Chaos Solitons & Fractals最新文献_第8页

Security-conscious dynamic output feedback control for periodic piecewise nonlinear systems under output-reliant disturbance observer 输出依赖干扰观测器下周期分段非线性系统的安全意识动态输出反馈控制

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2026-07-01 Epub Date: 2026-02-23 DOI: 10.1016/j.chaos.2026.118064

Thangavel Satheesh , Rathinasamy Sakthivel , Oh-Min Kwon

This article focuses on addressing the finite-time dynamic output feedback control problem for periodic piecewise nonlinear systems in the midst of multifaceted disturbances and cyber attack scenarios. Moreover, multifaceted disturbances encompass mismatched disturbances and multiple matched disturbances. Therein, the mismatched disturbances are presumed to be norm-bounded vectors, whereas multiple matched disturbances emanate from exogenous systems. Additionally, two instances of nonlinear dynamics in the plants are looked at, pertaining to both known and unknown cases. Furthermore, to estimate the multiple matched disturbances, output-reliant multiple periodic piecewise disturbance observers for both the nonlinear instances are framed separately on the grounds of the control input and the measurement output. Moreover, the mismatched disturbances are attenuated using the mixed

H_{\infty}

/passivity approach. From thereon, a holistic control framework blending these observers with a dynamic output feedback mechanism is put together to guarantee the intended closed-loop performance in a finite-time span by rejecting the matched disturbances. Therein, a probabilistic model with Bernoulli-distributed variable is laid out to characterize the incidence of deception attacks, potentially improving the resilience of the control. Following that, by means of Lyapunov stability theory, the stability of the target system is analyzed and the precise setups of the periodic piecewise controller and disturbance observer gains are put forward. Ultimately, simulation results are put forward to confirm the reliability and effectiveness of the control protocol.

本文主要研究周期分段非线性系统在多面干扰和网络攻击情况下的有限时间动态输出反馈控制问题。此外，多面干扰包括不匹配干扰和多个匹配干扰。其中，不匹配的干扰被假定为范数有界向量，而多个匹配的干扰来自外源系统。此外，研究了植物非线性动力学的两个实例，涉及已知和未知的情况。此外，为了估计多个匹配的扰动，基于控制输入和测量输出分别构建了两个非线性实例的依赖于输出的多周期分段扰动观测器。此外，采用混合H∞/无源方法对失匹配干扰进行了衰减。在此基础上，建立了一个将这些观测器与动态输出反馈机制混合在一起的整体控制框架，通过抑制匹配的干扰来保证在有限时间范围内的预期闭环性能。在此基础上，提出了一个具有伯努利分布变量的概率模型来表征欺骗攻击的发生率，从而潜在地提高了控制的弹性。然后，利用李雅普诺夫稳定性理论分析了目标系统的稳定性，提出了周期分段控制器和扰动观测器增益的精确设置。最后给出了仿真结果，验证了控制协议的可靠性和有效性。

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引用次数: 0

Control of the composite spiral waves: Noise can alter the structure of composite spiral waves and facilitate transitions between composite and classic spiral waves 复合螺旋波的控制：噪声可以改变复合螺旋波的结构，促进复合螺旋波与经典螺旋波之间的过渡

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2026-07-01 Epub Date: 2026-02-26 DOI: 10.1016/j.chaos.2026.118143

Jian Gao, Bin Xu, Cong Liu, Chuansheng Shen

Spiral waves are ubiquitous structures in a broad spectrum of natural and engineering systems, and effective control of spiral wave architectures carries both theoretical significance and potential application value. This study investigates the impact of noise on the composite spiral wave (CSW) dynamics and their structural transitions to classical spiral waves. Using a discrete spatiotemporal predator–prey reaction–diffusion model, we demonstrate that noise critically regulates the CSW structure through dual-threshold phase transitions. Numerical simulations and theoretical analyses reveal that when noise intensity exceeds critical thresholds, discontinuous first-order transitions occur: first altering the size of the small spiral wave (SSW) region, then driving global expansion into classical spiral waves. Mechanistically, noise induces local state mutations and modifies boundary propagation velocities, enabling chiral inversion of wave patterns. The system exhibits resilience, spontaneously recovering CSW structures upon noise removal. These findings establish noise as a potent tool for controlling nonlinear wave structures, with implications for cardiac arrhythmia mitigation and programmable metamaterial design.

