Proceedings of the Royal Society of Edinburgh Section A-Mathematics最新文献

{"title":"Structural stability of transonic shock flows with an external force","authors":"Shangkun Weng, Wengang Yang","doi":"10.1017/prm.2024.39","DOIUrl":"https://doi.org/10.1017/prm.2024.39","url":null,"abstract":"This paper is devoted to the structural stability of a transonic shock passing through a flat nozzle for two-dimensional steady compressible flows with an external force. We first establish the existence and uniqueness of one-dimensional transonic shock solutions to the steady Euler system with an external force by prescribing suitable pressure at the exit of the nozzle when the upstream flow is a uniform supersonic flow. It is shown that the external force helps to stabilize the transonic shock in flat nozzles and the shock position is uniquely determined. Then we are concerned with the structural stability of these transonic shock solutions when the exit pressure is suitably perturbed. One of the new ingredients in our analysis is to use the deformation-curl decomposition to the steady Euler system developed by Weng and Xin [Sci. Sinica Math., 49 (2019), pp. 307–320] to deal with the transonic shock problem.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"4 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583459","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":"Unconditional convergence of eigenfunction expansions for abstract and elliptic operators","authors":"Vladimir Mikhailets, Aleksandr Murach","doi":"10.1017/prm.2024.40","DOIUrl":"https://doi.org/10.1017/prm.2024.40","url":null,"abstract":"We study the most general class of eigenfunction expansions for abstract normal operators with pure point spectrum in a complex Hilbert space. We find sufficient conditions for such expansions to be unconditionally convergent in spaces with two norms and also estimate the degree of this convergence. Our result essentially generalizes and complements the known theorems of Krein and of Krasnosel'skiĭ and Pustyl'nik. We apply it to normal elliptic pseudodifferential operators on compact boundaryless <jats:inline-formula> <jats:alternatives> <jats:tex-math>$C^{infty }$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000404_inline1.png\" /> </jats:alternatives> </jats:inline-formula>-manifolds. We find generic conditions for eigenfunction expansions induced by such operators to converge unconditionally in the Sobolev spaces <jats:inline-formula> <jats:alternatives> <jats:tex-math>$W^{ell }_{p}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000404_inline2.png\" /> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:tex-math>$p&gt;2$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000404_inline3.png\" /> </jats:alternatives> </jats:inline-formula> or in the spaces <jats:inline-formula> <jats:alternatives> <jats:tex-math>$C^{ell }$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000404_inline4.png\" /> </jats:alternatives> </jats:inline-formula> (specifically, for the <jats:inline-formula> <jats:alternatives> <jats:tex-math>$p$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000404_inline5.png\" /> </jats:alternatives> </jats:inline-formula>-th mean or uniform convergence on the manifold). These conditions are sufficient and necessary for the indicated convergence on Sobolev or Hörmander function classes and are given in terms of parameters characterizing these classes. We also find estimates for the degree of the convergence on such function classes. These results are new even for differential operators on the circle and for multiple Fourier series.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"45 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583349","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":"An alternative approach to solenoidal Lipschitz truncation","authors":"Stefan Schiffer","doi":"10.1017/prm.2024.38","DOIUrl":"https://doi.org/10.1017/prm.2024.38","url":null,"abstract":"In this work, we present an alternative approach to obtain a solenoidal Lipschitz truncation result in the spirit of D. Breit, L. Diening and M. Fuchs [Solenoidal Lipschitz truncation and applications in fluid mechanics. <jats:italic>J. Differ. Equ.</jats:italic> 253 (2012), 1910–1942.]. More precisely, the goal of the truncation is to modify a function <jats:inline-formula> <jats:alternatives> <jats:tex-math>$u in W^{1,p}(mathbb {R}^N;mathbb {R}^N)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000386_inline1.png\" /> </jats:alternatives> </jats:inline-formula> that satisfies the additional constraint <jats:inline-formula> <jats:alternatives> <jats:tex-math>$operatorname {div} u=0$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000386_inline2.png\" /> </jats:alternatives> </jats:inline-formula>, such that its modification <jats:inline-formula> <jats:alternatives> <jats:tex-math>$tilde {u}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000386_inline3.png\" /> </jats:alternatives> </jats:inline-formula> is Lipschitz continuous and divergence-free. This approach is different to the approaches outlined in the aforementioned work and D. Breit, L. Diening and S. Schwarzacher [Solenoidal Lipschitz truncation for parabolic PDEs. <jats:italic>Math. Models Methods Appl. Sci.</jats:italic> 23 (2013), 2671–2700, Section 4] and is able to obtain the rather strong bound on the difference between <jats:inline-formula> <jats:alternatives> <jats:tex-math>$u$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000386_inline4.png\" /> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$tilde {u}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000386_inline5.png\" /> </jats:alternatives> </jats:inline-formula> from the former article. Finally, we outline how the approach pursued in this work may be generalized to closed differential forms.