Physica D: Nonlinear Phenomena最新文献

{"title":"Insights into coupling effects of double light square bubbles on shocked hydrodynamic instability","authors":"Salman Saud Alsaeed ,&nbsp;Satyvir Singh","doi":"10.1016/j.physd.2025.134646","DOIUrl":"10.1016/j.physd.2025.134646","url":null,"abstract":"<div><div>This study investigates the coupling effects of double light square bubbles on the evolution of Richtmyer–Meshkov instability under shock interactions. Using high-fidelity numerical simulations based on a high-order modal discontinuous Galerkin solver, we analyze the influence of initial separation distance, Atwood number, and Mach number on bubble interactions, vortex formation, and instability growth. The results reveal that the coupling strength between the bubbles increases significantly as the separation distance decreases, leading to enhanced vorticity production, strong coupling jets, and intensified mixing. At larger separations, the bubbles evolve independently with minimal interaction, whereas at smaller separations, the merging of inner vortex rings and rapid enstrophy growth characterize the flow. The study further establishes a scaling law to quantify the dependence of coupling strength on separation distance, Atwood number, and Mach number, providing predictive insights into peak enstrophy generation and turbulence enhancement. The findings have important implications for understanding shock-driven hydrodynamic instabilities in inertial confinement fusion, astrophysical flows, and high-energy-density physics.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134646"},"PeriodicalIF":2.7,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143737746","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":"Compressibility effects on mixing layer in Rayleigh–Taylor turbulence","authors":"Cheng-Quan Fu ,&nbsp;Zhiye Zhao ,&nbsp;Pei Wang ,&nbsp;Nan-Sheng Liu ,&nbsp;Xi-Yun Lu","doi":"10.1016/j.physd.2025.134643","DOIUrl":"10.1016/j.physd.2025.134643","url":null,"abstract":"<div><div>The compressibility effects on the mixing layer are examined in Rayleigh–Taylor (RT) turbulence via direct numerical simulation at a high Atwood number of 0.9 and three typical Mach numbers (0.32, 0.71, and 1). The focus has been on the evolution of the mixing layer and the generation of kinetic energy. Specifically, a novel finding emerges at high Atwood number, where enhanced compressibility with increasing Mach number leads to a mean flow directed opposite to gravity in front of the bubble mixing layer. This mean flow, induced by compressibility, causes the width of the bubble layer in compressible RT turbulence to deviate from the quadratic growth observed in the incompressible case. It is further established that this deviation can be modeled by dilatation within the mean flow region. Moreover, the compressibility significantly influences the generation of global kinetic energy at high Mach numbers. The global kinetic energy of RT turbulence with high compressibility is primarily derived from the conversion of internal energy through pressure-dilatation work, rather than from the conversion of potential energy. It is also revealed that the mean flow leads to the conversion of kinetic energy into potential energy, while the fluctuating flow converts the potential energy into kinetic energy within the mixing layer.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134643"},"PeriodicalIF":2.7,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143746557","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":"Oscillatory instabilities in dynamically active signaling compartments coupled via bulk diffusion in a 3-D spherical domain","authors":"Sarafa Iyaniwura ,&nbsp;Michael J. Ward","doi":"10.1016/j.physd.2025.134645","DOIUrl":"10.1016/j.physd.2025.134645","url":null,"abstract":"<div><div>For a coupled cell-bulk ODE-PDE model in a 3-D spherical domain, we analyze oscillatory dynamics in spatially segregated dynamically active signaling compartments that are coupled through a passive extracellular bulk diffusion field. Within the confining spherical domain, the signaling compartments are a collection of small spheres of a common radius <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>ɛ</mi><mo>)</mo></mrow><mo>≪</mo><mn>1</mn></mrow></math></span>. In our cell-bulk model, each cell secretes a signaling chemical into the extracellular bulk region, while also receiving a chemical feedback that is produced by all the other cells. This secretion and global feedback of chemical into the cells is regulated by permeability parameters on the cell membrane. In the near well-mixed limit corresponding to a large bulk diffusivity <span><math><mrow><mi>D</mi><mo>=</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>/</mo><mi>ɛ</mi><mo>≫</mo><mn>1</mn></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, the method of matched asymptotic expansions is used to reduce the cell-bulk model to a novel nonlinear ODE system for the intracellular concentrations and the spatially averaged bulk diffusion field. The novelty in this ODE system, as compared to the type of ODE system that typically is studied in the well-mixed limit, is that it involves <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and an <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>ɛ</mi><mo>)</mo></mrow></mrow></math></span> correction