螺旋波是广泛存在于自然和工程系统中的结构，对螺旋波结构进行有效控制具有重要的理论意义和潜在的应用价值。本文研究了噪声对复合螺旋波（CSW）动力学的影响及其向经典螺旋波的结构转变。利用一个离散的时空捕食者-猎物反应-扩散模型，我们证明了噪声通过双阈值相变对CSW结构进行了严格的调节。数值模拟和理论分析表明，当噪声强度超过临界阈值时，会发生不连续的一阶转换：首先改变小螺旋波（SSW）区域的大小，然后驱动全局扩展为经典螺旋波。从机制上讲，噪声诱导局部状态突变并改变边界传播速度，从而实现波形的手性反演。该系统表现出弹性，在去除噪声后自发恢复CSW结构。这些发现确立了噪声是控制非线性波结构的有力工具，对心律失常缓解和可编程超材料设计具有重要意义。

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引用次数: 0

Skewness-Kurtosis: Small samples and power-law behavior 偏峰度：小样本和幂律行为

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2026-07-01 Epub Date: 2026-02-14 DOI: 10.1016/j.chaos.2026.118066

Carlo De Michele , Samuele De Bartolo

Skewness and kurtosis are fundamental statistical moments commonly used to quantify asymmetry and tail heaviness and peakedness of probability distributions. Despite their widespread application in statistical mechanics, condensed matter physics, and complex systems, important aspects of their empirical behavior remain unclear — particularly in small samples and in relation to their hypothesized power-law scaling. In this work, we address both issues using a combination of empirical and synthetic data. First, we establish a lower bound for sample kurtosis as a function of sample size and skewness. In doing this, a remarkable deltoid-shaped domain appears for

n

= 4, which in case of discrete distributions exhibits fractal properties reflecting the underlying geometrical constraints of skewness–kurtosis space. Second, we examine the conditions under which the 4/3 power-law relationship between kurtosis and skewness emerges, effectively extending Taylor’s power-law to higher-order moments. Our results show that this scaling predominantly occurs in data sampled from heavy-tailed distributions and medium/large sample sizes, highlighting the interplay between tail behavior and sample size in shaping the empirical behavior of higher-order moments.

偏度和峰度是量化概率分布的不对称性、尾重和峰性的基本统计矩。尽管它们在统计力学、凝聚态物理和复杂系统中得到了广泛的应用，但它们的经验行为的重要方面仍然不清楚——特别是在小样本中，以及与它们假设的幂律缩放有关。在这项工作中，我们使用经验和合成数据的组合来解决这两个问题。首先，我们建立样本峰度的下界作为样本大小和偏度的函数。在此过程中，n = 4出现了一个显著的三角形区域，在离散分布的情况下，它表现出分形特性，反映了偏度-峰度空间的潜在几何约束。其次，我们研究了峰度和偏度之间出现4/3幂律关系的条件，有效地将泰勒幂律扩展到高阶矩。我们的研究结果表明，这种缩放主要发生在从重尾分布和中/大样本量中采样的数据中，突出了尾巴行为和样本量在塑造高阶矩的经验行为方面的相互作用。

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引用次数: 0

Adaptive neural network passivity-based control with state observer for partially unknown nonlinear systems 部分未知非线性系统的状态观测器自适应神经网络无源控制

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2026-06-01 Epub Date: 2026-01-30 DOI: 10.1016/j.chaos.2026.117989

Junjie Gong , Shengjie Guo , Jian Chen , Dengsheng Cai , Liang He , Wei Wei , Yu Long

