Pub Date : 2026-01-09DOI: 10.1016/j.nonrwa.2025.104597
Hao Chen , Rui Yang
In this paper, we study the global bifurcation diagrams and multiplicity of positive solutions for the one-dimensional Minkowski-curvature equation with singular nonlinearitywhere λ, p, L are three positive parameters. We determine the number of positive solutions corresponding to the vary of the values of these parameters based on time-map approach. For this open question, we observe that the global bifurcation diagrams differ significantly between two cases: Case 1. For 0 < L < 1, |u′| < 1 ensures u < L < 1, resulting in a non-singular problem; Case 2. For L ≥ 1, additional condition u < 1 is needed to avoid the occurrence of singularity. We show that the bifurcation curve of positive solutions either is strictly increasing or S-like shaped in the first case, while the bifurcation curve is more complicated, strictly increasing, S-like shaped, or ⊃-like shaped in the other one.
{"title":"Global bifurcation diagrams and multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with singular nonlinearity","authors":"Hao Chen , Rui Yang","doi":"10.1016/j.nonrwa.2025.104597","DOIUrl":"10.1016/j.nonrwa.2025.104597","url":null,"abstract":"<div><div>In this paper, we study the global bifurcation diagrams and multiplicity of positive solutions for the one-dimensional Minkowski-curvature equation with singular nonlinearity<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mrow><mo>−</mo><msup><mrow><mo>(</mo><mstyle><mfrac><msup><mi>u</mi><mo>′</mo></msup><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>u</mi><mrow><mo>′</mo><mn>2</mn></mrow></msup></mrow></msqrt></mfrac></mstyle><mo>)</mo></mrow><mo>′</mo></msup><mo>=</mo><mstyle><mfrac><mi>λ</mi><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>u</mi><mo>)</mo></mrow><mi>p</mi></msup></mfrac></mstyle><mo>,</mo></mrow></mtd><mtd><mrow><mtext>in</mtext><mspace></mspace><mo>(</mo><mo>−</mo><mi>L</mi><mo>,</mo><mi>L</mi><mo>)</mo><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>u</mi><mo>(</mo><mo>−</mo><mi>L</mi><mo>)</mo><mo>=</mo><mi>u</mi><mo>(</mo><mi>L</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math></span></span></span>where <em>λ, p, L</em> are three positive parameters. We determine the number of positive solutions corresponding to the vary of the values of these parameters based on time-map approach. For this open question, we observe that the global bifurcation diagrams differ significantly between two cases: Case 1. For 0 < <em>L</em> < 1, |<em>u</em>′| < 1 ensures <em>u</em> < <em>L</em> < 1, resulting in a non-singular problem; Case 2. For <em>L</em> ≥ 1, additional condition <em>u</em> < 1 is needed to avoid the occurrence of singularity. We show that the bifurcation curve of positive solutions either is strictly increasing or <em>S</em>-like shaped in the first case, while the bifurcation curve is more complicated, strictly increasing, <em>S</em>-like shaped, or ⊃-like shaped in the other one.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104597"},"PeriodicalIF":1.8,"publicationDate":"2026-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145926982","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}
Pub Date : 2026-01-07DOI: 10.1016/j.nonrwa.2025.104600
Andaluzia Matei
In the present paper we draw attention to a strongly coupled nonlinear system consisting of two variational inequalities. Such a system can arise from weak formulations of contact models with implicit material laws governed by non additively-separable g-bipotentials. A multi-contact model applying to an implicit standard material illustrates the theory. Firstly, we deliver abstract results. Then, we apply the abstract results to the well-posedness of the multi-contact model under consideration.
