Nonlinearity最新文献

{"title":"Parametric approximations of fast close encounters of the planar three-body problem as arcs of a focus-focus dynamics","authors":"Massimiliano Guzzo","doi":"10.1088/1361-6544/ad72c6","DOIUrl":"https://doi.org/10.1088/1361-6544/ad72c6","url":null,"abstract":"A gravitational close encounter of a small body with a planet may produce a substantial change of its orbital parameters which can be studied using the circular restricted three-body problem. In this paper we provide parametric representations of the fast close encounters with the secondary body of the planar CRTBP as arcs of non-linear focus-focus dynamics. The result is the consequence of a remarkable factorisation of the Birkhoff normal forms of the Hamiltonian of the problem represented with the Levi–Civita regularisation. The parameterisations are computed using two different sequences of Birkhoff normalisations of given order <italic toggle=\"yes\">N</italic>. For each value of <italic toggle=\"yes\">N</italic>, the Birkhoff normalisations and the parameters of the focus-focus dynamics are represented by polynomials whose coefficients can be computed iteratively with a computer algebra system; no quadratures, such as those needed to compute action-angle variables of resonant normal forms, are needed. We also provide some numerical demonstrations of the method for values of the mass parameter representative of the Sun–Earth and the Sun–Jupiter cases.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"41 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Simultaneous local normal forms of dynamical systems with singular underlying geometric structures","authors":"Kai Jiang, Tudor S Ratiu and Nguyen Tien Zung","doi":"10.1088/1361-6544/ad700d","DOIUrl":"https://doi.org/10.1088/1361-6544/ad700d","url":null,"abstract":"The aim of this paper is to develop, for the first time, a general theory of simultaneous local normalisation of couples , where X is a dynamical system (vector field) and is an underlying geometric structure preserved by X, even if both have singularities. Such couples appear naturally in many problems, e.g. Hamiltonian dynamics, where is a symplectic structure and one has the theory of Birkhoff normal forms, or constrained dynamics, where is a smooth, in general singular, distribution of tangent subspaces, etc. In this paper, the geometric structure is of the following types: volume form, symplectic form, contact form, Poisson tensor, as well as their singular versions. The paper addresses mainly the more difficult situations when both X and are singular at a point and its results prove the existence of natural simultaneous normal forms in these cases. In general, the normalisation is only formal, but when and X are (real or complex) analytic and X is analytically or Darboux integrable, then the simultaneous normalisation is also analytic. Our theory is based on a new approach, called the Toric Conservation Principle, as well as the classical step-by-step normalisation technique, and the equivariant path method.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"15 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201389","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains","authors":"Alberto Boscaggin, Francesca Colasuonno, Benedetta Noris, Tobias Weth","doi":"10.1088/1361-6544/ad74d0","DOIUrl":"https://doi.org/10.1088/1361-6544/ad74d0","url":null,"abstract":"We deal with the following semilinear equation in exterior domains <inline-formula>\u0000<tex-math><?CDATA $ -Delta u + u = aleft(xright)|u|^{p-2}u,qquad uin H^1_0left({A_R}right),$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mtable columnalign=\"left\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mstyle scriptlevel=\"0\"></mml:mstyle><mml:mi>u</mml:mi><mml:mo>∈</mml:mo><mml:msubsup><mml:mi>H</mml:mi><mml:mn>0</mml:mn><mml:mn>1</mml:mn></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>R</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><inline-graphic xlink:href=\"nonad74d0ueqn1.gif\"></inline-graphic></inline-formula> where <inline-formula>\u0000<tex-math><?CDATA ${A_R} : = {xinmathbb{R}^N:, |x| gt {R}}$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>R</mml:mi></mml:msub></mml:mrow><mml:mo>:=</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo>:</mml:mo><mml:mstyle scriptlevel=\"0\"></mml:mstyle><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>x</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>&gt;</mml:mo><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"nonad74d0ieqn1.gif\"></inline-graphic></inline-formula>, <inline-formula>\u0000<tex-math><?CDATA $Nunicode{x2A7E} 3$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mi>N</mml:mi><mml:mtext>⩾</mml:mtext><mml:mn>3</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href=\"nonad74d0ieqn2.gif\"></inline-graphic></inline-formula>, <italic toggle=\"yes\">R</italic> &gt; 0. Assuming that the weight <italic toggle=\"yes\">a</italic> is positive and satisfies some symmetry and monotonicity properties, we exhibit a positive solution having the same features as <italic toggle=\"yes\">a</italic>, for values of <italic toggle=\"yes\">p</italic> &gt; 2 in a suitable range that includes exponents greater than the standard Sobolev critical one. In the special case of radial weight <italic toggle=\"yes\">a</italic>, our existence result ensures multiplicity of nonradial solutions. We also provide an existence result for supercritical <italic toggle=\"yes\">p</italic> in nonradial exterior domains.