To overcome the limitations of interconnection and damping assignment passivity-based control (IDA-PBC) in partially unknown nonlinear systems, particularly the reliance on large damping coefficients that may cause control inputs to exceed physical constraints, this paper proposes a novel adaptive neural network passivity-based control strategy incorporating a passivity-based observer. In the proposed framework, neural networks are introduced to relax the strict requirement of the desired closed-loop port-controlled Hamiltonian (PCH) model and to construct dynamic compensation mechanisms. These mechanisms are embedded into the interconnection and damping assignment passivity-based observer (IDA-PBO), resulting in an adaptive neural network IDA-PBO (ANNIDA-PBO) with enhanced dynamic compensation capability. On the control side, neural-network-based adaptive laws and compensation terms are developed to accurately approximate and compensate for unmodeled dynamics. This design alleviates the inherent limitation of conventional IDA-PBC methods that depend on large damping coefficients to achieve robustness. As a result, the proposed framework enables a well-conditioned selection of control gains within physical constraints and effectively decouples the strong dependence between robustness performance and damping parameters in standard IDA-PBC designs. Furthermore, the closed-loop system is shown to be semi-globally uniformly ultimately bounded through port-controlled Hamiltonian modeling and Lyapunov stability analysis. Finally, simulation studies on two representative nonlinear systems are conducted to validate the proposed method, demonstrating improved control accuracy and enhanced robustness against external disturbances.

为了克服部分未知非线性系统中基于互连和阻尼分配的无源控制（IDA-PBC）的局限性，特别是对可能导致控制输入超出物理约束的大阻尼系数的依赖，本文提出了一种新的基于无源观测器的自适应神经网络无源控制策略。在该框架中，引入神经网络，放宽了对理想的闭环端口控制哈密顿模型的严格要求，并构建了动态补偿机制。将这些机制嵌入到基于互联和阻尼分配的无源观测器（IDA-PBO）中，形成具有增强动态补偿能力的自适应神经网络IDA-PBO （ANNIDA-PBO）。在控制方面，开发了基于神经网络的自适应律和补偿项，以精确地逼近和补偿未建模的动力学。该设计减轻了传统IDA-PBC方法依赖大阻尼系数来实现鲁棒性的固有局限性。因此，所提出的框架能够在物理约束下进行条件良好的控制增益选择，并有效地解耦了标准IDA-PBC设计中鲁棒性性能与阻尼参数之间的强依赖性。进一步，通过端口控制哈密顿模型和Lyapunov稳定性分析，证明了闭环系统是半全局一致最终有界的。最后，对两个具有代表性的非线性系统进行了仿真研究，验证了所提方法的有效性，表明该方法提高了控制精度，增强了对外部干扰的鲁棒性。

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引用次数: 0

Predicting the onset of period-doubling bifurcations via dominant eigenvalue extracted from autocorrelation 利用自相关提取的优势特征值预测倍周期分岔的发生

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2026-06-01 Epub Date: 2026-01-30 DOI: 10.1016/j.chaos.2026.118010

Zhiqin Ma , Chunhua Zeng , Ting Gao , Jinqiao Duan

Predicting the occurrence of transitions in the qualitative dynamics of many natural systems is crucial, yet it remains a challenging task. Generic early warning signals like variance and lag-1 autocorrelation identify critical slowing down near tipping points but lack practical thresholds for predicting imminent transitions. More recent studies found that the dynamical eigenvalue is rooted in the framework of empirical dynamical modeling and then estimates the dominant eigenvalue of a system from time series, providing a threshold (

|

DEV

|

= 1) to predict bifurcations and classify their types. However, its application requires careful calibration of the hyperparameters and focuses on reconstructing system dynamics directly from data. Here, we employ Ornstein–Uhlenbeck process to derive analytic approximations for the lag-

τ

autocorrelation function prior to period-doubling bifurcation thereby estimating the dominant eigenvalue of dynamical systems, named dominant eigenvalue extracted from autocorrelation (DE-AC), and revealing its dynamic behavior when approaching a period-doubling bifurcation. Theoretically, dominant eigenvalue tends to

- 1

when the system approaches a period-doubling bifurcation. In particular, we evaluated DE-AC on simulation data from cardiac alternans model and on experimental data from chick heart aggregates undergoing a period-doubling bifurcation. DE-AC reliably detected the beginning of the cardiac arrhythmia (period-doubling bifurcation) in most cases. Moreover, it demonstrated superior sensitivity and specificity as an early warning signal compared to the three widely used indicators—variance, lag-1 autocorrelation, and dynamical eigenvalue. Our theoretical and empirical results suggest that DE-AC represents a quantitative measure for predicting the onset of potentially dangerous alternating rhythms in the heart. The ability to better infer, detect, and distinguish the nature of impending transitions in complex systems will help humans manage critical transitions in biological systems.