term that incorporates the spatial configuration of the signaling compartments. For the case of Sel’kov intracellular kinetics, this new ODE system is used to study Hopf bifurcations that are triggered by the global coupling. In addition, the Kuramoto order parameter is used to study phase synchronization for the leading-order ODE system for a heterogeneous population of cells where some fraction of the cells have a random reaction-kinetic parameter. For a small collection of six cells, the spatial configuration of cells is also shown to influence both quorum-sensing behavior and diffusion-mediated communication that lead to synchronous intracellular oscillations. Moreover, we show that a single additional pacemaker cell can trigger intracellular oscillations in the other six cells, which otherwise would not occur. Finally, for the non well-mixed regime where <span><math><mrow><mi>D</mi><mo>=</mo><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, we use asymptotic analysis in the limit <span><math><mrow><mi>ɛ</mi><mo>→</mo><mn>0</mn></mrow></math></span> to derive a new integro-differential ODE system for the intracellular dynamics.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134645"},"PeriodicalIF":2.7,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143737749","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":"Small-amplitude periodic solutions in the polynomial jerk equation of arbitrary degree","authors":"Jaume Llibre ,&nbsp;Xianbo Sun","doi":"10.1016/j.physd.2025.134628","DOIUrl":"10.1016/j.physd.2025.134628","url":null,"abstract":"<div><div>A zero-Hopf singularity for a 3-dimensional differential system is a singularity for which the Jacobian matrix of the differential system evaluated at it has eigenvalues zero and <span><math><mo>±</mo></math></span>ω<span><math><mi>i</mi></math></span> with ω ≠ 0. In this paper we investigate the periodic orbits that bifurcate from a zero-Hopf singularity of the <span><math><mi>n</mi></math></span>th-degree polynomial jerk equation <span><math><mrow><mover><mrow><mi>x</mi></mrow><mrow><mo>⃛</mo></mrow></mover><mspace></mspace><mo>−</mo></mrow></math></span> ϕ<span><math><mrow><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mover><mrow><mi>x</mi></mrow><mrow><mo>̇</mo></mrow></mover><mo>,</mo><mover><mrow><mi>x</mi></mrow><mrow><mo>̈</mo></mrow></mover><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span>, where ϕ<span><math><mrow><mo>(</mo><mo>∗</mo><mo>,</mo><mo>∗</mo><mo>,</mo><mo>∗</mo><mo>)</mo></mrow></math></span> is an arbitrary <span><math><mi>n</mi></math></span>th-degree polynomial in three variables. We obtain sharp upper bounds on the maximum number of limit cycles that can emerge from such a zero-Hopf singularity using the averaging theory up to the second order. The result improves upon previous findings reported in the literature on zero-Hopf singularities and averaging theory. As an application we characterize small-amplitude periodic traveling waves in a class of generalized non-integrable Kawahara equations. This is accomplished by transforming the partial differential models into a five-dimensional dynamical system and subsequently analyzing a jerk system on a normally hyperbolic critical manifold, leveraging the averaging method and singular perturbation theory.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134628"},"PeriodicalIF":2.7,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143716160","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":"Application of tetragonal curves theory to the 4-field Błaszak–Marciniak lattice hierarchy","authors":"Qiulan Zhao,&nbsp;Caixue Li,&nbsp;Xinyue Li","doi":"10.1016/j.physd.2025.134638","DOIUrl":"10.1016/j.physd.2025.134638","url":null,"abstract":"<div><div>Through the paper, we explore the theory of tetragonal curves and derive the quasi-periodic solutions to the 4-field Błaszak–Marciniak lattice hierarchy. The hierarchy associated with a discrete fourth-order matrix spectral problem is derived from the zero-curvature equation and discrete Lenard equation. The tetragonal curve and its related Riemann theta functions are introduced through the characteristic polynomial of the Lax matrix. Additionally, the Baker-Akhiezer functions and a class of meromorphic functions on the tetragonal curve are investigated. Furthermore, the Abel map and Abelian differentials are used to straighten out various flows, leading ultimately to the quasi-periodic solutions of the 4-field Błaszak–Marciniak lattice hierarchy.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134638"},"PeriodicalIF":2.7,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143716158","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":"Effects of Atwood number and isothermal stratification strength on single-mode compressible Rayleigh–Taylor instability","authors":"Orkun Ustun ,&nbsp;Man Long Wong ,&nbsp;Denis Aslangil","doi":"10.1016/j.physd.2025.134644","DOIUrl":"10.1016/j.physd.2025.134644","url":null,"abstract":"<div><div>The coupled effects of the variable-density and compressible isothermal background stratification strength on the growth of the fully compressible single-mode two-dimensional two-fluids Rayleigh–Taylor instability (RTI) are examined using direct numerical simulations (DNS) with varying Atwood numbers, <span><math><mi>A</mi></math></span> = 