Pub Date : 2026-02-13DOI: 10.1016/j.chaos.2026.118061
Alexander Kuznetsov, Yuliya Sedova
A map derived by discretization of an ensemble consisting of five non-identical in control parameter van der Pol oscillators by replacing time derivatives with finite differences is studied. In this map, with a sufficiently large frequency detuning of the oscillators, a cascade of bifurcations leading to the birth of invariant tori with increasing dimensions is observed, that corresponds to the first steps of the Landau-Hopf scenario. The influence of the discretization parameter value on the observed structure is discussed. Using this model, the effect of noise on quasi-periodic Hopf bifurcations and invariant tori of varying dimensions is studied. The presence of several Lyapunov exponents is taken into account and corresponding one-parameter and two-parameter illustrations are given. The cases of noise effect on “the weakest” and “the strongest” oscillators in an ensemble are examined and compared. The possibility of alternative chaotization of tori and their stabilization by noise is demonstrated. Illustrations of such stabilization are given.
{"title":"On the effect of noise on invariant tori and quasi-periodic bifurcations of different dimensions","authors":"Alexander Kuznetsov, Yuliya Sedova","doi":"10.1016/j.chaos.2026.118061","DOIUrl":"https://doi.org/10.1016/j.chaos.2026.118061","url":null,"abstract":"A map derived by discretization of an ensemble consisting of five non-identical in control parameter van der Pol oscillators by replacing time derivatives with finite differences is studied. In this map, with a sufficiently large frequency detuning of the oscillators, a cascade of bifurcations leading to the birth of invariant tori with increasing dimensions is observed, that corresponds to the first steps of the Landau-Hopf scenario. The influence of the discretization parameter value on the observed structure is discussed. Using this model, the effect of noise on quasi-periodic Hopf bifurcations and invariant tori of varying dimensions is studied. The presence of several Lyapunov exponents is taken into account and corresponding one-parameter and two-parameter illustrations are given. The cases of noise effect on “the weakest” and “the strongest” oscillators in an ensemble are examined and compared. The possibility of alternative chaotization of tori and their stabilization by noise is demonstrated. Illustrations of such stabilization are given.","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"95 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2026-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146209837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-12DOI: 10.1016/j.chaos.2026.118071
Xindong Si, Wuquan Li
In this paper, the bipartite synchronization problem of coupled neural networks with sign-switching topology under replay attacks is investigated based on multi-rate sampled-data control. First, an error system model incorporating a Laplacian matrix coupling term is constructed based on graph theory and the properties of the switching topology. Second, a pinning multi-rate sampled-data controller is designed by incorporating the multi-rate sampled-data control scheme and the characteristics of replay attacks. Then, a looped-function that remains positive definite only at sampling instants is constructed. Considering the discrete-continuous Lyapunov stability theory and inequality techniques, a mean-square bipartite synchronization criterion is derived. Finally, the effectiveness and advantages of the control scheme and the constructed Lyapunov function are verified through numerical examples and the maximum allowable sampling interval algorithm.
{"title":"Securing bipartite synchronization of neural networks under multi-rate sampling and switching topologies","authors":"Xindong Si, Wuquan Li","doi":"10.1016/j.chaos.2026.118071","DOIUrl":"https://doi.org/10.1016/j.chaos.2026.118071","url":null,"abstract":"In this paper, the bipartite synchronization problem of coupled neural networks with sign-switching topology under replay attacks is investigated based on multi-rate sampled-data control. First, an error system model incorporating a Laplacian matrix coupling term is constructed based on graph theory and the properties of the switching topology. Second, a pinning multi-rate sampled-data controller is designed by incorporating the multi-rate sampled-data control scheme and the characteristics of replay attacks. Then, a looped-function that remains positive definite only at sampling instants is constructed. Considering the discrete-continuous Lyapunov stability theory and inequality techniques, a mean-square bipartite synchronization criterion is derived. Finally, the effectiveness and advantages of the control scheme and the constructed Lyapunov function are verified through numerical examples and the maximum allowable sampling interval algorithm.","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"95 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2026-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146209920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-12DOI: 10.1016/j.chaos.2026.118057
Sarra Senouci, Zhangchun Tang, Mohammed Raouf Senouci, Sid Ali Madoune, Abdelkader Senouci
Synchronizing chaotic systems is a fundamental challenge with significant applications, yet its practical implementation is often hindered by non-ideal channel characteristics such as noise and time delay. This paper introduces a new, robust framework for achieving high-fidelity synchronization in such challenging environments. The primary objective is to develop and validate a control strategy that is resilient to both limited state information and realistic channel imperfections. Our methodology pairs a Luenberger-style state observer with a novel, delay-aware Linear-Quadratic Regulator (LQR) within a master–slave system configuration. The controller explicitly compensates for input delay using a formulation derived from Lyapunov–Krasovskii theory and demonstrates the efficacy of actuating on a single, strategically selected state variable. Simulations conducted over a high-fidelity fiber-optic channel model confirm the framework’s performance. The results demonstrate rapid, precise synchronization that is robust to significant time-varying delays (up to 28ms), achieving settling times as low as 0.8 s while maintaining a steady-state error on the order of 10−5 amidst signal noise and quantization effects. The observer-based partial-state feedback approach successfully matches the performance of full-state feedback, validating its effectiveness. This work presents a comprehensive solution that significantly enhances the feasibility of applying chaos synchronization in real-world systems, proving its stability and robustness against a wide range of initial conditions and channel disturbances.
