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	1. Measure-theoretic Entropy for Weak-solvable Cancellative Left-amenable Semigroup
	HUANG ShiYao,HUANG XiaoJun
	Mathematics 16 March 2023
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	Abstract:The study of dynamical systems under semigroup actions is an important branch of topological dynamical systems. At the same time, the study of dynamical system entropy is also of great significance, among which metric entropy can measure the complexity of the motion of a dynamical system on a probability space and constitutes an invariant of isomorphic systems. The main research content of this article is to generalize the Fekete lemma to weakly solvable cancellative conformal semigroups, and prove that the limit related to the F$\phi$lner sequence in this semigroup satisfies the "infimum rule". Finally, the metric entropy of the dynamical system under this semigroup action is given.
	TO cite this article:HUANG ShiYao,HUANG XiaoJun. Measure-theoretic Entropy for Weak-solvable Cancellative Left-amenable Semigroup[OL].[16 March 2023] http://en.paper.edu.cn/en_releasepaper/content/4759489


	2. Inverse shadowing property of hyperspace for free semigroup actions
	QIU Lin,HUANG XiaoJun
	Mathematics 17 January 2022
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	Abstract:The study of hyperspace properties is a major branch of infinite-dimensional topology. Shadowing property is an important concept of dynamical system, there are chain shadowing, h-shadowing and inverse shadowing. Which is related to the chaos and stability of the dynamical system. The purpose of this paper is to extend the generally inverse shadowing to the free semigroup action $(F^{+}_{m},\mathcal{F})\curvearrowright X$. And showed the relationship of inverse shadowing between the hyperspace and the base space.
	TO cite this article:QIU Lin,HUANG XiaoJun. Inverse shadowing property of hyperspace for free semigroup actions[OL].[17 January 2022] http://en.paper.edu.cn/en_releasepaper/content/4756088

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	3. Eco-evolutionary dynamics with feedback driven by imitation
	Cao Lixuan,Bin Wu
	Mathematics 12 March 2021
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	Abstract:Eco-evolutionary dynamics with environmental feedback occur when individual action modifies both the relative abundance of strategies and surrounding environment. The feedback mechanism can be captured by a function from actions of the population to the environment. As for the evolution of strategies, replicator equation common models it based on the natural selection. However, strategies spread not only by natural selection but also by imitation in evolutionary game. Specifically, Fermi process, as a kind of imitation rule, is generally adopted by individuals. This work propose a class of ecoevolutionary dynamics where strategies coevolve with the environment by general imitation functions. Then the condition under which a limit cycle may occur in the system is obtained, which can never be present in coupled replicator dynamics with environment. Further, the dynamical outcomes can be devided into four cases and compare with the classical system in detail.
	TO cite this article:Cao Lixuan,Bin Wu. Eco-evolutionary dynamics with feedback driven by imitation[OL].[12 March 2021] http://en.paper.edu.cn/en_releasepaper/content/4754090


	4. Chaos Criterion Theorems on Specific 2n Order and 2n + 1 Order Polynomial Discrete Maps with Application
	Chu Jixun,Min Lequan
	Mathematics 29 December 2019
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	Abstract:Li and Yorke's seminal paper published in 1975 has provided a criterion for the existence of chaos in one-dimensional difference equations. Generally speaking, it is difficult to use this criterion to verify whether a high order polynomial discrete mapping is chaotic. Because one needs to solve several high order polynomial equations on the polynomial parameters. Firstly, this paper introduces two theorems of the existence of chaos on two kinds of specific 2n order and 2n+1 order polynomial discrete mappings. These two theorems provide chaotic parameter intervals of the specific polynomials, which satisfy the Li-Yorke criterion. Secondly, four examples are presented to verify the effectiveness of the theorems. Thirdly, by using a generalized synchronization theorem and the chaos mappings given in the four examples, a new 8-dimensional chaotic generalized synchronization system (8DCGSS) is constructed. Then a chaotic PRNG (CPRNG) is designed. The keyspace of the CPRNG is larger than 2^1117. Finally, by using the FIPS 140-2 randomness test and a generalized FIPS 140-2 randomness test, the randomness of the keystreams generated via the CPRNG is measured. The results show that the CPRNG is able to generate sound pseudorandom keystreams.
	TO cite this article:Chu Jixun,Min Lequan. Chaos Criterion Theorems on Specific 2n Order and 2n + 1 Order Polynomial Discrete Maps with Application[OL].[29 December 2019] http://en.paper.edu.cn/en_releasepaper/content/4750298


	5. Long time stability of solutions for the fourth order nonlinear Schrödinger equation
	GAO Meina,LIU Jianjun
	Mathematics 28 May 2018
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	Abstract:In this paper, a long time stability result is proved is proved for the cubic fourth order nonlinear Schrödinger equation with periodic boundary conditions$$\mathbf{i}u_t=\partial_x^4u\pm\|u\|^2u,\hspace{12pt}x\in\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}.$$
	TO cite this article:GAO Meina,LIU Jianjun. Long time stability of solutions for the fourth order nonlinear Schrödinger equation[OL].[28 May 2018] http://en.paper.edu.cn/en_releasepaper/content/4745259


