{"title":"具有第一积分和离散对称的四维系统中鞍形同斜轨道附近的动力学","authors":"Sajjad Bakrani","doi":"10.1016/j.jde.2025.113689","DOIUrl":null,"url":null,"abstract":"<div><div>We consider a <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-equivariant 4-dimensional system of ODEs with a smooth first integral <em>H</em> and a saddle equilibrium state <em>O</em>. We assume that there exists a transverse homoclinic orbit Γ to <em>O</em> that approaches <em>O</em> along the nonleading directions. Suppose <span><math><mi>H</mi><mo>(</mo><mi>O</mi><mo>)</mo><mo>=</mo><mi>c</mi></math></span>. In <span><span>[3]</span></span>, the dynamics near Γ in the level set <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>c</mi><mo>)</mo></math></span> was described. In particular, some criteria for the existence of the stable and unstable invariant manifolds of Γ were given. In the current paper, we describe the dynamics near Γ in the level set <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>h</mi><mo>)</mo></math></span> for <span><math><mi>h</mi><mo>≠</mo><mi>c</mi></math></span> close to <em>c</em>. We prove that when <span><math><mi>h</mi><mo><</mo><mi>c</mi></math></span>, there exists a unique saddle periodic orbit in each level set <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>h</mi><mo>)</mo></math></span>, and the forward (resp. backward) orbit of any point off the stable (resp. unstable) invariant manifold of this periodic orbit leaves a small neighborhood of Γ. We further show that when <span><math><mi>h</mi><mo>></mo><mi>c</mi></math></span>, the forward and backward orbits of any point in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>h</mi><mo>)</mo></math></span> near Γ leave a small neighborhood of Γ. We also prove analogous results for the scenario where two transverse homoclinics to <em>O</em> (homoclinic figure-eight) exist. The results of this paper, together with [3], give a full description of the dynamics in a small open neighborhood of Γ (and a small open neighborhood of a homoclinic figure-eight). An application to a system of coupled Schrödinger equations with cubic nonlinearity is also considered.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"446 ","pages":"Article 113689"},"PeriodicalIF":2.3000,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Dynamics near homoclinic orbits to a saddle in four-dimensional systems with a first integral and a discrete symmetry\",\"authors\":\"Sajjad Bakrani\",\"doi\":\"10.1016/j.jde.2025.113689\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We consider a <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-equivariant 4-dimensional system of ODEs with a smooth first integral <em>H</em> and a saddle equilibrium state <em>O</em>. We assume that there exists a transverse homoclinic orbit Γ to <em>O</em> that approaches <em>O</em> along the nonleading directions. Suppose <span><math><mi>H</mi><mo>(</mo><mi>O</mi><mo>)</mo><mo>=</mo><mi>c</mi></math></span>. In <span><span>[3]</span></span>, the dynamics near Γ in the level set <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>c</mi><mo>)</mo></math></span> was described. In particular, some criteria for the existence of the stable and unstable invariant manifolds of Γ were given. In the current paper, we describe the dynamics near Γ in the level set <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>h</mi><mo>)</mo></math></span> for <span><math><mi>h</mi><mo>≠</mo><mi>c</mi></math></span> close to <em>c</em>. We prove that when <span><math><mi>h</mi><mo><</mo><mi>c</mi></math></span>, there exists a unique saddle periodic orbit in each level set <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>h</mi><mo>)</mo></math></span>, and the forward (resp. backward) orbit of any point off the stable (resp. unstable) invariant manifold of this periodic orbit leaves a small neighborhood of Γ. We further show that when <span><math><mi>h</mi><mo>></mo><mi>c</mi></math></span>, the forward and backward orbits of any point in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>h</mi><mo>)</mo></math></span> near Γ leave a small neighborhood of Γ. We also prove analogous results for the scenario where two transverse homoclinics to <em>O</em> (homoclinic figure-eight) exist. The results of this paper, together with [3], give a full description of the dynamics in a small open neighborhood of Γ (and a small open neighborhood of a homoclinic figure-eight). An application to a system of coupled Schrödinger equations with cubic nonlinearity is also considered.</div></div>\",\"PeriodicalId\":15623,\"journal\":{\"name\":\"Journal of Differential Equations\",\"volume\":\"446 \",\"pages\":\"Article 113689\"},\"PeriodicalIF\":2.3000,\"publicationDate\":\"2025-08-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022039625007168\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022039625007168","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Dynamics near homoclinic orbits to a saddle in four-dimensional systems with a first integral and a discrete symmetry
We consider a -equivariant 4-dimensional system of ODEs with a smooth first integral H and a saddle equilibrium state O. We assume that there exists a transverse homoclinic orbit Γ to O that approaches O along the nonleading directions. Suppose . In [3], the dynamics near Γ in the level set was described. In particular, some criteria for the existence of the stable and unstable invariant manifolds of Γ were given. In the current paper, we describe the dynamics near Γ in the level set for close to c. We prove that when , there exists a unique saddle periodic orbit in each level set , and the forward (resp. backward) orbit of any point off the stable (resp. unstable) invariant manifold of this periodic orbit leaves a small neighborhood of Γ. We further show that when , the forward and backward orbits of any point in near Γ leave a small neighborhood of Γ. We also prove analogous results for the scenario where two transverse homoclinics to O (homoclinic figure-eight) exist. The results of this paper, together with [3], give a full description of the dynamics in a small open neighborhood of Γ (and a small open neighborhood of a homoclinic figure-eight). An application to a system of coupled Schrödinger equations with cubic nonlinearity is also considered.
期刊介绍:
The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools.
Research Areas Include:
• Mathematical control theory
• Ordinary differential equations
• Partial differential equations
• Stochastic differential equations
• Topological dynamics
• Related topics