{"title":"Periodic orbits and integrability of Rocard’s system","authors":"Xinhao Hu, Yilei Tang","doi":"10.1016/j.physd.2025.134594","DOIUrl":"10.1016/j.physd.2025.134594","url":null,"abstract":"<div><div>In 1941, based on Van der Pol’s relaxation oscillator equation, physicist Yves Rocard in the book (Rocard 1941) proposed a relaxation econometric oscillator to describe cyclical oscillations in the economy. Furthermore, it was later found that the model exhibits chaotic phenomenon. Rocard’s chaotic system predates Lorenz’s discovery by 22 years, which is a three-dimensional autonomous differential system. In this paper, we research periodic orbits and integrability of Rocard’s system. We study the zero-Hopf bifurcation near equilibria and center problem on center manifolds, proving that one or three periodic orbits can bifurcate through the application of the averaging method up to arbitrary finite order, while obtaining center conditions for all equilibria through Lyapunov method. Furthermore, we investigate the integrability of Rocard’s system, which has no algebraic first integrals, Darboux polynomials, or Darboux first integrals, and is not Liouvillian integrable.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"475 ","pages":"Article 134594"},"PeriodicalIF":2.7,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143549612","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":"Automatic classification of magnetic field line topology by persistent homology","authors":"N. Bohlsen , V. Robins , M. Hole","doi":"10.1016/j.physd.2025.134595","DOIUrl":"10.1016/j.physd.2025.134595","url":null,"abstract":"<div><div>A method for the automatic classification of the orbits of magnetic field lines into topologically distinct classes using the Vietoris–Rips persistent homology is presented. The input to the method is the Poincare map orbits of field lines and the output is a separation into three classes: islands, chaotic layers, and invariant tori. The classification is tested numerically for the case of a toy model of a perturbed tokamak represented initially in its geometric coordinates. The persistent <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> data is demonstrated to be sufficient to distinguish magnetic islands from the other orbits. When combined with persistent <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> information, describing the average spacing between points on the Poincare section, the larger chaotic orbits can then be separated from very thin chaotic layers and invariant tori. It is then shown that if straight field line coordinates exist for a nearby integrable field configuration, the performance of the classification can be improved by transforming into this natural coordinate system. The focus is the application to toroidal magnetic confinement but the method is sufficiently general to apply to generic <span><math><mrow><mn>1</mn><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span>d Hamiltonian systems.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"475 ","pages":"Article 134595"},"PeriodicalIF":2.7,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143549684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Giant vortex in a harmonically-trapped rotating dipolar 164Dy condensate","authors":"Luis E. Young-S. , S.K. Adhikari","doi":"10.1016/j.physd.2025.134590","DOIUrl":"10.1016/j.physd.2025.134590","url":null,"abstract":"<div><div>We demonstrate the formation of dynamically stable giant vortices in a harmonically-trapped strongly dipolar <sup>164</sup>Dy Bose–Einstein condensate under rotation around the polarization direction of dipolar atoms, employing the numerical solution of an improved mean-field model including a Lee-Huang-Yang-type interaction, meant to stop a collapse at high atom density. These giant vortices are stationary, obtainable by imaginary-time propagation using a Gaussian initial state, while the appropriate phase of the giant vortex is imprinted on the initial wave function. The dynamical stability of the giant vortices is established by real-time propagation during a long interval of time after a small change of a parameter.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"475 ","pages":"Article 134590"},"PeriodicalIF":2.7,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143549613","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":"Large time and distance asymptotics of the one-dimensional impenetrable Bose gas and Painlevé IV transition","authors":"Zhi-Xuan Meng , Shuai-Xia Xu , Yu-Qiu Zhao","doi":"10.1016/j.physd.2025.134589","DOIUrl":"10.1016/j.physd.2025.134589","url":null,"abstract":"<div><div>In the present paper, we study the time-dependent correlation function of the one-dimensional impenetrable Bose gas, which can be expressed in terms of the Fredholm determinant of a time-dependent sine kernel and the solutions of the separated NLS equations. We derive the large time and distance asymptotic expansions of this determinant and the solutions of the separated NLS equations in both the space-like region and time-like region of the <span><math><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></math></span>-plane. Furthermore, we observe a phase transition between the asymptotic expansions in these two different regions. The phase transition is then shown to be described by a particular solution of the Painlevé IV equation.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"475 ","pages":"Article 134589"},"PeriodicalIF":2.7,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143526610","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}