{"title":"A new framework for robust observer-based synchronization of chaotic systems with a delay-aware LQR controller","authors":"Sarra Senouci, Zhangchun Tang, Mohammed Raouf Senouci, Sid Ali Madoune, Abdelkader Senouci","doi":"10.1016/j.chaos.2026.118057","DOIUrl":"https://doi.org/10.1016/j.chaos.2026.118057","url":null,"abstract":"Synchronizing chaotic systems is a fundamental challenge with significant applications, yet its practical implementation is often hindered by non-ideal channel characteristics such as noise and time delay. This paper introduces a new, robust framework for achieving high-fidelity synchronization in such challenging environments. The primary objective is to develop and validate a control strategy that is resilient to both limited state information and realistic channel imperfections. Our methodology pairs a Luenberger-style state observer with a novel, delay-aware Linear-Quadratic Regulator (LQR) within a master–slave system configuration. The controller explicitly compensates for input delay using a formulation derived from Lyapunov–Krasovskii theory and demonstrates the efficacy of actuating on a single, strategically selected state variable. Simulations conducted over a high-fidelity fiber-optic channel model confirm the framework’s performance. The results demonstrate rapid, precise synchronization that is robust to significant time-varying delays (up to <mml:math altimg=\"si4.svg\" display=\"inline\"><mml:mrow><mml:mn>28</mml:mn><mml:mspace width=\"0.33em\"></mml:mspace><mml:mi mathvariant=\"normal\">ms</mml:mi></mml:mrow></mml:math>), achieving settling times as low as 0.8 s while maintaining a steady-state error on the order of <mml:math altimg=\"si206.svg\" display=\"inline\"><mml:mrow><mml:mn>1</mml:mn><mml:msup><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math> amidst signal noise and quantization effects. The observer-based partial-state feedback approach successfully matches the performance of full-state feedback, validating its effectiveness. This work presents a comprehensive solution that significantly enhances the feasibility of applying chaos synchronization in real-world systems, proving its stability and robustness against a wide range of initial conditions and channel disturbances.","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"10 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2026-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146209864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-11DOI: 10.1016/j.chaos.2026.118058
Yan Zhang, Xinbao Li, Fang Wang, Yuhang Meng, Shihong Yin
{"title":"Optimal control for a class of input-constrained nonlinear switched systems","authors":"Yan Zhang, Xinbao Li, Fang Wang, Yuhang Meng, Shihong Yin","doi":"10.1016/j.chaos.2026.118058","DOIUrl":"https://doi.org/10.1016/j.chaos.2026.118058","url":null,"abstract":"","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"316 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2026-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146152818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-11DOI: 10.1016/j.chaos.2026.118047
Yongxiang Zhang, Peng Zhang, Miguel A.F. Sanjuán
{"title":"Final state sensitivity and noise-induced extreme events in a nonsmooth dynamical system","authors":"Yongxiang Zhang, Peng Zhang, Miguel A.F. Sanjuán","doi":"10.1016/j.chaos.2026.118047","DOIUrl":"https://doi.org/10.1016/j.chaos.2026.118047","url":null,"abstract":"","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"47 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2026-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146152813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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