Generic global diffusion for analytic uncoupled a priori unstable systems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908496508747776 |
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| author | Delshams, Amadeu Zhang, Ke |
| author_facet | Delshams, Amadeu Zhang, Ke |
| contents | We show that given a general uncoupled a priori unstable Hamiltonian \[ \frac12 p^2 + V(q) + G(I) + εh(p, q, I, φ, t), \] where $h$ is a generic Mañé analytic function and $ε$ is small enough, there is an orbit for which the momentum $I$ changes by any arbitrarily prescribed value. We call this phenomenon as global diffusion since the size of the change in $I$ is independent of both $ε$ and $h$. The fact that the pendulum and rotor variables are uncoupled is used essentially in our proof. The proof is based on simple and constructive geometrical methods, carefully studying the reduced Poincaré functions of the problem which generate the corresponding scattering maps. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_11349 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generic global diffusion for analytic uncoupled a priori unstable systems Delshams, Amadeu Zhang, Ke Dynamical Systems 37J40, 70H09 We show that given a general uncoupled a priori unstable Hamiltonian \[ \frac12 p^2 + V(q) + G(I) + εh(p, q, I, φ, t), \] where $h$ is a generic Mañé analytic function and $ε$ is small enough, there is an orbit for which the momentum $I$ changes by any arbitrarily prescribed value. We call this phenomenon as global diffusion since the size of the change in $I$ is independent of both $ε$ and $h$. The fact that the pendulum and rotor variables are uncoupled is used essentially in our proof. The proof is based on simple and constructive geometrical methods, carefully studying the reduced Poincaré functions of the problem which generate the corresponding scattering maps. |
| title | Generic global diffusion for analytic uncoupled a priori unstable systems |
| topic | Dynamical Systems 37J40, 70H09 |
| url | https://arxiv.org/abs/2412.11349 |