The Nekhoroshev theorem has become an important tool for explaining the long-term stability of many quasi-integrable systems of interest in physics. The action variables of systems that satisfy the hypotheses of the Nekhoroshev theorem remain close to their initial value up to very long times, which grow exponentially as an inverse power of the perturbation's norm. In this paper we study some of the simplest systems that do not satisfy the hypotheses of the Nekhoroshev theorem. These systems can be represented by a perturbed Hamiltonian whose integrable part is a quadratic non-convex function of the action variables. We study numerically the possibility of action diffusion over short times for these systems (continuous or maps) and we compare it with the so-called Arnold diffusion. More precisely we find that, except for very special non-convex functions, for which the effect of non-convexity concerns low-order resonances, the diffusion coefficient decreases faster than a power law (and possibly exponentially) of the perturbation's norm. According to theory, we find that the diffusion coefficient as a function of the perturbation's norm decreases more slowly than in the convex case.

Diffusion and stability in perturbed non-convex integrable systems

GUZZO, MASSIMILIANO;
2006

Abstract

The Nekhoroshev theorem has become an important tool for explaining the long-term stability of many quasi-integrable systems of interest in physics. The action variables of systems that satisfy the hypotheses of the Nekhoroshev theorem remain close to their initial value up to very long times, which grow exponentially as an inverse power of the perturbation's norm. In this paper we study some of the simplest systems that do not satisfy the hypotheses of the Nekhoroshev theorem. These systems can be represented by a perturbed Hamiltonian whose integrable part is a quadratic non-convex function of the action variables. We study numerically the possibility of action diffusion over short times for these systems (continuous or maps) and we compare it with the so-called Arnold diffusion. More precisely we find that, except for very special non-convex functions, for which the effect of non-convexity concerns low-order resonances, the diffusion coefficient decreases faster than a power law (and possibly exponentially) of the perturbation's norm. According to theory, we find that the diffusion coefficient as a function of the perturbation's norm decreases more slowly than in the convex case.
2006
File in questo prodotto:
File Dimensione Formato  
glfvqr.pdf

accesso aperto

Tipologia: Preprint (submitted version)
Licenza: Accesso libero
Dimensione 879.33 kB
Formato Adobe PDF
879.33 kB Adobe PDF Visualizza/Apri
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/1562601
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 26
  • ???jsp.display-item.citation.isi??? 23
  • OpenAlex ND
social impact