Shane A. McQuarrie , Anirban Chaudhuri , Karen E. Willcox , Mengwu Guo
{"title":"Bayesian learning with Gaussian processes for low-dimensional representations of time-dependent nonlinear systems","authors":"Shane A. McQuarrie , Anirban Chaudhuri , Karen E. Willcox , Mengwu Guo","doi":"10.1016/j.physd.2025.134572","DOIUrl":"10.1016/j.physd.2025.134572","url":null,"abstract":"<div><div>This work presents a data-driven method for learning low-dimensional time-dependent physics-based surrogate models whose predictions are endowed with uncertainty estimates. We use the operator inference approach to model reduction that poses the problem of learning low-dimensional model terms as a regression of state space data and corresponding time derivatives by minimizing the residual of reduced system equations. Standard operator inference models perform well with accurate training data that are dense in time, but producing stable and accurate models when the state data are noisy and/or sparse in time remains a challenge. Another challenge is the lack of uncertainty estimation for the predictions from the operator inference models. Our approach addresses these challenges by incorporating Gaussian process surrogates into the operator inference framework to (1) probabilistically describe uncertainties in the state predictions and (2) procure analytical time derivative estimates with quantified uncertainties. The formulation leads to a generalized least-squares regression and, ultimately, reduced-order models that are described probabilistically with a closed-form expression for the posterior distribution of the operators. The resulting probabilistic surrogate model propagates uncertainties from the observed state data to reduced-order predictions. We demonstrate the method is effective for constructing low-dimensional models of two nonlinear partial differential equations representing a compressible flow and a nonlinear diffusion–reaction process, as well as for estimating the parameters of a low-dimensional system of nonlinear ordinary differential equations representing compartmental models in epidemiology.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"475 ","pages":"Article 134572"},"PeriodicalIF":2.7,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143511277","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":"Lp stability-based synchronization of delayed multi-weight neural networks under switching topologies","authors":"Yunxiao Jia, Xiaona Yang, Xian Zhang","doi":"10.1016/j.physd.2025.134577","DOIUrl":"10.1016/j.physd.2025.134577","url":null,"abstract":"<div><div>In this paper, the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> stability-based synchronization problem of delayed multi-weight neural networks under switching topologies is investigated. The involved delays include time-varying leakage, transmission and distributed delays. Firstly, a novel controller is designed to ensure <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> stability-based synchronization between the drive and response multi-weight neural networks. Secondly, a property of solutions of the considered error system is investigated, which forms a basis of obtaining a new criterion of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> stability-based synchronization. In contrast to the existing ones, the obtained criterion comprises just a small number of simple linear scalar inequalities, thereby amount of computations is greatly reduced. Finally, a numerical example related to communication networks is presented to demonstrate the applicability of the obtained <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> stability-based synchronization criterion. It is worth noting that the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> stability-based synchronization control problem of delayed multi-weight neural networks under switching topologies is solved for the first time, and the proposed method is directly based on the definitions of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> stability-based synchronization, which is easily extended to some switching delayed system models.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"475 ","pages":"Article 134577"},"PeriodicalIF":2.7,"publicationDate":"2025-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510672","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":"Exponential synchronization of high-dimensional Kuramoto models on the complex sphere based on directed graphs","authors":"Xinyun Liu , Wei Li , Xueyan Li , Yushi Shi","doi":"10.1016/j.physd.2025.134578","DOIUrl":"10.1016/j.physd.2025.134578","url":null,"abstract":"<div><div>Synchronization of populations is a common phenomenon in nature. The high-dimensional Kuramoto model is one of the most typical continuous system models for studying synchronization phenomena in multi-individual systems. Due to Lohe’s remarkable work on models of multi-individual systems, the high-dimensional Kuramoto models are also called