{"title":"Nonlinear variational systems related to contact models with implicit material laws","authors":"Andaluzia Matei","doi":"10.1016/j.nonrwa.2025.104600","DOIUrl":"10.1016/j.nonrwa.2025.104600","url":null,"abstract":"<div><div>In the present paper we draw attention to a strongly coupled nonlinear system consisting of two variational inequalities. Such a system can arise from weak formulations of contact models with implicit material laws governed by non additively-separable g-bipotentials. A multi-contact model applying to an implicit standard material illustrates the theory. Firstly, we deliver abstract results. Then, we apply the abstract results to the well-posedness of the multi-contact model under consideration.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104600"},"PeriodicalIF":1.8,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145926981","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}
Pub Date : 2026-01-06DOI: 10.1016/j.nonrwa.2025.104599
Youseung Cho, Minsuk Yang
We study smooth solutions to the three-dimensional stationary Navier–Stokes equations and establish new Liouville-type theorems under refined decay assumptions. Building on the work of Cho et al., we introduce a refinement to previously known integrability criteria and analyze the associated averaged quantities. Our main result shows that if the Lp growth rate of a solution remains bounded for some 3/2 < p < 3, then the solution must be trivial. The proof combines averaged decay estimates, energy inequalities, and an iteration scheme.
我们研究了三维平稳Navier-Stokes方程的光滑解，并在精细衰变假设下建立了新的liouville型定理。在Cho等人的工作基础上，我们引入了对先前已知的可积性准则的改进，并分析了相关的平均量。我们的主要结果表明，如果一个解的Lp增长率在3/2的范围内保持有界 <； p <； 3，那么该解一定是平凡的。该证明结合了平均衰减估计、能量不等式和迭代方案。
{"title":"Refined Liouville-type theorems for the stationary Navier–Stokes equations","authors":"Youseung Cho, Minsuk Yang","doi":"10.1016/j.nonrwa.2025.104599","DOIUrl":"10.1016/j.nonrwa.2025.104599","url":null,"abstract":"<div><div>We study smooth solutions to the three-dimensional stationary Navier–Stokes equations and establish new Liouville-type theorems under refined decay assumptions. Building on the work of Cho et al., we introduce a refinement to previously known integrability criteria and analyze the associated averaged quantities. Our main result shows that if the <em>L<sup>p</sup></em> growth rate of a solution remains bounded for some 3/2 < <em>p</em> < 3, then the solution must be trivial. The proof combines averaged decay estimates, energy inequalities, and an iteration scheme.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104599"},"PeriodicalIF":1.8,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145926980","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}
Pub Date : 2026-01-03DOI: 10.1016/j.nonrwa.2025.104598
Fizay-Noah Lee
We consider the steady state Nernst-Planck system for multiple species with nonequilibrium boundary conditions, describing electrodiffusion of ions or charged particles. We show the existence and regularity of solutions and also establish a sufficient condition for uniqueness.
{"title":"Existence, uniqueness and regularity of nonequilibrium steady states in multispecies ion transport","authors":"Fizay-Noah Lee","doi":"10.1016/j.nonrwa.2025.104598","DOIUrl":"10.1016/j.nonrwa.2025.104598","url":null,"abstract":"<div><div>We consider the steady state Nernst-Planck system for multiple species with nonequilibrium boundary conditions, describing electrodiffusion of ions or charged particles. We show the existence and regularity of solutions and also establish a sufficient condition for uniqueness.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104598"},"PeriodicalIF":1.8,"publicationDate":"2026-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885260","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}
Pub Date : 2025-12-30DOI: 10.1016/j.nonrwa.2025.104582
Shri Harine P, Ankit Kumar
Classical prey-predator models often assume that the predator’s handling time is constant. However, in real ecosystems, a predator’s handling time can vary due to several biotic and abiotic factors. Based on this, we modified the Holling Type II functional response by incorporating a nonlinear handling time function. Fear in prey can lead to notable population reductions, predominantly through decreased foraging and reproduction. Considering these essential factors, we developed a prey-predator model encompassing temporal dynamics, self-diffusion and cross-diffusion. For the temporal model, we investigated the non-negativity, boundedness, and stability conditions of the existing steady states. Furthermore, bifurcations such as Hopf, transcritical, and Bautin were observed with respect to parameters like the cost of fear and the maximal achievable handling time. Bistability behaviour was observed through the analysis involving these two parameters. Sensitivity analysis was conducted to understand the influence of parameters contributing to the coexistence of prey and predator populations. Stability conditions for both spatiotemporal models (with self and cross-diffusion) were established, highlighting the role of cross-diffusion coefficients in inducing Turing instability and pattern formation. Spatial patterns such as spots and vertically aligned chains were observed. An increase in the maximal achievable handling time was found to support prey occupation in high-density regions, promoting coexistence, whereas excessively high maximal handling time can lead to predator extinction.
