Academic Journal
Exponentially Small Splitting of Separatrices Associated to 3D Whiskered Tori with Cubic Frequencies
| Title: | Exponentially Small Splitting of Separatrices Associated to 3D Whiskered Tori with Cubic Frequencies |
|---|---|
| Authors: | Amadeu Delshams, Marina Gonchenko, Pere Gutiérrez |
| Contributors: | Universitat Politècnica de Catalunya. Departament de Matemàtiques, Universitat Politècnica de Catalunya. SD - Sistemes Dinàmics de la UPC, Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions |
| Source: | Recercat. Dipósit de la Recerca de Catalunya instname UPCommons. Portal del coneixement obert de la UPC Universitat Politècnica de Catalunya (UPC) |
| Publication Status: | Preprint |
| Publisher Information: | Springer Science and Business Media LLC, 2020. |
| Publication Year: | 2020 |
| Subject Terms: | Dinamics, Transverse homoclinic orbits, Cubic frequency vectors, Dynamical Systems (math.DS), 01 natural sciences, Splitting of separatrices, Sistemes dinàmics, Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics, 0103 physical sciences, 70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics [Classificació AMS], FOS: Mathematics, Sistemes hamiltonians, and nonholonomic systems, Hamiltonian systems, Mathematics - Dynamical Systems, 0101 mathematics, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems, Lagrangian, Melnikov integrals, Àrees temàtiques de la UPC::Matemàtiques i estadística, Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics, Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics [Àrees temàtiques de la UPC], Matemàtiques i estadística [Àrees temàtiques de la UPC], 37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems [Classificació AMS], Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, contact |
| Description: | We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional torus with a fast frequency vector $��/\sqrt\varepsilon$, with $��=(1,��,\widetilde��)$ where $��$ is a cubic irrational number whose two conjugates are complex, and the components of $��$ generate the field $\mathbb Q(��)$. A paradigmatic case is the cubic golden vector, given by the (real) number $��$ satisfying $��^3=1-��$, and $\widetilde��=��^2$. For such 3-dimensional frequency vectors, the standard theory of continued fractions cannot be applied, so we develop a methodology for determining the behavior of the small divisors $\langle k,��\rangle$, $k\in{\mathbb Z}^3$. Applying the Poincar��-Melnikov method, this allows us to carry out a careful study of the dominant harmonic (which depends on $\varepsilon$) of the Melnikov function, obtaining an asymptotic estimate for the maximal splitting distance, which is exponentially small in $\varepsilon$, and valid for all sufficiently small values of~$\varepsilon$. This estimate behaves like $\exp\{-h_1(\varepsilon)/\varepsilon^{1/6}\}$ and we provide, for the first time in a system with 3 frequencies, an accurate description of the (positive) function $h_1(\varepsilon)$ in the numerator of the exponent, showing that it can be explicitly constructed from the resonance properties of the frequency vector $��$, and proving that it is a quasiperiodic function (and not periodic) with respect to $\ln\varepsilon$. In this way, we emphasize the strong dependence of the estimates for the splitting on the arithmetic properties of the frequencies. arXiv admin note: text overlap with arXiv:1507.07397 |
| Document Type: | Article Report Other literature type |
| File Description: | application/pdf |
| Language: | English |
| ISSN: | 1432-0916 0010-3616 |
| DOI: | 10.1007/s00220-020-03832-y |
| DOI: | 10.48550/arxiv.1906.01439 |
| Access URL: | https://upcommons.upc.edu/bitstream/2117/331406/1/220_2020_3832_OnlinePDF.pdf http://arxiv.org/abs/1906.01439 http://hdl.handle.net/2117/135375 https://hdl.handle.net/2117/135375 http://hdl.handle.net/2117/331406 http://ui.adsabs.harvard.edu/abs/2020CMaPh.378.1931D/abstract https://upcommons.upc.edu/bitstream/2117/331406/1/220_2020_3832_OnlinePDF.pdf https://link.springer.com/article/10.1007/s00220-020-03832-y https://upcommons.upc.edu/handle/2117/135375 https://arxiv.org/abs/1906.01439 https://arxiv.org/pdf/1906.01439.pdf |
| Rights: | Springer Nature TDM CC BY NC ND arXiv Non-Exclusive Distribution |
| Accession Number: | edsair.doi.dedup.....2c7d792cf12bafd3633c06b7f5655afe |
| Database: | OpenAIRE |
| ISSN: | 14320916 00103616 |
|---|---|
| DOI: | 10.1007/s00220-020-03832-y |