the Lohe models, and the Lohe Hermitian sphere (LHS) model is a generalization of the Lohe models in the complex space. In this paper, we study the exponential synchronization problem of the LHS models based on directed graphs. By introducing the synchronization error function, we have developed a set of synchronization error dynamic equations for the identical oscillators using matrix Riccati differential equations. The system of synchronization error dynamic equations is studied, a total error function is constructed, and exponential synchronization of the LHS model on the unit complex sphere is demonstrated. An approximate linearization of the error dynamics equations is performed, to obtain the exponential decay rate of the system. For the LHS model with nonidentical oscillators on the unit complex sphere, using the synchronization error function, it is shown that practical synchronization can be achieved when the connection graph of the system is strongly connected.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"475 ","pages":"Article 134578"},"PeriodicalIF":2.7,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510673","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":"On periodic solutions for the Maxwell–Bloch equations","authors":"A.I. Komech","doi":"10.1016/j.physd.2025.134581","DOIUrl":"10.1016/j.physd.2025.134581","url":null,"abstract":"<div><div>We consider the Maxwell–Bloch system which is a finite-dimensional approximation of the coupled nonlinear Maxwell–Schrödinger equations. The approximation consists of one-mode Maxwell field coupled to <span><math><mrow><mi>N</mi><mo>≥</mo><mn>1</mn></mrow></math></span> two-level molecules. Our main result is the existence of solutions with time-periodic Maxwell field. For the proof we construct time-periodic solutions to the reduced system with respect to the symmetry gauge group <span><math><mrow><mi>U</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. The solutions correspond to fixed points of the Poincaré map, which are constructed using the contraction of high-amplitude Maxwell field and the Lefschetz theorem. The theorem is applied to a suitable <em>modification</em> of the reduced equations which defines a smooth dynamics on the <em>compactified</em> phase space. The crucial role is played by the fact that the Euler characteristic of the compactified space is strictly greater than the same of the infinite component.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"475 ","pages":"Article 134581"},"PeriodicalIF":2.7,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143463836","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":"On the nonexistence of NLS breathers","authors":"Miguel Á. Alejo , Adán J. Corcho","doi":"10.1016/j.physd.2025.134580","DOIUrl":"10.1016/j.physd.2025.134580","url":null,"abstract":"<div><div>In this work, a rigorous proof of the nonexistence of breather solutions for NLS equations is presented. By using suitable virial functionals, we are able to characterize the nonexistence of breather solutions, different from standing waves, by only using their inner energy and the power of the corresponding nonlinearity of the equation. We extend this result for several NLS models with different power nonlinearities and even the derivative and logarithmic NLS equations.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"475 ","pages":"Article 134580"},"PeriodicalIF":2.7,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474342","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}
Farel William Viret Kharchandy, Vamsinadh Thota, Jitraj Saha
{"title":"Existence, stability and nonlinear estimates of stationary-state solutions to the nonlinear aggregation with collision-induced fragmentation model","authors":"Farel William Viret Kharchandy, Vamsinadh Thota, Jitraj Saha","doi":"10.1016/j.physd.2025.134579","DOIUrl":"10.1016/j.physd.2025.134579","url":null,"abstract":"<div><div>Existence and uniqueness of a stationary-state solution to the nonlinear aggregation and collision-induced fragmentation equation is proved over a weighted <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-space. The assumption of a detailed balance condition is relaxed to attain the existence of the solution. Aggregation and fragmentation kernels are considered to exhibit linear and quadratic growth rates respectively which encompass a wide range of physically significant kernels. Asymptotic properties of the time-dependent solution are analyzed in detail and convergence of the same to the stationary-state solution is also examined. Exponential rate of convergence is obtained by proving the asymptotic stability of the stationary-state solution. Further, nonlinear estimates of the solution are obtained using semigroup theory of operators. The study is further extended to analyze the nonexistence of a stationary-state solution for a particular choice of kinetic kernels over a suitably constructed solution space. A numerical example is provided in order to visualize the nonexistence of a stationary-state solution and other physical quantities.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"475 ","pages":"Article 134579"},"PeriodicalIF":2.7,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143464731","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}