0.1, 0.3, and 0.5; and different background isothermal Mach numbers, <span><math><mrow><mi>M</mi><mi>a</mi></mrow></math></span> = 0.3, 0.9, and 1.5, respectively, in the problem Reynolds number, <span><math><mrow><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span>, range of 6375 to 51000. The results show that higher stratification strength leads to more suppression of the RTI growth for the cases with a low Atwood number. However, when the Atwood number is high, the suppression effect of compressible background stratification on the RTI growth becomes nonlinear with <span><math><mrow><mi>M</mi><mi>a</mi></mrow></math></span>, and in general, it becomes weaker. Furthermore, for the case with the highest background stratification strength and highest Atwood number, we observe local supersonic regions and even shock waves with increasing <span><math><mrow><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span> at late time during the mixing. Additionally, a relevant transport equation for mixing is studied, and it is found that diffusion and production terms are dominant, and the redistribution term becomes more important with a larger Atwood number.</div><div>Vortex dynamics are also analyzed using normalized vorticity and its transport equation. It is observed that for cases at various Atwood numbers, increasing Mach number generally suppresses the growth of the vortical structures. Examining the vorticity transport equation, it is shown that the baroclinicity and viscous diffusion terms are the major contributors to the change of vorticity in cases with different combinations of <span><math><mi>A</mi></math></span> and <span><math><mrow><mi>M</mi><mi>a</mi></mrow></math></span>. In addition, with increasing <span><math><mrow><mi>M</mi><mi>a</mi></mrow></math></span>, the vorticity-dilatation term becomes more significant due to the flow compressibility effects. It is also noticeable that small-scale vortical structures become more pronounced with increasing <span><math><mrow><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span> for all Atwood numbers.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134644"},"PeriodicalIF":2.7,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761163","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":"Pair space in classical Mechanics II. N-body central configurations","authors":"Alon Drory","doi":"10.1016/j.physd.2025.134634","DOIUrl":"10.1016/j.physd.2025.134634","url":null,"abstract":"<div><div>A previous work introduced pair space, which is spanned by the center of mass of a system and the relative positions (pair positions) of its constituent bodies. Here, I show that in the <span><math><mi>N</mi></math></span>-body Newtonian problem, a configuration that does not remain on a fixed line in space is a central configuration if and only if it conserves all pair angular momenta. For collinear systems, I obtain a set of equations for the ratios of the relative distances of the bodies, from which I derive some bounds on the minimal length of the line. For the non-collinear case I derive some geometrical relations, independent of the masses of the bodies. These are necessary conditions for a non-collinear configuration to be central. They generalize, to arbitrary <span><math><mi>N</mi></math></span>, a consequence of the Dziobek relation, which holds for <span><math><mrow><mi>N</mi><mo>=</mo><mn>4</mn></mrow></math></span>.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134634"},"PeriodicalIF":2.7,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143737747","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":"Novel solitary patterns in a class of Klein–Gordon equations","authors":"Philip Rosenau ,&nbsp;Slava Krylov","doi":"10.1016/j.physd.2025.134640","DOIUrl":"10.1016/j.physd.2025.134640","url":null,"abstract":"<div><div>We study the emergence, stability and evolution of solitons and compactons in a class of Klein–Gordon equations <span><span><span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>−</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><mi>u</mi><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>n</mi></mrow></msup><mo>−</mo><msub><mrow><mi>κ</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>n</mi></mrow></msub><msup><mrow><mi>u</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>n</mi></mrow></msup><mo>,</mo><mspace></mspace><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>&lt;</mo><mi>n</mi><mo>,</mo></mrow></math></span></span></span>endowed with both trivial and non-trivial stable equilibria, and demonstrate that similarly to the classical <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>n</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math></span> cases, solitons are linearly unstable, but the instability weakens as <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>n</mi></mrow></msub><mi>↑</mi></mrow></math></span>, and vanishes at <span><math><mrow><msubsup><mrow><mi>κ</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>n</mi></mrow><mrow><mi>c</mi><mi>r</mi><mi>i</mi><mi>t</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>n</mi></mrow><mrow><msup><mrow><mrow><mo>(</mo><mn>2</mn><mo>+</mo><mi>n</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math></span>, where solitons disappear and kink forms.