Topological behavior, such as chaos, irreducibility, and mixing of a one-sided shift of finite type, is well elucidated. Meanwhile, the investigation of multidimensional shifts, for instance, textile systems, is difficult and only a few results have been obtained so far. This paper studies shifts defined on infinite trees, which are called tree-shifts. Infinite trees have a natural structure of one-sided symbolic dynamical systems equipped with multiple shift maps and constitute an intermediate class between one-sided shifts and multidimensional shifts. We have shown not only an irreducible tree-shift of finite type but also a mixing tree-shift that is chaotic in the sense of Devaney. Furthermore, the graph and labeled graph representations of tree-shifts are revealed so that the verification of irreducibility and mixing of a tree-shift is equivalent to determining the irreducibility and mixing of matrices, respectively. This extends the classical results of one-sided symbolic dynamics. A necessary and sufficient condition for the irreducibility and mixing of tree-shifts of finite type is demonstrated. Most important of all, the examination can be done in finite steps with an upper bound.

This study presents a current sharing method to actively balance the output currents of a parallel DC– DC converters (PDCC) system regarding the demanded power. First, the operating principle of the PDCC system with parallel-connected bidirectional converters is studied. To regulate the output voltage in DC bus and share the output currents of the individual converters, a dual loop control architecture comprising an outer voltage control loop and multiple inner current control loops is designed based on the automatic master-slave control scheme. Moreover, a feedback type two degree-of-freedom proportional-integral-derivative (FB2PID) controller is introduced to obtain the pulse-width modulation control signals for the converters. In order to improve the dynamic response and robustness of the active current-sharing control performances of the FB2PID controlled PDCC system, a bat algorithm (BA) optimized FB2PID control system is further proposed to concurrently and dynamically optimize the control parameters of the FB2PID controller in current control loop. Thus, the output current of each converter can be controlled to share the demand power equally in the presence of uncertainties. Finally, simulation and experimental results reveal that the proposed BA-optimized FB2PID control system outperforms conventional PID and FB2PID control systems with regard to the voltage regulation and current sharing performances under time-varying electric load condition.

This paper studies the entropy of tree-shifts of finite type with and without boundary conditions. We demonstrate that computing the entropy of a tree-shift of finite type is equivalent to solving a system of nonlinear recurrence equations. Furthermore, the entropy of the binary Markov tree-shifts over two symbols is either 0 or ln ⁡ 2 . Meanwhile, the realization of a class of reals including multinacci numbers is elaborated, which indicates that tree-shifts are capable of rich phenomena. By considering the influence of three different types of boundary conditions, say, the periodic, Dirichlet, and Neumann boundary conditions, the necessary and sufficient conditions for the coincidence of entropy with and without boundary conditions are addressed.

The uncertainty of multidimensional shift spaces draws attracted attention of researchers. For example, the emptiness problem is undecidable; there exist aperiodic shifts of finite type; there is a nonempty shift of finite type exhibiting nonextensible local patterns. This paper investigates symbolic dynamics on Cayley trees and gives affirmative answers to the above questions in tree-shifts. Beyond that, with introducing block gluing tree-shift, a sufficient condition of exhibiting positive topological entropy is revealed.