Physica D: Nonlinear Phenomena最新文献

{"title":"Observability of complex systems via conserved quantities","authors":"Bhargav R. Karamched ,&nbsp;Jack Schmidt ,&nbsp;David Murrugarra","doi":"10.1016/j.physd.2025.134714","DOIUrl":"10.1016/j.physd.2025.134714","url":null,"abstract":"<div><div>Many systems in biology, physics, and engineering are modeled by nonlinear dynamical systems where the states are usually unknown and only a subset of the state variables can be physically measured. Can we understand the full system from what we measure? In the mathematics literature, this question is framed as the observability problem. It has to do with recovering information about the state variables from the observed states (the measurements). In this paper, we relate the observability problem to another structural feature of many models relevant in the physical and biological sciences: the conserved quantity. For models based on systems of differential equations, conserved quantities offer desirable properties such as dimension reduction which simplifies model analysis. Here, we use differential embeddings to show that conserved quantities involving a set of special variables provide more flexibility in what can be measured to address the observability problem for systems of interest in biology. Specifically, we provide conditions under which a collection of conserved quantities make the system observable. We apply our methods to provide alternate measurable variables in models where conserved quantities have been used for model analysis historically in biological contexts.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"477 ","pages":"Article 134714"},"PeriodicalIF":2.7,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144072496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Asymmetric dual cascade in gravitational wave turbulence","authors":"Benoît Gay,&nbsp;Sébastien Galtier","doi":"10.1016/j.physd.2025.134712","DOIUrl":"10.1016/j.physd.2025.134712","url":null,"abstract":"<div><div>We numerically simulate, in both the forced and decay regimes, a fourth-order nonlinear diffusion equation derived from the kinetic equation of gravitational wave turbulence in the limit of strongly local quartic interactions. When a forcing is applied to an intermediate wavenumber <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, we observe a dual cascade of energy and wave action. In the stationary state, the associated flux ratio is proportional to <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, and the Kolmogorov–Zakharov spectra are recovered. In decaying turbulence, the study reveals that the wave action spectrum can extend to wavenumbers greater than the initial excitation <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> with constant negative flux, while the energy flux is positive with a power law dependence in <span><math><mi>k</mi></math></span>. This leads to an unexpected result: a single inertial range with a Kolmogorov–Zakharov wave action spectrum extending progressively to wavenumbers larger than <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. We also observe a wave action decay in time in <span><math><msup><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></math></span> while the front of the energy spectrum progresses according to a <span><math><msup><mrow><mi>t</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></math></span> law. These properties can be understood with simple theoretical arguments.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"477 ","pages":"Article 134712"},"PeriodicalIF":2.7,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fractal characteristics of interfaces in Richtmyer–Meshkov turbulence","authors":"Yue Zhang ,&nbsp;Zhangbo Zhou ,&nbsp;Wan Cheng ,&nbsp;Juchun Ding ,&nbsp;Xisheng Luo","doi":"10.1016/j.physd.2025.134711","DOIUrl":"10.1016/j.physd.2025.134711","url":null,"abstract":"<div><div>This work investigates the fractal dimension of the spike front and the bubble front in Richtmyer–Meshkov (RM) turbulence with initial broadband and narrowband perturbations. The fractal dimension <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> of interfaces is found to be intimately associated with interfacial mixing, exhibiting a gradual increase with interface evolution until it attains a stable value, between <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>≈</mo><mn>2</mn><mo>.</mo><mn>2</mn><mo>∼</mo><mn>2</mn><mo>.</mo><mn>33</mn></mrow></math></span>, upon entering the self-similar stage. The fractal dimension on the spike front is closer to the universal fractal dimension of fully-developed turbulence, indicating stronger turbulent fluctuations there. Large-scale perturbations render the interfaces more two-dimensional thereby resulting in a lower fractal dimension. The results demonstrate that the fractal dimension, <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>f</mi><mn>3</mn></mrow></msub></math></span>, of a surface in three dimensions can be faithfully reflected by the fractal dimension, <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>f</mi><mn>2</mn></mrow></msub></math></span>, of the interface of its two-dimensional projection. This finding validates the additive rule of <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>f</mi><mn>3</mn></mrow></msub><mo>=</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>f</mi><mn>2</mn></mrow></msub><mo>+</mo><mn>1</mn></mrow></math></span> in RM turbulence, which is of considerable utility for experimental research. The filtering approach is furthermore employed to illustrate the scale-invariance of the fractal interfaces. The fractal features of the early spike and bubble interfaces can be described by a simple model fractal of the Koch curve, with distinct scale factors.