	6. Quasi-periodic solutions for the fourth order nonlinear Schr$\ddot{\mbox{o}}$dinger equation
	GAO Meina,LIU Jianjun
	Mathematics 15 May 2018
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	Abstract:This paper is concerned with the cubic fourth order nonlinearSchr$\ddot{\mbox{o}}$dinger equation with periodic boundary conditions$$\mathbf{i}u_t=\partial_x^4u\pm\|u\|^2u,\hspace{12pt}x\in\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}.$$We show the above equation possesses Cantor families of quasi-periodic solutions of small amplitude.
	TO cite this article:GAO Meina,LIU Jianjun. Quasi-periodic solutions for the fourth order nonlinear Schr$\ddot{\mbox{o}}$dinger equation[OL].[15 May 2018] http://en.paper.edu.cn/en_releasepaper/content/4744994


	7. Fuzzy Adaptive Iterative Learning Control for Consensus of Multi-Agent Systems with Imprecise Communication Topology Structure
	CHEN Jiaxi,LI Junmin,LI Jinsha
	Mathematics 21 April 2017
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	Abstract:This paper investigates the adaptive consensus problem of first-order linearly para -meterized multi-agent systems (MASs) with imprecise communication topology structure. T -S fuzzy models are presented to describe leader-followers MASs with imprecise communicati -on topology structure, and a fuzzy distributed adaptive iterative learning control protocol is proposed. With the dynamic of leader unknown to any of the agent, the proposed protocol gu -arantees that the follower agents can track the leader uniformly on $left[ {0,T} ight]$ for consensus prob -lem. A sufficient condition of the consensus for the closed-loop multi-agent system (MAS) is given based on Lyapunov stability theory. Finally, a simulation example is given to illustrate the effectiveness of the proposed method in this sudy.
	TO cite this article:CHEN Jiaxi,LI Junmin,LI Jinsha. Fuzzy Adaptive Iterative Learning Control for Consensus of Multi-Agent Systems with Imprecise Communication Topology Structure[OL].[21 April 2017] http://en.paper.edu.cn/en_releasepaper/content/4727883


	8. Tail Pressure for Sub-additive Potentials
	PENG Jinglan,ZHOU Yunhua
	Mathematics 20 December 2016
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	Abstract:This paper define the topological tail pressure for sub-additive potentials and, under some assumptions, get a variational principle which exhibits the relationship between topological tail pressure and measure-theoretic tail entropy. Then, this paper definea new measure-theoretic tail pressure for invariant measures of sub-additive potentials on a compact metric space and an interesting property of it is obtained.
	TO cite this article:PENG Jinglan,ZHOU Yunhua. Tail Pressure for Sub-additive Potentials [OL].[20 December 2016] http://en.paper.edu.cn/en_releasepaper/content/4714492


	9. Lag group consensus for the second-order nonlinear multi-agentsystems via adaptive control approach
	GUO Wanli
	Mathematics 01 November 2016
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	Abstract:This paper studies the lag group consensus problem ofthe second-order nonlinear multi-agent systems. A protocol mixedwith the adaptive control approach is presented for agents to behindthe leader in each group at different times, so as to avoidcongestion. Without assuming that the topology is connected orweakly connected even there is rooted directed spanning tree,sufficient conditions for the lag group consensus are proposedanalytically. Moreover, it is shown that with the given protocol, nomatter what topology of the multi-agent system is, the lag groupconsensus can be obtained always. Finally, a simulation example isgiven to illustrate our theoretical analysis.
	TO cite this article:GUO Wanli. Lag group consensus for the second-order nonlinear multi-agentsystems via adaptive control approach[OL].[ 1 November 2016] http://en.paper.edu.cn/en_releasepaper/content/4708194


	10. Pinning adaptive-impulsive consensus of the multi-agent systems with uncertain perturbation
	GUO Wanli
	Mathematics 01 November 2016
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	Abstract:This paper investigates the problem of consensus forthe multi-agent systems with uncertain perturbations. A noveladaptive-impulsive protocol is designed for reaching the consensus.Particularly, this protocol is only imposed on a small fraction ofagents to pin the whole system to a prescribed value. Two consensuscriteria are established in cases whether the underlying graph ofthe multi-agent system is strongly connected or not. Two numericalexamples are given to demonstrate the effectiveness of the proposedcriteria.
	TO cite this article:GUO Wanli. Pinning adaptive-impulsive consensus of the multi-agent systems with uncertain perturbation[OL].[ 1 November 2016] http://en.paper.edu.cn/en_releasepaper/content/4708200

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