{"title":"Modelling spatiotemporal prey-predator interactions incorporating fear effect and variable handling time","authors":"Shri Harine P, Ankit Kumar","doi":"10.1016/j.nonrwa.2025.104582","DOIUrl":"10.1016/j.nonrwa.2025.104582","url":null,"abstract":"<div><div>Classical prey-predator models often assume that the predator’s handling time is constant. However, in real ecosystems, a predator’s handling time can vary due to several biotic and abiotic factors. Based on this, we modified the Holling Type II functional response by incorporating a nonlinear handling time function. Fear in prey can lead to notable population reductions, predominantly through decreased foraging and reproduction. Considering these essential factors, we developed a prey-predator model encompassing temporal dynamics, self-diffusion and cross-diffusion. For the temporal model, we investigated the non-negativity, boundedness, and stability conditions of the existing steady states. Furthermore, bifurcations such as Hopf, transcritical, and Bautin were observed with respect to parameters like the cost of fear and the maximal achievable handling time. Bistability behaviour was observed through the analysis involving these two parameters. Sensitivity analysis was conducted to understand the influence of parameters contributing to the coexistence of prey and predator populations. Stability conditions for both spatiotemporal models (with self and cross-diffusion) were established, highlighting the role of cross-diffusion coefficients in inducing Turing instability and pattern formation. Spatial patterns such as spots and vertically aligned chains were observed. An increase in the maximal achievable handling time was found to support prey occupation in high-density regions, promoting coexistence, whereas excessively high maximal handling time can lead to predator extinction.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104582"},"PeriodicalIF":1.8,"publicationDate":"2025-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885263","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}
Pub Date : 2025-12-30DOI: 10.1016/j.nonrwa.2025.104587
Samuel Carlos S. Ferreira , Bruno R. Freitas , João Carlos R. Medrado
We analyze a 3D discontinuous piecewise linear dynamical system, , with a plane Σ as its switching manifold, which contains two-fold intersection straight lines. The eigenvalues associated with DX and DY are composed of one real eigenvalue and a pair of complex conjugate eigenvalues. A canonical form is obtained using changes in variables and parameters. Two half-return Poincaré maps are generated from two closing equations derived from exponential matrices, leading to a displacement map Δ. Using the Weierstrass Preparation Theorem, we prove the existence of a subclass within this family that admits at least one large amplitude limit cycle. When the real part of the complex eigenvalues is non-zero, the restriction of Δ to space W, formed by the concatenation of bi-dimensional focal planes associated with the complex eigenvalues, can have up to three positive zeros on W ∩ Σ, corresponding to three large amplitude limit cycles. We provide examples with one, two, and three limit cycles.
{"title":"Limit cycles of 3D piecewise linear systems with concurrent tangent lines","authors":"Samuel Carlos S. Ferreira , Bruno R. Freitas , João Carlos R. Medrado","doi":"10.1016/j.nonrwa.2025.104587","DOIUrl":"10.1016/j.nonrwa.2025.104587","url":null,"abstract":"<div><div>We analyze a 3D discontinuous piecewise linear dynamical system, <span><math><mrow><mi>Z</mi><mo>=</mo><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></mrow></math></span>, with a plane Σ as its switching manifold, which contains two-fold intersection straight lines. The eigenvalues associated with <em>DX</em> and <em>DY</em> are composed of one real eigenvalue and a pair of complex conjugate eigenvalues. A canonical form is obtained using changes in variables and parameters. Two half-return Poincaré maps are generated from two closing equations derived from exponential matrices, leading to a displacement map Δ. Using the Weierstrass Preparation Theorem, we prove the existence of a subclass within this family that admits at least one large amplitude limit cycle. When the real part of the complex eigenvalues is non-zero, the restriction of Δ to space <em>W</em>, formed by the concatenation of bi-dimensional focal planes associated with the complex eigenvalues, can have up to three positive zeros on <em>W</em> ∩ Σ, corresponding to three large amplitude limit cycles. We provide examples with one, two, and three limit cycles.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104587"},"PeriodicalIF":1.8,"publicationDate":"2025-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885261","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}
Pub Date : 2025-12-27DOI: 10.1016/j.nonrwa.2025.104586
Chun-Bo Lian , Bin Ge , Qing-Hai Cao , Qing-Mei Zhou
In the present paper, we study the existence of a radial sign-changing solution to a class of quasilinear double phase problems in . Without assuming the usual strictly increasing condition on , we provide some sufficient conditions under which the above problems have at least one sign-changing radial ground states solution. We generalize the result of Liu and Dai [J. Math. Phys. 61(2020) 091508].