预测在许多自然系统的定性动力学转变的发生是至关重要的，但它仍然是一个具有挑战性的任务。通用的早期预警信号，如方差和lag-1自相关，可以识别临界点附近的关键减速，但缺乏预测即将发生的转变的实际阈值。最近的研究发现，动态特征值植根于经验动力学建模的框架，然后从时间序列中估计系统的主导特征值，提供一个阈值（|DEV| = 1）来预测分岔并对其类型进行分类。然而，它的应用需要仔细校准超参数，并侧重于直接从数据重建系统动力学。本文采用Ornstein-Uhlenbeck过程推导了倍周期分岔前lag-τ自相关函数的解析近似，从而估计了动力系统的显性特征值，称为从自相关中提取的显性特征值（DE-AC），并揭示了其在接近倍周期分岔时的动态行为。理论上，当系统接近倍周期分岔时，优势特征值趋向于−1。特别是，我们在心脏交替模型的模拟数据和鸡心脏聚合体经历倍期分叉的实验数据上评估了DE-AC。DE-AC在大多数病例中可靠地检测到心律失常（双周期分叉）的开始。此外，与方差、lag-1自相关和动态特征值这三种广泛使用的指标相比，它作为预警信号表现出更高的灵敏度和特异性。我们的理论和实证结果表明，DE-AC代表了预测心脏潜在危险交替节律发作的定量措施。更好地推断、检测和区分复杂系统中即将发生的转变的性质的能力将有助于人类管理生物系统中的关键转变。

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引用次数: 0

Stabilization of fractional parabolic equations through designed robust Robin boundary controller 基于鲁棒Robin边界控制器的分数阶抛物方程镇定

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2026-06-01 Epub Date: 2026-01-30 DOI: 10.1016/j.chaos.2026.118003

Hassen Arfaoui

The paper deals with the boundary stabilization for time-fractional parabolic equations (TFPEs) with non-constant coefficients and distributed disturbance by backstepping approach. Firstly, boundary stabilization is investigated through right Robin boundary control (RRBC) with homogeneous left Robin boundary condition at the lower end of the spatial domain. In this context, we proved a Mittag-Leffler stability result for the solution of the TFPEs. To achieve this objective, a robust boundary feedback control law has been designed for the stabilization of the TFPEs by backstepping approach. To the best of our knowledge, the stabilization of TFPEs with left homogeneous Robin boundary condition through right Robin-type boundary feedback control law via backstepping method is novel. Some numerical examples are presented at the end of this paper to confirm the theoretical results.

本文用反步法研究了具有非常系数和分布扰动的时间分数型抛物方程的边界镇定问题。首先，利用空间域下端齐次左罗宾边界条件的右罗宾边界控制（RRBC）研究边界稳定问题；在这种情况下，我们证明了tpes解的Mittag-Leffler稳定性结果。为了实现这一目标，设计了一种鲁棒的边界反馈控制律，并采用回溯法实现了tfp的镇定。据我们所知，利用右罗宾型边界反馈控制律反步法稳定具有左齐次罗宾边界条件的TFPEs是一种新颖的方法。最后给出了一些数值算例来验证理论结果。

引用次数: 0

Bright photorefractive solitons supported by simultaneous linear and quadratic electro-optic effect with both drift and diffusion charge transport 同时具有漂移和扩散电荷输运的线性和二次电光效应支持的明亮光折变孤子

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2026-06-01 Epub Date: 2026-01-29 DOI: 10.1016/j.chaos.2026.117940

