Pub Date : 2023-11-09DOI: 10.1080/10236198.2023.2279628
Bo Zheng, Jianshe Yu
AbstractIn this paper, we establish a discrete model with periodic parameters to depict the Wolbachia spread dynamics in mosquito populations in cyclic environments. This work modifies the models established in the existing literature that did not take into account the variation of parameters with environmental periodic changes due to seasonality and other factors. When the parameters in our model are constants, it has been extensively studied and widely used. We present a conjecture about the existence of at most two periodic solutions worthy of further study, and show that the conjecture is true for the special case of 2-periodic parameters. Numerical simulations are also provided to illustrate the occurrence of periodic phenomena.Keywords: Mosquito-borne diseasesWolbachianon-autonomous discrete modelperiodic solutionscyclic environmentsMSC(2020):: 92B0592D3037N25 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was supported by National Natural Science Foundation of China (Nos: 11971127, 12331017, 12071095, 12371484).
{"title":"<i>Wolbachia</i> spread dynamics in mosquito populations in cyclic environments","authors":"Bo Zheng, Jianshe Yu","doi":"10.1080/10236198.2023.2279628","DOIUrl":"https://doi.org/10.1080/10236198.2023.2279628","url":null,"abstract":"AbstractIn this paper, we establish a discrete model with periodic parameters to depict the Wolbachia spread dynamics in mosquito populations in cyclic environments. This work modifies the models established in the existing literature that did not take into account the variation of parameters with environmental periodic changes due to seasonality and other factors. When the parameters in our model are constants, it has been extensively studied and widely used. We present a conjecture about the existence of at most two periodic solutions worthy of further study, and show that the conjecture is true for the special case of 2-periodic parameters. Numerical simulations are also provided to illustrate the occurrence of periodic phenomena.Keywords: Mosquito-borne diseasesWolbachianon-autonomous discrete modelperiodic solutionscyclic environmentsMSC(2020):: 92B0592D3037N25 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was supported by National Natural Science Foundation of China (Nos: 11971127, 12331017, 12071095, 12371484).","PeriodicalId":15616,"journal":{"name":"Journal of Difference Equations and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135241207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-09DOI: 10.1080/10236198.2023.2279632
Ryusuke Kon
AbstractThe dynamics of discrete-time two-species systems may not be simple even if they have no positive fixed points. To reveal the behaviour of such systems, we focus on a special class of discrete-time two-species systems whose degenerate cases have a line segment of positive fixed points. As a main result, we show that the line segment of positive fixed points persists as an invariant curve under a small perturbation of the degenerate case. The absence of a positive fixed point ensures that such an invariant curve consists of heteroclinic orbits connecting two axial fixed points. The general result is applied to a discrete-time competition model of Ricker type with reproductive delay. The application reveals the existence of heteroclinic orbits connecting two axial 2-cycles, not only heteroclinic orbits connecting two axial fixed points.Keywords: Heteroclinic orbitinvariant curvecompetition systemRicker map2-cycleMathematic Subject classifications: 92-1092B0539A6039A3037N25 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was supported by JSPS KAKENHI Grant Number 20K03735, Japan.