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"117 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583454","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":"Unit fractions with shifted prime denominators","authors":"Thomas F. Bloom","doi":"10.1017/prm.2024.42","DOIUrl":"https://doi.org/10.1017/prm.2024.42","url":null,"abstract":"<p>We prove that any positive rational number is the sum of distinct unit fractions with denominators in <span><span><span data-mathjax-type=\"texmath\"><span>${p-1 : ptextrm { prime}}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240401120835823-0350:S0308210524000428:S0308210524000428_inline1.png\"/></span></span>. The same conclusion holds for the set <span><span><span data-mathjax-type=\"texmath\"><span>${p-h : ptextrm { prime}}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240401120835823-0350:S0308210524000428:S0308210524000428_inline2.png\"/></span></span> for any <span><span><span data-mathjax-type=\"texmath\"><span>$hin mathbb {Z}backslash {0}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240401120835823-0350:S0308210524000428:S0308210524000428_inline3.png\"/></span></span>, provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"47 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583447","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":"Skew group categories, algebras associated to Cartan matrices and folding of root lattices","authors":"Xiao-Wu Chen, Ren Wang","doi":"10.1017/prm.2024.34","DOIUrl":"https://doi.org/10.1017/prm.2024.34","url":null,"abstract":"<p>For an action of a finite group on a finite EI quiver, we construct its ‘orbifold’ quotient EI quiver. The free EI category associated to the quotient EI quiver is equivalent to the skew group category with respect to the given group action. Specializing the result to a finite group action on a finite acyclic quiver, we prove that, under reasonable conditions, the skew group category of the path category is equivalent to a finite EI category of Cartan type. If the ground field is of characteristic <span><span><span data-mathjax-type=\"texmath\"><span>$p$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327103329614-0176:S0308210524000349:S0308210524000349_inline1.png\"/></span></span> and the acting group is a cyclic <span><span><span data-mathjax-type=\"texmath\"><span>$p$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327103329614-0176:S0308210524000349:S0308210524000349_inline2.png\"/></span></span>-group, we prove that the skew group algebra of the path algebra is Morita equivalent to the algebra associated to a Cartan matrix, defined in [C. Geiss, B. Leclerc, and J. Schröer, <span>Quivers with relations for symmetrizable Cartan matrices I: Foundations</span>, Invent. Math. <span>209</span> (2017), 61–158]. We apply the Morita equivalence to construct a categorification of the folding projection between the root lattices with respect to a graph automorphism. In the Dynkin cases, the restriction of the categorification to indecomposable modules corresponds to the folding of positive roots.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"117 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583462","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 reconstruction of unknown driving forces from low-mode observations in the 2D Navier–Stokes equations","authors":"Vincent R. Martinez","doi":"10.1017/prm.2024.31","DOIUrl":"https://doi.org/10.1017/prm.2024.31","url":null,"abstract":"<p>This article is concerned with the problem of determining an unknown source of non-potential, external time-dependent perturbations of an incompressible fluid from large-scale observations on the flow field. A relaxation-based approach is proposed for accomplishing this, which makes use of a nonlinear property of the equations of motions to asymptotically enslave small scales to large scales. In particular, an algorithm is introduced that systematically produces approximations of the flow field on the unobserved scales in order to generate an approximation to the unknown force; the process is then repeated to generate an improved approximation of the unobserved scales, and so on. A mathematical proof of convergence of this algorithm is established in the context of the two-dimensional Navier–Stokes equations with periodic boundary conditions under the assumption that the force belongs to the observational subspace of phase space; at each stage in the algorithm, it is shown that the model error, represented as the difference between the approximating and true force, asymptotically decreases to zero in a geometric fashion provided that sufficiently many scales are observed and certain parameters of the algorithm are appropriately tuned.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"54 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140602045","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":"Tropical graph curves","authors":"Madhusudan Manjunath","doi":"10.1017/prm.2024.32","DOIUrl":"https://doi.org/10.1017/prm.2024.32","url":null,"abstract":"<p>We study tropical line arrangements associated to a three-regular graph <span><span><span data-mathjax-type=\"texmath\"><span>$G$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328124918952-0562:S0308210524000325:S0308210524000325_inline1.png\"/></span></span> that we refer to as <span>tropical graph curves</span>. Roughly speaking, the tropical graph curve associated to <span><span><span data-mathjax-type=\"texmath\"><span>$G$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328124918952-0562:S0308210524000325:S0308210524000325_inline2.png\"/></span></span>, whose genus is <span><span><span