Aavishkar Katti , Andrzej Ziółkowski

We have predicted a new and large family of optical spatial solitons in photorefractive crystals having both linear and quadratic electro-optic effects by considering for the first time an equal significance of both drift and diffusion on the charge transport. We solve the ensuing modified Nonlinear Schrödinger equation employing the paraxial approximation in conjunction with a variational solution. Consequently, we have derived the existence curve of solitons together with the evolution equations for soliton width and deflection. Bistable states have been found, revealing two threshold powers for soliton generation associated with a certain soliton width. Distinct power regimes have been identified on this basis, and the evolution of soliton breadth and self-deflection has been examined within these contexts. The external electric field affects the soliton trajectory, with self-deflection escalating as the electric field intensifies. The interaction between the linear and quadratic electro-optic effects has been examined by analyzing two scenarios: one in which the linear electro-optic effect predominates, and another in which the quadratic effect prevails. We analyze and differentiate the impact of each electro-optic effect on the existence curve of solitons and their trajectories.

通过首次考虑漂移和扩散对电荷输运的同等重要性，我们预测了光折变晶体中具有线性和二次电光效应的一种新的光学空间孤子族。我们利用傍轴近似结合变分解来求解随后的修正非线性Schrödinger方程。由此导出了孤子的存在曲线以及孤子宽度和挠度的演化方程。发现了双稳态，揭示了与一定孤子宽度相关的孤子产生的两个阈值功率。在此基础上确定了不同的功率制度，并在这些背景下研究了孤子宽度和自偏转的演变。外加电场影响孤子的运动轨迹，随着电场的增强，孤子的自偏转逐渐增大。通过分析两种情况，考察了线性电光效应和二次电光效应之间的相互作用：一种是线性电光效应占主导地位，另一种是二次效应占主导地位。我们分析和区分了每种电光效应对孤子存在曲线及其轨迹的影响。

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引用次数: 0

Limit cycles of planar discontinuous piecewise linear Hamiltonian systems in three regions of Y-type 平面间断分段线性哈密顿系统在y型三区域上的极限环

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2026-06-01 Epub Date: 2026-01-30 DOI: 10.1016/j.chaos.2026.117997

Jaume Llibre , Regilene Oliveira

In recent years there has been a significant interest in studying discontinuous piecewise differential systems, mainly due to the wide range of applications in modeling natural phenomena. To understand the dynamics of these systems in the plane one challenge is to control their number of limit cycles. In this paper we study the existence of limit cycles in planar discontinuous piecewise linear Hamiltonian systems with three zones separated by the line

Y = {(x, y) : x \geq 0 and y = 0} \cup {(x, y) : x = 0 and y \geq 0} \cup {(x, y) : x \leq 0 and y = 0} .

We provide the maximum number of crossing limit cycles intersecting each branch of

Y

in one point, and intersecting two branches of the

Y

each one in two points. So we have solved the extension of the 16th Hilbert problem to this class of differential systems.

近年来，研究不连续分段微分系统引起了极大的兴趣，这主要是由于它在模拟自然现象方面的广泛应用。要理解这些系统在平面上的动力学，一个挑战是控制它们的极限环数。本文研究了平面不连续分段线性哈密顿系统极限环的存在性，该系统的三个区域由直线Y={(x， Y):x≥0andy=0}∪{(x， Y):x=0andy≥0}∪{(x， Y):x≤0andy=0}分隔。我们给出了与Y的每个分支相交于一点的最大极限环数，以及与Y的两个分支相交于两点的最大极限环数。我们已经解决了第16个希尔伯特问题对这类微分系统的推广。

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引用次数: 0

Modeling cross-modal interactions via a nonlinear information-density-aware network for MAFLD risk assessment 基于非线性信息密度感知网络的MAFLD风险评估跨模态相互作用建模

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2026-06-01 Epub Date: 2026-01-31 DOI: 10.1016/j.chaos.2026.117978

Xiaohua Hu , Xiang Zhu , Jia Shi , Zhaoxian Yan , Shuangshuang Qian , Hanlin Qi , Juan Du , He Yan , Changquan Ling