{"title":"Invariant curves in a discrete-time two-species system","authors":"Ryusuke Kon","doi":"10.1080/10236198.2023.2279632","DOIUrl":"https://doi.org/10.1080/10236198.2023.2279632","url":null,"abstract":"AbstractThe dynamics of discrete-time two-species systems may not be simple even if they have no positive fixed points. To reveal the behaviour of such systems, we focus on a special class of discrete-time two-species systems whose degenerate cases have a line segment of positive fixed points. As a main result, we show that the line segment of positive fixed points persists as an invariant curve under a small perturbation of the degenerate case. The absence of a positive fixed point ensures that such an invariant curve consists of heteroclinic orbits connecting two axial fixed points. The general result is applied to a discrete-time competition model of Ricker type with reproductive delay. The application reveals the existence of heteroclinic orbits connecting two axial 2-cycles, not only heteroclinic orbits connecting two axial fixed points.Keywords: Heteroclinic orbitinvariant curvecompetition systemRicker map2-cycleMathematic Subject classifications: 92-1092B0539A6039A3037N25 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was supported by JSPS KAKENHI Grant Number 20K03735, Japan.","PeriodicalId":15616,"journal":{"name":"Journal of Difference Equations and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135290894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-03DOI: 10.1080/10236198.2023.2277714
Masafumi Kan, Akiko Fukuda, Sennosuke Watanabe
The recursion formula of the quotient difference algorithm for computing matrix eigenvalues corresponds with the discrete Toda equation, which is well-known in discrete integrable systems. Previous studies have revealed that the ultradiscrete Toda equation computes eigenvalues of tridiagonal matrices over the min-plus algebra. The min-plus algebra is a commutative semiring in which minimum and plus operations are introduced into the union of the set of real numbers and positive infinity. The discrete hungry Toda equation, which is a generalization of the discrete Toda equation, can compute the eigenvalues of lower Hessenberg banded matrices. This study focuses on the ultradiscrete hungry Toda equation and show that the time evolution of the equation yields the eigenvalues of lower Hessenberg banded matrices over the min-plus algebra.
{"title":"Ultradiscrete hungry Toda equation and eigenvalues over min-plus algebra","authors":"Masafumi Kan, Akiko Fukuda, Sennosuke Watanabe","doi":"10.1080/10236198.2023.2277714","DOIUrl":"https://doi.org/10.1080/10236198.2023.2277714","url":null,"abstract":"The recursion formula of the quotient difference algorithm for computing matrix eigenvalues corresponds with the discrete Toda equation, which is well-known in discrete integrable systems. Previous studies have revealed that the ultradiscrete Toda equation computes eigenvalues of tridiagonal matrices over the min-plus algebra. The min-plus algebra is a commutative semiring in which minimum and plus operations are introduced into the union of the set of real numbers and positive infinity. The discrete hungry Toda equation, which is a generalization of the discrete Toda equation, can compute the eigenvalues of lower Hessenberg banded matrices. This study focuses on the ultradiscrete hungry Toda equation and show that the time evolution of the equation yields the eigenvalues of lower Hessenberg banded matrices over the min-plus algebra.","PeriodicalId":15616,"journal":{"name":"Journal of Difference Equations and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135818736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-17DOI: 10.1080/10236198.2023.2270700
M. R. S. Kulenović, J. Marcotte, O. Merino
AbstractSufficient conditions are given for planar cooperative maps to have the qualitative global dynamics determined solely on local stability information obtained from fixed and minimal period-two points. The results are given for a class of strongly cooperative planar maps of class C1 on an order interval. The maps are assumed to have a finite number of strongly ordered fixed points, and also the strongly ordered minimal period-two points. Some applications are included.Keywords: Attractivitybasin of attractioncooperative mapdifference equationinvariant setsperiodic pointsAMS 2020 Mathematics Subject Classification:: 37C2537E3039A30 AcknowledgmentsThe Authors are grateful to two anonymous referees for their constructive comments and suggestions.Disclosure statementNo potential conflict of interest was reported by the author(s).
{"title":"Global dynamics results for a class of planar cooperative maps","authors":"M. R. S. Kulenović, J. Marcotte, O. Merino","doi":"10.1080/10236198.2023.2270700","DOIUrl":"https://doi.org/10.1080/10236198.2023.2270700","url":null,"abstract":"AbstractSufficient conditions are given for planar cooperative maps to have the qualitative global dynamics determined solely on local stability information obtained from fixed and minimal period-two points. The results are given for a class of strongly cooperative planar maps of class C1 on an order interval. The maps are assumed to have a finite number of strongly ordered fixed points, and also the strongly ordered minimal period-two points. Some applications are included.Keywords: Attractivitybasin of attractioncooperative mapdifference equationinvariant setsperiodic pointsAMS 2020 Mathematics Subject Classification:: 37C2537E3039A30 AcknowledgmentsThe Authors are grateful to two anonymous referees for their constructive comments and suggestions.Disclosure statementNo potential conflict of interest was reported by the author(s).","PeriodicalId":15616,"journal":{"name":"Journal of Difference Equations and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135993573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-17DOI: 10.1080/10236198.2023.2270536
Jian Cao, Victor J. W. Guo, Xiao Yu
AbstractWe give a new extension of a ‘divergent’ Ramanujan-type supercongruence of Guillera and Zudilin by establishing a q-analogue of this result. Our proof makes use of the ‘creative microscoping’ method, which was introduced by the second author and Zudilin in 2019. We also present a similar extension of the (L.2) supercongruence of Van Hamme in the modulus p2 case.Keywords: Supercongruencebasic hypergeometric seriescyclotomic polynomialscreative microscoping2020 Mathematics Subject Classifications: 33D1511A0711B65 Disclosure statementNo potential conflict of interest was reported by the author(s).