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"477 ","pages":"Article 134711"},"PeriodicalIF":2.7,"publicationDate":"2025-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143947152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Aperiodically intermittent H∞ synchronization of coupled complex-valued neural networks with and without parameter uncertainties","authors":"Yanli Huang,&nbsp;Yuqing Jia","doi":"10.1016/j.physd.2025.134694","DOIUrl":"10.1016/j.physd.2025.134694","url":null,"abstract":"<div><div>In this paper, we investigate aperiodically intermittent <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span> synchronization and robustly aperiodically intermittent <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span> synchronization for both coupled complex-valued neural networks (CCVNNs) and coupled complex-valued delayed neural networks (CCVDNNs). Initially, certain synchronization criteria for CCVNNs are established using suitable Lyapunov functionals in conjunction with a variety of inequality techniques. Furthermore, we also examine robust synchronization of CCVNNs, taking into account the impact of parameter uncertainties in neural networks. In addition, the corresponding aperiodically intermittent <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span> synchronization and robustly aperiodically intermittent <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span> synchronization results are generalized to the network of CCVDNNs, in which the activation function is time-varying delayed. Finally, two numerical examples prove the reliability and the benefits of the proposed results.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"477 ","pages":"Article 134694"},"PeriodicalIF":2.7,"publicationDate":"2025-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143942296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A bound state attractor in optical turbulence","authors":"Clément Colléaux ,&nbsp;Jonathan Skipp ,&nbsp;Sergey Nazarenko ,&nbsp;Jason Laurie","doi":"10.1016/j.physd.2025.134687","DOIUrl":"10.1016/j.physd.2025.134687","url":null,"abstract":"<div><div>We study numerically the nonintegrable dynamics of coherent, solitonic, nonlinear waves, in a spatially nonlocal nonlinear Schrödinger equation relevant to realistic modelling of optical systems: the Schrödinger–Helmholtz equation. We observe a single oscillating, coherent solitary wave emerging from a variety of initial conditions. Using the direct scattering transform of the (integrable) cubic nonlinear Schrödinger equation, we find that this structure is a bound state, comprising of a primary and secondary soliton whose amplitudes oscillate in phase opposition. We interpret this as the solitons periodically exchanging mass. We also observe bound states comprising of three oscillating solitons, hinting at the existence of a family of multi-soliton bound states. Focusing on the two-soliton bound state, we observe it self-organising from an initial state of incoherent turbulence, and from solitonic structures launched into the system. When a single (primary) solitonic structure is launched, a resonance process between it and waves in the system generates the secondary soliton, resulting in the bound state. Further, when two solitons are initially launched, we show that they can merge if their phases are synchronised when they collide. When the system is launched from a turbulent state comprised of many initial solitons, we propose that the bound state formation is preceded by a sequence of binary collisions, in which the mass is transferred on average from the weak soliton to the strong one, with occasional soliton mergers. Both processes lead to increasingly stronger and fewer dominant solitons. The final state – a solitary bound state surrounded by weakly nonlinear waves – is robust and ubiquitous. We propose that for nonlocal media, a bound state comprising of at least two solitons is a more typical statistical attractor than the single-soliton attractor suggested in previous literature.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"477 ","pages":"Article 134687"},"PeriodicalIF":2.7,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143934567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Model reduction of nonlinear time-delay systems via ODE approximation and spectral submanifolds","authors":"Yuan Tang,&nbsp;Mingwu Li","doi":"10.1016/j.physd.2025.134701","DOIUrl":"10.1016/j.physd.2025.134701","url":null,"abstract":"<div><div>Time-delay dynamical systems inherently embody infinite-dimensional dynamics, thereby amplifying their complexity. This aspect is especially notable in nonlinear dynamical systems, which frequently defy analytical solutions and necessitate approximations or numerical methods. These requirements present considerable challenges for the real-time simulation and analysis of their nonlinear dynamics. To address these challenges, we present a model reduction framework for nonlinear time-delay systems using spectral submanifolds (SSMs). We first approximate the time-delay systems as ordinary differential equations (ODEs) without delay and then compute the SSMs and their associated reduced-order models (ROMs) of the ODE approximations. These SSM-based ROMs successfully predict the nonlinear dynamical behaviors of the time-delay systems, including free and forced vibrations, and accurately identify critical features such as isolated branches in the forced response curves and bifurcations of periodic and quasi-periodic orbits. The efficiency and accuracy of the ROMs are demonstrated through examples of increasing complexity.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"477 ","pages":"Article 134701"},"PeriodicalIF":2.7,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143934658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Detecting causality in the frequency domain with Cross-Mapping Coherence","authors":"Zsigmond Benkő ,&nbsp;Bálint Varga ,&nbsp;Marcell Stippinger ,&nbsp;Zoltán Somogyvári","doi":"10.1016/j.physd.2025.134708","DOIUrl":"10.1016/j.physd.2025.134708","url":null,"abstract":"<div><div>Understanding causal relationships within a system is crucial for uncovering its underlying mechanisms. Causal discovery methods, which facilitate the construction of such models from time series data, hold the potential to significantly advance scientific and engineering fields.</div><div>This study introduces the Cross-Mapping Coherence (CMC) method, designed to reveal causal connections in the frequency domain between time series. CMC builds upon nonlinear state-space reconstruction and extends the Convergent Cross-Mapping algorithm to the frequency domain by utilizing coherence metrics for evaluation. We tested the CMC method using simulations of logistic maps, Lorenz systems, Kuramoto oscillators, and the Wilson–Cowan model of the visual cortex. CMC accurately identified the direction of causal connections in these simulated scenarios. When applied to the Wilson–Cowan model, CMC was able to disentangle feedforward alpha and feedback gamma coupling between the V1 and V4 areas, supporting the results of previous analysis.