</div><div>As the growing unstable soliton approaches the non-trivial equilibrium, it morphs into a ’mesaton’, a robust box shaped sharp pulse with a flat-top plateau, which expands at a sonic speed. In the <span><math><msubsup><mrow><mi>κ</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>n</mi></mrow><mrow><mi>c</mi><mi>r</mi><mi>i</mi><mi>t</mi></mrow></msubsup></math></span> vicinity, where instability is suppressed, whereas the internal modes have hardly changed, solitons persist for a very long time but then, rather than turn into mesaton, convert into a breather-like formation.</div><div>Linear damping tempers the conversion and slows mesaton’s propagation. When <span><math><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>&lt;</mo><mi>n</mi><mo>&lt;</mo><mn>0</mn></mrow></math></span>, compactons emerge and being unstable morph either into a mesaton or into a breather-like formation.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134640"},"PeriodicalIF":2.7,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143724106","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":"Shock waves in an ideal gas with variable density, the radiative and conductive heat fluxes in the presence of gravitational force and magnetic field via the Lie group technique","authors":"Gorakh Nath,&nbsp;Abhay Maurya","doi":"10.1016/j.physd.2025.134637","DOIUrl":"10.1016/j.physd.2025.134637","url":null,"abstract":"<div><div>In our study, we have investigated the spherical (or cylindrical) shock waves propagation in an ideal gas with heat conduction and radiation heat flux in the presence of gravitational force and azimuthal magnetic field via the Lie group transformation technique. In this article, the heat conduction is described using Fourier’s law for heat conduction. In the case of thick gray gas model, the radiation is treated as of the diffusion type. The absorption coefficient and the thermal conductivity are considered to depend on some specific powers of density and temperature. By utilizing the Lie group transformation technique, four potential similarity solution cases were identified, in which the similarity solution exist in only one case (i.e., Case I). In this case, the shock radius follows a power law dependence on time. For this case, the similarity solutions are derived for the flow region behind the shock front, and the impact of problem physical parameters on the flow variables and on the shock strength are studied in detail. The results of this study offer a clear understanding of the influence of radiative and conductive heat transfer parameters, the gravitational parameter, the similarity exponent, and the magnetic field on the shock and on the flow dynamics behind the shock front. It is found that the shock wave decays with an increment in shock Cowling number or the heat transfer parameters or gravitational parameter. On increasing the value of similarity exponent, the strength of the shock wave increases in the non-magnetic case; whereas in the magnetic case, the shock strength reduces. It is also observed that, the shock strength is enhanced, when we change the geometry from cylindrical to spherical.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134637"},"PeriodicalIF":2.7,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143737748","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":"Dynamics of McMillan mappings I. McMillan multipoles","authors":"Tim Zolkin ,&nbsp;Sergei Nagaitsev ,&nbsp;Ivan Morozov","doi":"10.1016/j.physd.2025.134620","DOIUrl":"10.1016/j.physd.2025.134620","url":null,"abstract":"<div><div>In this article, we consider two dynamical systems: the McMillan sextupole and octupole integrable mappings, originally proposed by Edwin McMillan. Both represent the simplest symmetric McMillan maps, characterized by a single intrinsic parameter. While these systems find numerous applications across various domains of mathematics and physics, some of their dynamical properties remain unexplored. We aim to bridge this gap by providing a comprehensive description of all stable trajectories, including the parametrization of invariant curves, Poincaré rotation numbers, and canonical action–angle variables.</div><div>In the second part, we establish connections between these maps and general chaotic maps in standard form. Our investigation reveals that the McMillan sextupole and octupole serve as first-order approximations of the dynamics around the fixed point, akin to the linear map and quadratic invariant (known as the Courant–Snyder invariant in accelerator physics), which represents zeroth-order approximations (referred to as linearization). Furthermore, we propose a novel formalism for nonlinear Twiss parameters, which accounts for the dependence of rotation number on amplitude. This stands in contrast to conventional betatron phase advance used in accelerator physics, which remains independent of amplitude. Notably, in the context of accelerator physics, this new formalism demonstrates its capability in predicting dynamical aperture around low-order resonances for flat beams, a critical aspect in beam injection/extraction scenarios.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"476 ","pages":"Article 134620"},"PeriodicalIF":2.7,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143716159","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}