{"title":"A new extension of a “divergent” Ramanujan-type supercongruence","authors":"Jian Cao, Victor J. W. Guo, Xiao Yu","doi":"10.1080/10236198.2023.2270536","DOIUrl":"https://doi.org/10.1080/10236198.2023.2270536","url":null,"abstract":"AbstractWe give a new extension of a ‘divergent’ Ramanujan-type supercongruence of Guillera and Zudilin by establishing a q-analogue of this result. Our proof makes use of the ‘creative microscoping’ method, which was introduced by the second author and Zudilin in 2019. We also present a similar extension of the (L.2) supercongruence of Van Hamme in the modulus p2 case.Keywords: Supercongruencebasic hypergeometric seriescyclotomic polynomialscreative microscoping2020 Mathematics Subject Classifications: 33D1511A0711B65 Disclosure statementNo potential conflict of interest was reported by the author(s).","PeriodicalId":15616,"journal":{"name":"Journal of Difference Equations and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136033567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1080/10236198.2023.2265511
William T. Jamieson, Orlando Merino
AbstractFor a Kolmogorov map on the positive cone of Rn or an order interval determined by the origin and a positive element of Rn, sufficient conditions are given for the existence of a carrying simplex and a modified carrying simplex. Also, for a parametrized family of Kolmogorov maps that exhibits a bifurcation at the origin and that has a parametrized curve of positive fixed points defined for parameter values close to but above a critical value, sufficient conditions are given for the existence of modified carrying simplices or carrying simplices on order intervals determined by the origin and axial fixed points. In addition, sufficient conditions are given for global attractivity of the positive fixed point and of the boundary of the carrying simplex with respect to the order interval. The results are applied to the LPA (Larvae–Pupae–Adults) model investigated by J. Cushing in 2003.KEYWORDS: Discrete competitive modelretrotone mapcarrying simplexbifurcationattractivityKolmogorov mapLPA mapLyapunov function2020 MATHEMATICS SUBJECT CLASSIFICATIONS: Primary: 37C25.Secondary: 37C7037G35 AcknowledgmentsThe authors are indebted to two anonymous reviewers who did a painstaking and thoroughly in-depth job spotting mathematical errors, unclear or unsupported statements, typos, and grammatically incorrect sentences. The quality of this paper greatly improved with the reviewers' help. The suggestions given for Lemma 3.1(b) and the proof of Theorem 2.5 were particularly helpful. Also, one of the reviewers told us about the interesting references [Citation4,Citation13,Citation23], and [Citation31].Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 Definitions of retrotone map and carrying simplex in the literature often do not require the map to be a Kolmogorov map, see [Citation31]. The same level of generality is needed in Theorem 2.2 of this work.2 Definition 2.1 in [Citation20] of modified carrying simplex Σ requires Σ to be compact, but this requirement is unnecessary since it is also required that Σ is homeomorphic to the compact set {x∈R+n:x1+⋯+xn=1}.3 The condition ‘S|Σ is a homeomorphism’ is a common requirement for carrying simplices [Citation4] [Citation31,Citation34]. However, Definition 2.1 in [Citation20] of modified carrying simplex Σ does not require it, and the author of [Citation20] did not comment on why it was left out. Theorem 2.3 in [Citation20] together with Remark 2.1(b) in [Citation20] imply this condition is satisfied by Σ, so the conclusion of Theorem 2.3 is valid if the condition is added to the definition of modified carrying simplex. Our Theorem 2.2 requires the condition, so we chose to include it in the definition of modified carrying simplex given here. The resulting definition has the condition in common with Definition 6.1 in [Citation34].4 Whenever it is convenient, vectors in Rn are displayed in row vector form. However in formulas containing a multiplication of a matrix and a