{"title":"Quasilinear double phase problem on the whole space","authors":"Chun-Bo Lian , Bin Ge , Qing-Hai Cao , Qing-Mei Zhou","doi":"10.1016/j.nonrwa.2025.104586","DOIUrl":"10.1016/j.nonrwa.2025.104586","url":null,"abstract":"<div><div>In the present paper, we study the existence of a radial sign-changing solution to a class of quasilinear double phase problems in <span><math><msup><mi>R</mi><mi>N</mi></msup></math></span>. Without assuming the usual strictly increasing condition on <span><math><mfrac><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><msup><mrow><mo>|</mo><mi>t</mi><mo>|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup></mfrac></math></span>, we provide some sufficient conditions under which the above problems have at least one sign-changing radial ground states solution. We generalize the result of Liu and Dai [J. Math. Phys. 61(2020) 091508].</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104586"},"PeriodicalIF":1.8,"publicationDate":"2025-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885262","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}
Pub Date : 2025-12-25DOI: 10.1016/j.nonrwa.2025.104572
Guanghui Wang , Mingying Zhong
The pointwise estimates of the Green’s function for the incompressible Navier–Stokes–Maxwell system with Ohm’s law in 3D are given in this paper. It is shown that the Green’s function consists of the heat kernels, the diffusive waves at low-frequency, the hyperbolic waves at high-frequency with time decaying exponentially, and the singular short waves. In addition, we establish the pointwise estimate of the global solution to the nonlinear incompressible Navier–Stokes–Maxwell system with Ohm’s law based on the Green’s function. To solve the new problem that the nonlinear terms contain the nonlocal operators and which arise from the fluid-electromagnetic decomposition, we develop some new estimates of the nonlocal operators.
{"title":"The pointwise estimates for the incompressible Navier–Stokes–Maxwell system with Ohm’s law","authors":"Guanghui Wang , Mingying Zhong","doi":"10.1016/j.nonrwa.2025.104572","DOIUrl":"10.1016/j.nonrwa.2025.104572","url":null,"abstract":"<div><div>The pointwise estimates of the Green’s function for the incompressible Navier–Stokes–Maxwell system with Ohm’s law in 3D are given in this paper. It is shown that the Green’s function consists of the heat kernels, the diffusive waves at low-frequency, the hyperbolic waves at high-frequency with time decaying exponentially, and the singular short waves. In addition, we establish the pointwise estimate of the global solution to the nonlinear incompressible Navier–Stokes–Maxwell system with Ohm’s law based on the Green’s function. To solve the new problem that the nonlinear terms contain the nonlocal operators <span><math><mrow><mi>∇</mi><mrow><mrow><mi>d</mi></mrow><mi>i</mi><mi>v</mi></mrow><mstyle><mi>Δ</mi></mstyle><msup><mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span> and <span><math><mrow><mi>∇</mi><mo>×</mo><mi>∇</mi><mo>×</mo><mstyle><mi>Δ</mi></mstyle><msup><mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span> which arise from the fluid-electromagnetic decomposition, we develop some new estimates of the nonlocal operators.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104572"},"PeriodicalIF":1.8,"publicationDate":"2025-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841570","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}
Pub Date : 2025-12-24DOI: 10.1016/j.nonrwa.2025.104560
Liyan Pang , Xiao Zhang
In this paper, the long time behavior for a two-species Lotka-Volterra reaction-diffusion system with strong competition in a periodic medium is concerned. We prove that under the compactly supported initial values, the solutions of Cauchy problem converge to a pair of diverging pulsating fronts. Further, we obtain a sufficient condition for solutions to converge to 1 with two different speeds to the left and right. Due to the spatial heterogeneity, the pulsating fronts depend on its direction and any pair of rightward and leftward wave speeds be asymmetrical. Therefore, our analysis mainly depends on constructing appropriate super- and subsolutions and using the comparison principle and asymptotic behavior of bistable pulsating fronts.