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"23 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spectral triples and Dixmier trace representations of Gibbs measures: theory and examples","authors":"L Cioletti, L Y Hataishi, A O Lopes, M Stadlbauer","doi":"10.1088/1361-6544/ad7009","DOIUrl":"https://doi.org/10.1088/1361-6544/ad7009","url":null,"abstract":"In this paper we study spectral triples and non-commutative expectations associated to expanding and weakly expanding maps. In order to do so, we generalise the Perron–Frobenius–Ruelle theorem and obtain a polynomial decay of the operator, which allows to prove differentiability of a dynamically defined <italic toggle=\"yes\">ζ</italic>-function at its critical parameter. We then generalise Sharp’s construction of spectral triples to this setting and provide criteria when the associated spectral metric is non-degenerate and when the non-commutative expectation of the spectral triple is colinear to the integration with respect to the associated equilibrium state from thermodynamic formalism. Due to our general setting, we are able to simultaneously analyse expanding maps on manifolds or connected fractals, subshifts of finite type as well as the Dyson model from statistical physics, which underlines the unifying character of noncommutative geometry. Furthermore, we derive an explicit representation of the <italic toggle=\"yes\">ζ</italic>-function associated to a particular class of pathological continuous potentials, giving rise to examples where the representation as a non-commutative expectation via the associated zeta function holds, and others where it does not hold.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"92 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New notion of nonuniform exponential dichotomy with applications to the theory of pullback and forward attractors","authors":"José A Langa, Rafael Obaya, Alexandre N Oliveira-Sousa","doi":"10.1088/1361-6544/ad700a","DOIUrl":"https://doi.org/10.1088/1361-6544/ad700a","url":null,"abstract":"In this work we study nonuniform exponential dichotomies and existence of pullback and forward attractors for evolution processes associated to nonautonomous differential equations. We define a new concept of nonuniform exponential dichotomy, for which we provide several examples, study the relation with the standard notion, and establish a robustness under perturbations. We provide a dynamical interpretation of admissibility pairs related with exponential dichotomies to obtain existence of pullback and forward attractors. We apply these abstract results for ordinary and parabolic differential equations.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"32 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Continuity of the Lyapunov exponents of non-invertible random cocycles with constant rank","authors":"Pedro Duarte, Catalina Freijo","doi":"10.1088/1361-6544/ad72c7","DOIUrl":"https://doi.org/10.1088/1361-6544/ad72c7","url":null,"abstract":"In this paper we establish uniform large deviations estimates of exponential type and Hölder continuity of the Lyapunov exponents for random non-invertible cocycles with constant rank.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"6 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201416","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Singular limit of a chemotaxis model with indirect signal production and phenotype switching","authors":"Philippe Laurençot, Christian Stinner","doi":"10.1088/1361-6544/ad6bdf","DOIUrl":"https://doi.org/10.1088/1361-6544/ad6bdf","url":null,"abstract":"Convergence of solutions to a partially diffusive chemotaxis system with indirect signal production and phenotype switching is shown in a two-dimensional setting when the switching rate increases to infinity, thereby providing a rigorous justification of formal computations performed in the literature. The expected limit system being the classical parabolic–parabolic Keller–Segel system, the obtained convergence is restricted to a finite time interval for general initial conditions but valid for arbitrary bounded time intervals when the mass of the initial condition is appropriately small. Furthermore, if the solution to the limit system blows up in finite time, then neither of the two phenotypes in the partially diffusive system can be uniformly bounded with respect to the <italic toggle=\"yes\">L</italic>₂-norm beyond that time.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"27 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Extending discrete geometric singular perturbation theory to non-hyperbolic points","authors":"S Jelbart, C Kuehn","doi":"10.1088/1361-6544/ad72c5","DOIUrl":"https://doi.org/10.1088/1361-6544/ad72c5","url":null,"abstract":"We extend the recently developed <italic toggle=\"yes\">discrete geometric singular perturbation theory</italic> to the non-normally hyperbolic regime. Our primary tool is the <italic toggle=\"yes\">Takens embedding theorem</italic>, which provides a means of approximating the dynamics of particular maps with the time-1 map of a formal vector field. First, we show that the so-called <italic toggle=\"yes\">reduced map</italic>, which governs the slow dynamics near slow manifolds in the normally hyperbolic regime, can be locally approximated by the time-1 map of the reduced vector field which appears in continuous-time geometric singular perturbation theory. In the non-normally hyperbolic regime, we show that the dynamics of fast-slow maps with a unipotent linear part can be locally approximated by the time-1 map induced by a fast-slow vector field in the same dimension, which has a nilpotent singularity of the corresponding type. The latter result is used to describe (i) the local dynamics of two-dimensional fast-slow maps with non-normally singularities of regular fold, transcritical and pitchfork type, and (ii) dynamics on a (potentially high-dimensional) local center manifold in <italic toggle=\"yes\">n</italic>-dimensional fast-slow maps with regular contact or fold submanifolds of the critical manifold. In general, our results show that the dynamics near a large and important class of singularities in fast-slow maps can be described via the use of formal embedding theorems which allow for their approximation by the time-1 map of a fast-slow vector field featuring a loss of normal hyperbolicity.