{"title":"<i>n</i> -dimensional Kolmogorov maps, carrying simplices, and bifurcations at the origin","authors":"William T. Jamieson, Orlando Merino","doi":"10.1080/10236198.2023.2265511","DOIUrl":"https://doi.org/10.1080/10236198.2023.2265511","url":null,"abstract":"AbstractFor a Kolmogorov map on the positive cone of Rn or an order interval determined by the origin and a positive element of Rn, sufficient conditions are given for the existence of a carrying simplex and a modified carrying simplex. Also, for a parametrized family of Kolmogorov maps that exhibits a bifurcation at the origin and that has a parametrized curve of positive fixed points defined for parameter values close to but above a critical value, sufficient conditions are given for the existence of modified carrying simplices or carrying simplices on order intervals determined by the origin and axial fixed points. In addition, sufficient conditions are given for global attractivity of the positive fixed point and of the boundary of the carrying simplex with respect to the order interval. The results are applied to the LPA (Larvae–Pupae–Adults) model investigated by J. Cushing in 2003.KEYWORDS: Discrete competitive modelretrotone mapcarrying simplexbifurcationattractivityKolmogorov mapLPA mapLyapunov function2020 MATHEMATICS SUBJECT CLASSIFICATIONS: Primary: 37C25.Secondary: 37C7037G35 AcknowledgmentsThe authors are indebted to two anonymous reviewers who did a painstaking and thoroughly in-depth job spotting mathematical errors, unclear or unsupported statements, typos, and grammatically incorrect sentences. The quality of this paper greatly improved with the reviewers' help. The suggestions given for Lemma 3.1(b) and the proof of Theorem 2.5 were particularly helpful. Also, one of the reviewers told us about the interesting references [Citation4,Citation13,Citation23], and [Citation31].Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 Definitions of retrotone map and carrying simplex in the literature often do not require the map to be a Kolmogorov map, see [Citation31]. The same level of generality is needed in Theorem 2.2 of this work.2 Definition 2.1 in [Citation20] of modified carrying simplex Σ requires Σ to be compact, but this requirement is unnecessary since it is also required that Σ is homeomorphic to the compact set {x∈R+n:x1+⋯+xn=1}.3 The condition ‘S|Σ is a homeomorphism’ is a common requirement for carrying simplices [Citation4] [Citation31,Citation34]. However, Definition 2.1 in [Citation20] of modified carrying simplex Σ does not require it, and the author of [Citation20] did not comment on why it was left out. Theorem 2.3 in [Citation20] together with Remark 2.1(b) in [Citation20] imply this condition is satisfied by Σ, so the conclusion of Theorem 2.3 is valid if the condition is added to the definition of modified carrying simplex. Our Theorem 2.2 requires the condition, so we chose to include it in the definition of modified carrying simplex given here. The resulting definition has the condition in common with Definition 6.1 in [Citation34].4 Whenever it is convenient, vectors in Rn are displayed in row vector form. However in formulas containing a multiplication of a matrix and a ","PeriodicalId":15616,"journal":{"name":"Journal of Difference Equations and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135854325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1080/10236198.2023.2265499
B. Boldin, O. Diekmann, J. A. J. Metz
Traditionally, population models distinguish individuals on the basis of their current state. Given a distribution, a discrete time model then specifies (precisely in deterministic models, probabilistically in stochastic models) the population distribution at the next time point. The renewal equation alternative concentrates on newborn individuals and the model specifies the production of offspring as a function of age. This has two advantages: (i) as a rule, there are far fewer birth states than individual states in general, so the dimension is often low; (ii) it relates seamlessly to the next-generation matrix and the basic reproduction number. Here we start from the renewal equation for the births and use results of Feller and Thieme to characterize the asymptotic large time behaviour. Next we explicitly elaborate the relationship between the two bookkeeping schemes. This allows us to transfer the characterization of the large time behaviour to traditional structured-population models.
