{"title":"通过周期倍增实现异维循环级联","authors":"Nelson Wong, Bernd Krauskopf, Hinke M. Osinga","doi":"10.1016/j.cnsns.2024.108328","DOIUrl":null,"url":null,"abstract":"<div><p>A heterodimensional cycle is formed by the intersection of stable and unstable manifolds of two saddle periodic orbits that have unstable manifolds of different dimensions: connecting orbits exist from one periodic orbit to the other, and vice versa. The difference in dimensions of the invariant manifolds can only be achieved in vector fields of dimension at least four. At least one of the connecting orbits of the heterodimensional cycle will necessarily be structurally unstable, meaning that is does not persist under small perturbations. Nevertheless, the theory states that the existence of a heterodimensional cycle is generally a <em>robust</em> phenomenon: any sufficiently close vector field (in the <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-topology) also has a heterodimensional cycle.</p><p>We investigate a particular four-dimensional vector field that is known to have a heterodimensional cycle. We continue this cycle as a codimension-one invariant set in a two-parameter plane. Our investigations make extensive use of advanced numerical methods that prove to be an important tool for uncovering the dynamics and providing insight into the underlying geometric structure. We study changes in the family of connecting orbits as two parameters vary and Floquet multipliers of the periodic orbits in the heterodimensional cycle change. In particular the Floquet multipliers of one of the periodic orbits change from real positive to real negative prior to a period-doubling bifurcation. We then focus on the transitions that occur near this period-doubling bifurcation and find that it generates new families of heterodimensional cycles with different geometric properties. Our careful numerical study suggests that further two-parameter continuation of the ‘period-doubled heterodimensional cycles’ gives rise to an abundance of heterodimensional cycles of different types in the limit of a period-doubling cascade.</p><p>Our results for this particular example vector field make a contribution to the emerging bifurcation theory of heterodimensional cycles. In particular, the bifurcation scenario we present can be viewed as a specific mechanism behind so-called stabilisation of a heterodimensional cycle via the embedding of one of its constituent periodic orbits into a more complex invariant set.</p></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4000,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1007570424005136/pdfft?md5=f38edea7a0d479894640291d3ce40bfc&pid=1-s2.0-S1007570424005136-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Cascades of heterodimensional cycles via period doubling\",\"authors\":\"Nelson Wong, Bernd Krauskopf, Hinke M. Osinga\",\"doi\":\"10.1016/j.cnsns.2024.108328\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A heterodimensional cycle is formed by the intersection of stable and unstable manifolds of two saddle periodic orbits that have unstable manifolds of different dimensions: connecting orbits exist from one periodic orbit to the other, and vice versa. The difference in dimensions of the invariant manifolds can only be achieved in vector fields of dimension at least four. At least one of the connecting orbits of the heterodimensional cycle will necessarily be structurally unstable, meaning that is does not persist under small perturbations. Nevertheless, the theory states that the existence of a heterodimensional cycle is generally a <em>robust</em> phenomenon: any sufficiently close vector field (in the <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-topology) also has a heterodimensional cycle.</p><p>We investigate a particular four-dimensional vector field that is known to have a heterodimensional cycle. We continue this cycle as a codimension-one invariant set in a two-parameter plane. Our investigations make extensive use of advanced numerical methods that prove to be an important tool for uncovering the dynamics and providing insight into the underlying geometric structure. We study changes in the family of connecting orbits as two parameters vary and Floquet multipliers of the periodic orbits in the heterodimensional cycle change. In particular the Floquet multipliers of one of the periodic orbits change from real positive to real negative prior to a period-doubling bifurcation. We then focus on the transitions that occur near this period-doubling bifurcation and find that it generates new families of heterodimensional cycles with different geometric properties. Our careful numerical study suggests that further two-parameter continuation of the ‘period-doubled heterodimensional cycles’ gives rise to an abundance of heterodimensional cycles of different types in the limit of a period-doubling cascade.</p><p>Our results for this particular example vector field make a contribution to the emerging bifurcation theory of heterodimensional cycles. In particular, the bifurcation scenario we present can be viewed as a specific mechanism behind so-called stabilisation of a heterodimensional cycle via the embedding of one of its constituent periodic orbits into a more complex invariant set.</p></div>\",\"PeriodicalId\":50658,\"journal\":{\"name\":\"Communications in Nonlinear Science and Numerical Simulation\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":3.4000,\"publicationDate\":\"2024-09-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S1007570424005136/pdfft?md5=f38edea7a0d479894640291d3ce40bfc&pid=1-s2.0-S1007570424005136-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Nonlinear Science and Numerical Simulation\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1007570424005136\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Nonlinear Science and Numerical Simulation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1007570424005136","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Cascades of heterodimensional cycles via period doubling
A heterodimensional cycle is formed by the intersection of stable and unstable manifolds of two saddle periodic orbits that have unstable manifolds of different dimensions: connecting orbits exist from one periodic orbit to the other, and vice versa. The difference in dimensions of the invariant manifolds can only be achieved in vector fields of dimension at least four. At least one of the connecting orbits of the heterodimensional cycle will necessarily be structurally unstable, meaning that is does not persist under small perturbations. Nevertheless, the theory states that the existence of a heterodimensional cycle is generally a robust phenomenon: any sufficiently close vector field (in the -topology) also has a heterodimensional cycle.
We investigate a particular four-dimensional vector field that is known to have a heterodimensional cycle. We continue this cycle as a codimension-one invariant set in a two-parameter plane. Our investigations make extensive use of advanced numerical methods that prove to be an important tool for uncovering the dynamics and providing insight into the underlying geometric structure. We study changes in the family of connecting orbits as two parameters vary and Floquet multipliers of the periodic orbits in the heterodimensional cycle change. In particular the Floquet multipliers of one of the periodic orbits change from real positive to real negative prior to a period-doubling bifurcation. We then focus on the transitions that occur near this period-doubling bifurcation and find that it generates new families of heterodimensional cycles with different geometric properties. Our careful numerical study suggests that further two-parameter continuation of the ‘period-doubled heterodimensional cycles’ gives rise to an abundance of heterodimensional cycles of different types in the limit of a period-doubling cascade.
Our results for this particular example vector field make a contribution to the emerging bifurcation theory of heterodimensional cycles. In particular, the bifurcation scenario we present can be viewed as a specific mechanism behind so-called stabilisation of a heterodimensional cycle via the embedding of one of its constituent periodic orbits into a more complex invariant set.
期刊介绍:
The journal publishes original research findings on experimental observation, mathematical modeling, theoretical analysis and numerical simulation, for more accurate description, better prediction or novel application, of nonlinear phenomena in science and engineering. It offers a venue for researchers to make rapid exchange of ideas and techniques in nonlinear science and complexity.
The submission of manuscripts with cross-disciplinary approaches in nonlinear science and complexity is particularly encouraged.
Topics of interest:
Nonlinear differential or delay equations, Lie group analysis and asymptotic methods, Discontinuous systems, Fractals, Fractional calculus and dynamics, Nonlinear effects in quantum mechanics, Nonlinear stochastic processes, Experimental nonlinear science, Time-series and signal analysis, Computational methods and simulations in nonlinear science and engineering, Control of dynamical systems, Synchronization, Lyapunov analysis, High-dimensional chaos and turbulence, Chaos in Hamiltonian systems, Integrable systems and solitons, Collective behavior in many-body systems, Biological physics and networks, Nonlinear mechanical systems, Complex systems and complexity.
No length limitation for contributions is set, but only concisely written manuscripts are published. Brief papers are published on the basis of Rapid Communications. Discussions of previously published papers are welcome.