data-mathjax-type=\"texmath\"><span>$g$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328124918952-0562:S0308210524000325:S0308210524000325_inline3.png\"/></span></span>, is an arrangement of <span><span><span data-mathjax-type=\"texmath\"><span>$2g-2$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328124918952-0562:S0308210524000325:S0308210524000325_inline4.png\"/></span></span> lines in tropical projective space that contains <span><span><span data-mathjax-type=\"texmath\"><span>$G$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328124918952-0562:S0308210524000325:S0308210524000325_inline5.png\"/></span></span> (more precisely, the topological space associated to <span><span><span data-mathjax-type=\"texmath\"><span>$G$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328124918952-0562:S0308210524000325:S0308210524000325_inline6.png\"/></span></span>) as a deformation retract. We show the existence of tropical graph curves when the underlying graph is a three-regular, three-vertex-connected planar graph. Our method involves explicitly constructing an arrangement of lines in projective space, i.e. a graph curve whose tropicalization yields the corresponding tropical graph curve and in this case, solves a topological version of the tropical lifting problem associated to canonically embedded graph curves. We also show that the set of tropical graph curves that we construct are connected via certain local operations. These local operations are inspired by Steinitz’ theorem in polytope theory.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"21 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583616","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":"Torsion in classifying spaces of gauge groups","authors":"Masaki Kameko","doi":"10.1017/prm.2024.33","DOIUrl":"https://doi.org/10.1017/prm.2024.33","url":null,"abstract":"<p>We determine when the integral homology of the classifying space of a <span><span><span data-mathjax-type=\"texmath\"><span>$PU(n)$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327181421654-0442:S0308210524000337:S0308210524000337_inline1.png\"/></span></span>-gauge group over the sphere <span><span><span data-mathjax-type=\"texmath\"><span>$S^2$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327181421654-0442:S0308210524000337:S0308210524000337_inline2.png\"/></span></span> has torsion.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"39 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583449","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":"Generalized second order vectorial ∞-eigenvalue problems","authors":"Ed Clark, Nikos Katzourakis","doi":"10.1017/prm.2024.27","DOIUrl":"https://doi.org/10.1017/prm.2024.27","url":null,"abstract":"We consider the problem of minimizing the <jats:inline-formula> <jats:alternatives> <jats:tex-math>$L^infty$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000271_inline2.png\" /> </jats:alternatives> </jats:inline-formula> norm of a function of the hessian over a class of maps, subject to a mass constraint involving the <jats:inline-formula> <jats:alternatives> <jats:tex-math>$L^infty$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000271_inline3.png\" /> </jats:alternatives> </jats:inline-formula> norm of a function of the gradient and the map itself. We assume zeroth and first order Dirichlet boundary data, corresponding to the “hinged” and the “clamped” cases. By employing the method of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$L^p$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000271_inline4.png\" /> </jats:alternatives> </jats:inline-formula> approximations, we establish the existence of a special <jats:inline-formula> <jats:alternatives> <jats:tex-math>$L^infty$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000271_inline5.png\" /> </jats:alternatives> </jats:inline-formula> minimizer, which solves a divergence PDE system with measure coefficients as parameters. This is a counterpart of the Aronsson-Euler system corresponding to this constrained variational problem. Furthermore, we establish upper and lower bounds for the eigenvalue.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"7 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140312876","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":"Limit cycles in a rotated family of generalized Liénard systems allowing for finitely many switching lines","authors":"Hebai Chen, Yilei Tang, Weinian Zhang","doi":"10.1017/prm.2024.21","DOIUrl":"https://doi.org/10.1017/prm.2024.21","url":null,"abstract":"Analytic rotated vector fields have four significant properties: as the rotated parameter <jats:inline-formula> <jats:alternatives> <jats:tex-math>$alpha$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000210_inline1.png\" /> </jats:alternatives> </jats:inline-formula> changes, the amplitude of each stable (or unstable) limit cycle varies monotonically, each semi-stable limit cycle bifurcates at most two limit cycles, the isolated homoclinic loop (if exists) disappears while a unique limit cycle with the same stability arises or no closed orbits arise oppositely, and a unique limit cycle arises near the weak focus (if exists). In this paper, we prove that the four properties remain true for a rotated family of generalized Liénard systems having finitely many switching lines. Furthermore, we discuss variational exponent and use it to formulate multiplicity of limit cycles. Then we apply our results to give exact number of limit cycles to a continuous piecewise linear system with three zones and answer to a question on the maximum number of limit cycles in an SD oscillator.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"143 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140201014","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}