Modern multimodal learning often requires handling heterogeneous data types whose structures and information densities differ substantially. To address this challenge in the context of metabolic dysfunction-associated fatty liver disease (MAFLD) prediction, we propose an information-density-aware multimodal framework (NID-Net). Instead of relying on simple concatenation or shallow fusion, the model processes each modality using methods that align with its structural characteristics. Structured indicators with high information density are first processed by an XGBoost module optimized via Lagrange remainder correction, which enhances the nonlinearity of the loss landscape and improves robustness to data sparsity and imbalance. Meanwhile, tongue images with relatively low information density are encoded using a region-enhanced Swin Transformer, where adaptive regional biases guide the model toward informative local representations. The resulting modality-specific embeddings are fused within a Mixture-of-Experts (MoE) architecture, enabling selective specialization and nonlinear decision boundaries across modalities. Extensive experiments on real-world medical datasets demonstrate that NID-Net not only surpasses existing multimodal fusion approaches in predictive performance but also provides interpretable insights into cross-modal feature interactions. This work highlights the fundamental role of nonlinear design in achieving efficient, balanced, and explainable multimodal prediction systems.

现代多模态学习通常需要处理结构和信息密度差异很大的异构数据类型。为了在代谢功能障碍相关脂肪肝（MAFLD）预测的背景下解决这一挑战，我们提出了一个信息密度感知的多模式框架（NID-Net）。该模型不依赖于简单的连接或浅融合，而是使用与其结构特征相一致的方法处理每种模态。高信息密度的结构化指标首先通过拉格朗日余数校正优化的XGBoost模块进行处理，增强了损失域的非线性，提高了对数据稀疏性和不平衡的鲁棒性。同时，使用区域增强的Swin Transformer对信息密度相对较低的舌头图像进行编码，其中自适应区域偏差将模型引导到信息丰富的局部表示。生成的特定于模态的嵌入被融合到专家混合（MoE）体系结构中，从而实现了跨模态的选择性专门化和非线性决策边界。在真实医学数据集上的大量实验表明，NID-Net不仅在预测性能上超越了现有的多模态融合方法，而且还提供了对跨模态特征交互的可解释性见解。这项工作强调了非线性设计在实现高效、平衡和可解释的多模态预测系统中的基本作用。

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引用次数: 0

Observation of partial replica symmetry breaking in polarization space for modulation instability competed by self-phase modulation 偏振空间中自相位调制不稳定性的部分复制对称性破缺观察

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2026-06-01 Epub Date: 2026-01-31 DOI: 10.1016/j.chaos.2026.118005

Yu Long, Lei Gao, Ai Liu, Shiyun Dai, Yujia Li, Ligang Huang, Tao Zhu

Replica symmetry breaking provides a profound framework for understanding complex systems. However, a direct experimental confirmation of such complex statistical behavior within the polarization space of a fundamental nonlinear process like modulation instability has remained elusive. Here, exploiting advances in ultrafast optical polarization measurement, we report the first experimental observation of partial replica symmetry breaking in the polarization space, where polarization serves as a direct analogue of spin. By performing wavelength-resolved polarization measurements of modulation instability sidebands, we identify the replica symmetry breaking for specific wavelengths by calculating the overlapping parameter of fluctuations between any two arbitrary points in polarization space, while other wavelengths maintain replica symmetric behavior. The partial replica symmetry breaking in polarization space is attributed to the competition of modulation instability and self-phase modulation. Our results establish polarization as a natural platform for probing complex energy landscapes in disordered photonic systems.

复制对称破缺为理解复杂系统提供了一个深刻的框架。然而，在调制不稳定性等基本非线性过程的极化空间中，这种复杂的统计行为的直接实验证实仍然难以捉摸。在这里，利用超快光学偏振测量的进展，我们报告了偏振空间中部分复制对称破缺的首次实验观察，其中偏振作为自旋的直接模拟。通过对调制不稳定边带进行波长分辨偏振测量，我们通过计算偏振空间中任意两点之间波动的重叠参数来识别特定波长的复制对称性破缺，而其他波长保持复制对称性行为。偏振空间中的部分复模对称性破缺是调制不稳定性和自相位调制相互竞争的结果。我们的研究结果建立了极化作为探测无序光子系统中复杂能量景观的天然平台。

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引用次数: 0