of intuitionistic fuzzy divergence measure. Inspired by this, a new Jensen–Shannon divergence measure for intuitionistic fuzzy sets (IFSs) is introduced and some basic properties for this new divergence measure are obtained. In particular, this divergence measure, and its induced similarity measure, and induced entropy measure satisfy the axiomatic definitions of divergence, similarity, and entropy for IFSs. Based on our proposed divergence measure, entropy measure, and entropy-weight method, a new TOPSIS method is introduced to deal with multi-attribute decision making (MADM) problems under the intuitionistic fuzzy framework. Finally, a practical example on the credit evaluation of potential strategic partners and a comparative analysis with other TOPSIS methods is developed to illustrate the efficiency of the proposed TOPSIS method.