{"title":"Long time behavior for a Lotka-Volterra competition diffusion system in periodic medium","authors":"Liyan Pang , Xiao Zhang","doi":"10.1016/j.nonrwa.2025.104560","DOIUrl":"10.1016/j.nonrwa.2025.104560","url":null,"abstract":"<div><div>In this paper, the long time behavior for a two-species Lotka-Volterra reaction-diffusion system with strong competition in a periodic medium is concerned. We prove that under the compactly supported initial values, the solutions of Cauchy problem converge to a pair of diverging pulsating fronts. Further, we obtain a sufficient condition for solutions to converge to <strong><em>1</em></strong> with two different speeds to the left and right. Due to the spatial heterogeneity, the pulsating fronts depend on its direction and any pair of rightward and leftward wave speeds be asymmetrical. Therefore, our analysis mainly depends on constructing appropriate super- and subsolutions and using the comparison principle and asymptotic behavior of bistable pulsating fronts.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104560"},"PeriodicalIF":1.8,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841573","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}
Pub Date : 2025-12-24DOI: 10.1016/j.nonrwa.2025.104581
L.F. Gonçalves, A.C.T. Sánchez, D.J. Tonon
In this work, we establish an upper bound for the number of crossing limit cycles in a class of piecewise smooth dynamical systems. The system is formed by a linear rigid center and a rigid center governed by a homogeneous polynomial of even degree n, separated by the straight line . Our results complement the work of [1], which addressed the odd-degree case. Specifically, we prove that if the parameters satisfy , the system admits at most limit cycles. Furthermore, for the specific case , assuming d2 ≠ M2 and , we show that the system has at most one limit cycle, and this upper bound is attained. This study advances the analysis of this family of systems by covering the even-degree case under certain conditions on the affine transformation.
{"title":"Limit cycles on rigid piecewise smooth dynamical systems governed by even polynomials","authors":"L.F. Gonçalves, A.C.T. Sánchez, D.J. Tonon","doi":"10.1016/j.nonrwa.2025.104581","DOIUrl":"10.1016/j.nonrwa.2025.104581","url":null,"abstract":"<div><div>In this work, we establish an upper bound for the number of crossing limit cycles in a class of piecewise smooth dynamical systems. The system is formed by a linear rigid center and a rigid center governed by a homogeneous polynomial of even degree <em>n</em>, separated by the straight line <span><math><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span>. Our results complement the work of [1], which addressed the odd-degree case. Specifically, we prove that if the parameters satisfy <span><math><mrow><msub><mi>d</mi><mn>2</mn></msub><mo>=</mo><msub><mi>M</mi><mn>2</mn></msub></mrow></math></span>, the system admits at most <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo><mo>/</mo><mn>2</mn></mrow></math></span> limit cycles. Furthermore, for the specific case <span><math><mrow><mi>n</mi><mo>=</mo><mn>4</mn></mrow></math></span>, assuming <em>d</em><sub>2</sub> ≠ <em>M</em><sub>2</sub> and <span><math><mrow><msub><mi>d</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn></mrow></math></span>, we show that the system has at most one limit cycle, and this upper bound is attained. This study advances the analysis of this family of systems by covering the even-degree case under certain conditions on the affine transformation.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104581"},"PeriodicalIF":1.8,"publicationDate":"2025-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841571","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}
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