</div><div>Furthermore, CMC could detect weak connections (<span><math><mrow><mi>C</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>05</mn></mrow></math></span>), demonstrated sample efficiency (<span><math><mrow><mi>L</mi><mo>=</mo><mn>700</mn></mrow></math></span>), and maintained robustness in the presence of noise up to <span><math><mrow><mi>SNR</mi><mo>=</mo><mn>10</mn></mrow></math></span> on unidirectionally coupled logistic map systems.</div><div>In conclusion, the capability to determine directed causal influences across different frequency bands allows CMC to provide valuable insights into the dynamics of complex, nonlinear systems.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"477 ","pages":"Article 134708"},"PeriodicalIF":2.7,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144071343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The motion and mass growth of droplets with phase transitions in a homogeneous medium","authors":"Peiyu Zhang ,&nbsp;Aifang Qu ,&nbsp;Hairong Yuan","doi":"10.1016/j.physd.2025.134683","DOIUrl":"10.1016/j.physd.2025.134683","url":null,"abstract":"<div><div>In this paper, we focus on the motion and mass growth of droplets with phase transitions in a homogeneous medium. We characterize the problem by the unsteady non-isentropic compressible Euler system together with its Radon measure-valued solutions. That is, the gas is described by the regular part of Radon measure, while the droplets are illustrated by the atomic part. The difficulty lies in finding a suitable formulation of the constitutive equation in the sense of measure, such that it is physically meaningful and mathematically reasonable. We overcome it by proposing one, which can express the process of heat release by liquefaction and heat absorption by vaporization. Then we prove the local-in-time and global-in-time existence of a single droplet with different initial data. Also, we analyze the collision of two droplets and deduce the state of the new droplet formed by collisions. This provides a downscaling new approach to investigating the two-phase flows with phase transitions.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"477 ","pages":"Article 134683"},"PeriodicalIF":2.7,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143934660","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The three-body problem on the sphere and its reversing symmetries","authors":"Abimael Bengochea,&nbsp;Ernesto Pérez-Chavela,&nbsp;Carlos Barrera-Anzaldo","doi":"10.1016/j.physd.2025.134707","DOIUrl":"10.1016/j.physd.2025.134707","url":null,"abstract":"<div><div>In this work, we establish the concept of reversing symmetry in the three-body problem on the sphere, a novel approach that has not been previously explored. We introduce three reversing symmetries: one valid for arbitrary masses, and two that require two equal masses. We also provide a thorough characterization of their fixed points, which are crucial for understanding the dynamics of the system due to their connection with the symmetric periodic orbits of the system. Using two reversing symmetries, we numerically compute a choreography in the three-body problem on the sphere, a particular type of symmetric periodic orbit. This orbit is closely related to the classical figure-eight choreography, a well-known symmetric periodic orbit in the Newtonian planar three-body problem.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"477 ","pages":"Article 134707"},"PeriodicalIF":2.7,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143942297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Perturbation and determinacy of nonsmooth systems","authors":"Sal Catsis,&nbsp;Cameron L. Hall,&nbsp;Mike R. Jeffrey","doi":"10.1016/j.physd.2025.134679","DOIUrl":"10.1016/j.physd.2025.134679","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Take a system where several variables &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; (for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;) cause decision states &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; to be set independently to values &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, at any instant, and the outcome then affects how each &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; evolves according to a differential equation. We show here that the probability that the system lies in a given decision state at any instant cannot be determined solely from these differential equations, but is determined by the emergence of a dynamical attractor. Moreover this attractor is sensitive to small perturbations in &lt;em&gt;how&lt;/em&gt; the decisions are enacted, and even how the system’s evolution is calculated. If the probability that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; decides ‘&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;’ is &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; decides ‘&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;’ is &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, for instance, the probability that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; decides ‘&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;’ and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; decides ‘&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;’ at any moment is not generally &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, despite the independence of their decisions (nor is it any other determinable quantity such as &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;). Only certain weighted sums of probabilities of being in different decision states are determined by the logic of the system.&lt;/div&gt;&lt;div&gt;This result comes from formulating this simple decisio","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"477 ","pages":"Article 134679"},"PeriodicalIF":2.7,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143934657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}