{"title":"Population growth in discrete time: a renewal equation oriented survey","authors":"B. Boldin, O. Diekmann, J. A. J. Metz","doi":"10.1080/10236198.2023.2265499","DOIUrl":"https://doi.org/10.1080/10236198.2023.2265499","url":null,"abstract":"Traditionally, population models distinguish individuals on the basis of their current state. Given a distribution, a discrete time model then specifies (precisely in deterministic models, probabilistically in stochastic models) the population distribution at the next time point. The renewal equation alternative concentrates on newborn individuals and the model specifies the production of offspring as a function of age. This has two advantages: (i) as a rule, there are far fewer birth states than individual states in general, so the dimension is often low; (ii) it relates seamlessly to the next-generation matrix and the basic reproduction number. Here we start from the renewal equation for the births and use results of Feller and Thieme to characterize the asymptotic large time behaviour. Next we explicitly elaborate the relationship between the two bookkeeping schemes. This allows us to transfer the characterization of the large time behaviour to traditional structured-population models.","PeriodicalId":15616,"journal":{"name":"Journal of Difference Equations and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135855544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-11DOI: 10.1080/10236198.2023.2263099
Robert Stephen Cantrell, Chris Cosner, Ying Zhou
AbstractIn this paper, we develop an integrodifference equation model that incorporates spatial memory and learning so that each year, a fraction of the population use the same dispersal kernel as the previous year, and the remaining individuals return to where they bred or were born. In temporally static environments, the equilibrium of the system corresponds to an ideal free dispersal strategy, which is evolutionarily stable. We prove local stability of this equilibrium in a special case, and we observe convergence towards this equilibrium in numerical computations. When there are periodic or stochastic temporal changes in the environment, the population is less able to match the environment, but is able to do so to some extent depending on the parameters. Overall, the mechanism proposed in this model shows a possible way for the dispersal kernel of a population to evolve towards an ideal free dispersal kernel.KEYWORDS: Integrodifference equationmathematical biologyevolution of dispersalideal free distributionspatial memorymigrationMSC: 37N2592-1092D 40 AcknowledgementWe thank the reviewers for their feedback and suggestions, which helped improve the paper.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingR. S. Cantrell and C. Cosner received support from NSF Grant DMS-18-53478.
{"title":"Evolution of dispersal by memory and learning in integrodifference equation models","authors":"Robert Stephen Cantrell, Chris Cosner, Ying Zhou","doi":"10.1080/10236198.2023.2263099","DOIUrl":"https://doi.org/10.1080/10236198.2023.2263099","url":null,"abstract":"AbstractIn this paper, we develop an integrodifference equation model that incorporates spatial memory and learning so that each year, a fraction of the population use the same dispersal kernel as the previous year, and the remaining individuals return to where they bred or were born. In temporally static environments, the equilibrium of the system corresponds to an ideal free dispersal strategy, which is evolutionarily stable. We prove local stability of this equilibrium in a special case, and we observe convergence towards this equilibrium in numerical computations. When there are periodic or stochastic temporal changes in the environment, the population is less able to match the environment, but is able to do so to some extent depending on the parameters. Overall, the mechanism proposed in this model shows a possible way for the dispersal kernel of a population to evolve towards an ideal free dispersal kernel.KEYWORDS: Integrodifference equationmathematical biologyevolution of dispersalideal free distributionspatial memorymigrationMSC: 37N2592-1092D 40 AcknowledgementWe thank the reviewers for their feedback and suggestions, which helped improve the paper.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingR. S. Cantrell and C. Cosner received support from NSF Grant DMS-18-53478.","PeriodicalId":15616,"journal":{"name":"Journal of Difference Equations and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136212658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-09DOI: 10.1080/10236198.2023.2265495
D. J. W. Simpson
AbstractIn diverse physical systems stable oscillatory solutions devolve into more complicated solutions through border-collision bifurcations. Mathematically these occur when a stable fixed point of a piecewise-smooth map collides with a switching manifold as parameters are varied. The purpose of this paper is to highlight the extreme complexity possible in the subsequent dynamics. By perturbing instances of the n-dimensional border-collision normal form for which the nth iterate is a direct product of chaotic skew tent maps, it is shown that many chaotic attractors can arise. Burnside's lemma is used to count the attractors; chaoticity is proved by demonstrating that some iterate of the map is piecewise-expanding. The resulting transition from a stable fixed point to many coexisting chaotic attractors occurs throughout open subsets of parameter space and is not destroyed by higher order terms, hence can be expected to occur generically in mathematical models.Keywords: Piecewise-linearpiecewise-smoothborder-collision bifurcationrobust chaosBurnside's lemmaMathematics Subject Classifications: 37G3539A28 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was supported by Marsden Fund contract MAU1809, managed by Royal Society Te Apārangi. The author thanks Paul Glendinning and Chris Tuffley for discussions that helped improve the results.