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"41 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Complex-plane singularity dynamics for blow up in a nonlinear heat equation: analysis and computation","authors":"M Fasondini, J R King, J A C Weideman","doi":"10.1088/1361-6544/ad700b","DOIUrl":"https://doi.org/10.1088/1361-6544/ad700b","url":null,"abstract":"Blow-up solutions to a heat equation with spatial periodicity and a quadratic nonlinearity are studied through asymptotic analyses and a variety of numerical methods. The focus is on tracking the dynamics of the singularities in the complexified space domain all the way from the initial time until the blow-up time, which occurs when the singularities reach the real axis. This widely applicable approach gives forewarning of the possibility of blow up and an understanding of the influence of singularities on the solution behaviour on the real axis, aiding the (perhaps surprisingly involved) asymptotic analysis of the real-line behaviour. The analysis provides a distinction between small and large nonlinear effects, as well as insight into the various time scales over which blow up is approached. The solution to the nonlinear heat equation in the complex spatial plane is shown to be related asymptotically to a nonlinear ordinary differential equation. This latter equation is studied in detail, including its computation on multiple Riemann sheets, providing further insight into the singularities of blow-up solutions of the nonlinear heat equation when viewed as multivalued functions in the complex space domain and illustrating the potential intricacy of singularity dynamics in such (non-integrable) nonlinear contexts.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"5 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201419","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Nijenhuis geometry IV: conservation laws, symmetries and integration of certain non-diagonalisable systems of hydrodynamic type in quadratures","authors":"Alexey V Bolsinov, Andrey Yu Konyaev, Vladimir S Matveev","doi":"10.1088/1361-6544/ad6acc","DOIUrl":"https://doi.org/10.1088/1361-6544/ad6acc","url":null,"abstract":"The paper contains two lines of results: the first one is a study of symmetries and conservation laws of <inline-formula>\u0000<tex-math><?CDATA $mathrm{gl}$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mrow><mml:mi>gl</mml:mi></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href=\"nonad6accieqn1.gif\"></inline-graphic></inline-formula>-regular Nijenhuis operators. We prove the splitting theorem for symmetries and conservation laws of Nijenhuis operators, show that the space of symmetries of a <inline-formula>\u0000<tex-math><?CDATA $mathrm{gl}$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mrow><mml:mi>gl</mml:mi></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href=\"nonad6accieqn2.gif\"></inline-graphic></inline-formula>-regular Nijenhuis operator forms a commutative algebra with respect to (pointwise) matrix multiplication. Moreover, all the elements of this algebra are strong symmetries of each other. We establish a natural relationship between symmetries and conservation laws of a <inline-formula>\u0000<tex-math><?CDATA $mathrm{gl}$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mrow><mml:mi>gl</mml:mi></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href=\"nonad6accieqn3.gif\"></inline-graphic></inline-formula>-regular Nijenhuis operator and systems of the first and second companion coordinates. Moreover, we show that the space of conservation laws is naturally related to the space of symmetries in the sense that any conservation laws can be obtained from a single conservation law by multiplication with an appropriate symmetry. In particular, we provide an explicit description of all symmetries and conservation laws for <inline-formula>\u0000<tex-math><?CDATA $mathrm{gl}$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mrow><mml:mi>gl</mml:mi></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href=\"nonad6accieqn4.gif\"></inline-graphic></inline-formula>-regular operators at algebraically generic points. The second line of results contains an application of the theoretical part to a certain system of partial differential equations of hydrodynamic type, which was previously studied by different authors, but mainly in the diagonalisable case. We show that this system is integrable in quadratures, i.e. its solutions can be found for almost all initial curves by integrating closed 1-forms and solving some systems of functional equations. The system is not diagonalisable in general, and construction and integration of such systems is an actively studied and explicitly stated problem in the literature.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"70 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}