A note on uniform convergence and transitivity

Distributional chaos in a sequence and topologically weak mixing for nonautonomous discrete dynamical systems

“…Chaos of nonautonomous discrete dynamical systems has been extensively studied (see [2,4,6,7,21,25]). For some related concepts and properties for autonomous discrete dynamical systems we refer the reader to [8][9][10][11][12][13][14][15][17][18][19]. A continuous self-map on a metric space is said to be chaotic in the sense of Devaney [1] if it satisfies the following three properties are satisfied: (1) topological transitivity; (2) the denseness of periodic points; (3) sensitive dependence on initial conditions.…”

Research on reactive power compensation for distribution network based on biogeography-based optimization algorithm

“…It is an important way to improve power quality and system voltage stability. As intelligent algorithms simulating the law of nature have been widely used and recognized in the engineering field, such as simulation of biological evolution genetic algorithm [1][2], PSO simulate birds foraging [3], the simulation algorithm bacterial potency flooding [4], the simulation of plant growth and plant growth phototropism algorithm [5] and other methods, those methods can also be used in reactive power optimization. Dan Simon proposed the BBO (Biogeography-based optimization) [6] in IEEE Transactions on Evolutionary Computation in 2008.…”

Furstenberg families and chaos on uniform limit maps

“…In [12], we obtained an equivalence condition for the uniform limit map f to be topologically transitive or syndetically transitive or topologically weak mixing or topological mixing and a necessary condition for the uniform limit map f to be sensitive or cofinitely sensitive or multi-sensitive. In [14], we gave the correct proofs of Theorems 3.4-3.7 in [12] and presented an equivalence condition for the uniform limit map f to be syndetically sensitive or cofinitely sensitive or multi-sensitive or ergodically sensitive and a sufficient condition for the uniform limit map f to be totally transitive or topologically weak mixing, where a sequence (f n ) of continuous selfmaps of a compact metric space X converging uniformly to a continuous selfmap f of the compact metric space X.…”

“…Finding conditions assuring the preservation of a chaotic property under limit operations is an interesting problem (see [2,6,7,12,14,17,19]). In [17] the author proved that if the f n are continuous functions acting on a metric space (X, d) converging uniformly to a function f and f n is topologically transitive for all n 1, then f is not necessarily topologically transitive, and he gave some sufficient conditions for the uniform limit function f to be topologically transitive.…”

“…In [14], we gave the correct proofs of Theorems 3.4-3.7 in [12] and presented an equivalence condition for the uniform limit map f to be syndetically sensitive or cofinitely sensitive or multi-sensitive or ergodically sensitive and a sufficient condition for the uniform limit map f to be totally transitive or topologically weak mixing, where a sequence (f n ) of continuous selfmaps of a compact metric space X converging uniformly to a continuous selfmap f of the compact metric space X.…”