{"title":"Border-collision bifurcations from stable fixed points to any number of coexisting chaotic attractors","authors":"D. J. W. Simpson","doi":"10.1080/10236198.2023.2265495","DOIUrl":"https://doi.org/10.1080/10236198.2023.2265495","url":null,"abstract":"AbstractIn diverse physical systems stable oscillatory solutions devolve into more complicated solutions through border-collision bifurcations. Mathematically these occur when a stable fixed point of a piecewise-smooth map collides with a switching manifold as parameters are varied. The purpose of this paper is to highlight the extreme complexity possible in the subsequent dynamics. By perturbing instances of the n-dimensional border-collision normal form for which the nth iterate is a direct product of chaotic skew tent maps, it is shown that many chaotic attractors can arise. Burnside's lemma is used to count the attractors; chaoticity is proved by demonstrating that some iterate of the map is piecewise-expanding. The resulting transition from a stable fixed point to many coexisting chaotic attractors occurs throughout open subsets of parameter space and is not destroyed by higher order terms, hence can be expected to occur generically in mathematical models.Keywords: Piecewise-linearpiecewise-smoothborder-collision bifurcationrobust chaosBurnside's lemmaMathematics Subject Classifications: 37G3539A28 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was supported by Marsden Fund contract MAU1809, managed by Royal Society Te Apārangi. The author thanks Paul Glendinning and Chris Tuffley for discussions that helped improve the results.","PeriodicalId":15616,"journal":{"name":"Journal of Difference Equations and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135146485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-05DOI: 10.1080/10236198.2023.2260891
Ronald E. Mickens, Sandra A. Rucker
AbstractIt is known that many physical systems undergoing (nuclear, atomic, etc.) decay do not obey the standard decreasing exponential formula which corresponds to the solution of a first-order, linear ODE having constant coefficients. We propose and solve a new functional equation mathematical model whose solutions are consistent with current experimental data. The basis of our functional representation is centred on the critical role played by the concept of the decay half-life.Keywords: Exponential decaynon-exponential decaylinear functional equationsquantum mechanicsMathematics Subject Classification: 34-06 AcknowledgmentsDr. Ronald E. Mickens (REM) wishes to thank Dr. Pedro Jordan, Stennis Space Center, MI, for many useful discussions on mathematical modelling. Both REM and SAR acknowledge the critical help of Imani Beverly and Bryan Briones, Atlanta University Center, Robert W. Woodruff Library, in locating and reproducing various publications required for this investigation.Disclosure statementNo potential conflict of interest was reported by the author(s).
摘要:众所周知，许多物理系统(核、原子等)的衰变不符合标准的指数递减公式，该公式对应于一阶常系数线性ODE的解。我们提出并求解了一个新的泛函方程数学模型，其解与现有实验数据一致。我们的功能表示的基础集中在衰变半衰期概念所起的关键作用上。关键词:指数衰减非指数衰减线性泛函方程量子力学数学学科分类:34-06Ronald E. Mickens (REM)谨感谢密歇根州斯坦尼斯航天中心的Pedro Jordan博士就数学建模进行了许多有益的讨论。REM和SAR都感谢亚特兰大大学中心、Robert W. Woodruff图书馆的Imani Beverly和Bryan Briones在定位和复制本研究所需的各种出版物方面提供的重要帮助。披露声明作者未报告潜在的利益冲突。
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