Applied Mathematics and Statistics

Editorial for the First Issue of Applied Mathematics and Statistics (AMS)

Jinde Cao — 2024-07-30

Editorial

Editorial for the First Issue of Applied Mathematics and Statistics (AMS)

Jinde Cao

School of Mathematics, Southeast University, Nanjing 211189, China; jdcao@seu.edu.cn

Hopf Bifurcation of the Oregonator Oscillator with Distributed Delay

Yu Wang — 2024-08-19

Article

Hopf Bifurcation of the Oregonator Oscillator with Distributed Delay

Yu Wang ^1,* and Luca Guerrini ²

¹School of Mathematics, Southeast University, Nanjing 210096, China

²Department of Management, Polytechnic University of Marche, 60121 Ancona, Italy

* Correspondence: 230218180@seu.edu.cn

Received: 2 June 2024; Revised: 10 July 2024; Accepted: 30 July 2024; Published: 19 August 2024

Abstract: This paper investigates the bifurcation problem of the Oregonator oscillator with distributed time delay, and two cases are considered, namely weak and strong kernels. First, theoretical approaches are provided to analyze the stability properties of the equilibrium in these systems using the chain trick method. Near the positive equilibrium point, the Routh-Hurwitz criteria are employed to establish precise conditions for stability and Hopf bifurcation and determine the bifurcation direction. Additionally, this paper explores the implications of inffnite memory within a distributed delay to gain insights into the dynamic behavior. Moreover, extensive numerical simulations are conducted to support our theoretical analysis. The main simulations illustrate the bifurcation waveform and phase diagrams and reveal complex dynamic behavior, including stable and unstable oscillations.

Independent Domination in Claw-Free Cubic Graphs

Linyu Li — 2024-08-22

Article

Independent Domination in Claw-Free Cubic Graphs

Linyu Li ¹ and Jun Yue ^2,*

¹School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China

²School of Mathematics Science, Tiangong University, Tianjin 300387, China

* Correspondence: yuejun06@126.com

Received: 7 June 2024; Revised: 5 August 2024; Accepted: 13 August 2024; Published: 22 August 2024

Abstract: A vertex set S of a graph G is called an independent dominating set if S is an independent set and each vertex in V(G)\S is adjacent to a vertex in S. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set in G. This paper first proves that if G is a connected \(K_{1,3}\)-free cubic graph, then \(i(G)\leq \frac{1}{3}|V(G)|\). Meanwhile, \(i(G)=\frac{1}{3}|V(G)|\) if and only if \(G\in \mathcal{H}\), where \(\mathcal{H}\) is an infinite cubic family with each graph being a \(C_6^+\)-necklace. Then, it is shown that if G is a \(\{K_{1,3}, K_4^-, C_6^+\}\)-free cubic graph with no \(C_3\Box K_2\)-component, then \(i(G)\leq \frac{5}{18}|V(G)|\). This result is tight.

Adaptive Resilience via Probabilistic Automaton: Safeguarding Multi-Agent Systems from Leader Missing Attacks

Kuanxiang Wang — 2024-10-25

Article

Adaptive Resilience via Probabilistic Automaton: Safeguarding Multi-Agent Systems from Leader Missing Attacks

Kuanxiang Wang ¹ and Xin Gong ^2,*

¹The School of Information Science and Engineering, East China University of Science and Technology, Shanghai 200237, China

²The School of Cyber Science and Engineering, Southeast University, Nanjing 210096, China

* Correspondence: xingong@seu.edu.cn; Tel.: +86-185-9806-0508

Received: 26 August 2024; Revised: 3 October 2024; Accepted: 18 October 2024; Published: 25 October 2024

Abstract: The resilience of leader-following structures has been a hotspot in both academic and industrial research. Existing studies mainly focus on maintaining follower coherence, usually assuming that the leader can always function properly. However, these studies neglect the risk of system paralysis if the leader is compromised. To resolve this problem, this paper leverages probabilistic automata grammar reasoning to investigate how followers can gradually infer their operational rules within the system over time. First, a grammatical inference module is implemented on the followers to enable them to deduce their rules once they receive commands from the leader. Then, this paper proposes three probabilistic automata reasoning methods for this inference: the Algorithm for Learning Regular Grammars with Inference Assistance (ALERGIA), Distinguished String Automata Inference (DSAI), and Minimum Divergent Inference (MDI). By using these methods, a follower can reason about deterministic finite automata from multiple commands issued by the leader, which are then utilized to construct deterministic probabilistic finite automata for representing the follower's rules. Finally, several examples are provided to validate the correctness of these reasoning methods and compare their efficiency in learning probabilistic automata. The results indicate that all three methods achieve an accuracy of 98.535% in learning the correct automata transformation function, and ALERGIA runs slightly faster. These findings suggest that even if the leader is compromised, the agent can still perform tasks autonomously using the inferred rules, thereby avoiding system paralysis.

Bandit-Based Multi-Agent Source Seeking with Safety Guarantees

Zhibin Ji — 2024-12-27

Article

Bandit-Based Multi-Agent Source Seeking with Safety Guarantees

Zhibin Ji ^† , Dingqi Zhu ^† and Bin Du *

College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211016, China

* Correspondence: iniesdu@nuaa.edu.cn

† These authors contributed equally to this work.

Received: 2 November 2024; Revised: 24 December 2024;Accepted: 26 December 2024; Published: 27 December 2024

Abstract: In this paper, we focus on a multi-agent source seeking problem where the safety of agents is characterized by a set of linear constraints. In particular, the safety constraints are also dependent on the unknown environment states, which makes the source seeking problem challenging to solve. To overcome such a challenge, we introduce a new notion of measurable path and then specify the reachability condition for all agents. A time-sequence of exploration is further introduced to help the agents to escape the stuck positions. To provide a performance guarantee for our source seeking algorithm, we perform the regret analysis and show a sub-linear cumulative regret. Finally, we evaluate the effectiveness of our SafeSearch algorithm through a set of simulations.