Lyapunov stability under $q$-dilatation and $q$-contraction of coordinates
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917850899283968 |
|---|---|
| author | de Oliveira, Tulio Meneghelli Wiggers, Vinicius Scafi, Eduardo Zanin, Silvio Manchein, Cesar Beims, Marcus Werner |
| author_facet | de Oliveira, Tulio Meneghelli Wiggers, Vinicius Scafi, Eduardo Zanin, Silvio Manchein, Cesar Beims, Marcus Werner |
| contents | This study examines the Lyapunov stability under coordinate $q$-contraction and $q$-dilatation in three dynamical systems: the discrete-time dissipative Hénon map, and the conservative, non-integrable, continuous-time Hénon-Heiles and diamagnetic Kepler problems. The stability analysis uses the $q$-deformed Jacobian and $q$-derivative, with trajectory stability assessed for $q > 1$ (dilatation) and $q < 1$ (contraction). Analytical curves in the parameter space mark boundaries of distinct low-periodic motions in the Hénon map. Numerical simulations compute the maximal Lyapunov exponent across the parameter space, in Poincaré surfaces of section, and as a function of total energy in the conservative systems. Simulations show that $q$-contraction ($q$-dilatation) generally decreases (increases) positive Lyapunov exponents relative to the $q = 1$ case, while both transformations tend to increase Lyapunov exponents for regular orbits. Some exceptions to this trend remain unexplained regarding Kolmogorov-Arnold-Moser (KAM) tori stability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18691 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lyapunov stability under $q$-dilatation and $q$-contraction of coordinates de Oliveira, Tulio Meneghelli Wiggers, Vinicius Scafi, Eduardo Zanin, Silvio Manchein, Cesar Beims, Marcus Werner Chaotic Dynamics 34C28, 37D45 This study examines the Lyapunov stability under coordinate $q$-contraction and $q$-dilatation in three dynamical systems: the discrete-time dissipative Hénon map, and the conservative, non-integrable, continuous-time Hénon-Heiles and diamagnetic Kepler problems. The stability analysis uses the $q$-deformed Jacobian and $q$-derivative, with trajectory stability assessed for $q > 1$ (dilatation) and $q < 1$ (contraction). Analytical curves in the parameter space mark boundaries of distinct low-periodic motions in the Hénon map. Numerical simulations compute the maximal Lyapunov exponent across the parameter space, in Poincaré surfaces of section, and as a function of total energy in the conservative systems. Simulations show that $q$-contraction ($q$-dilatation) generally decreases (increases) positive Lyapunov exponents relative to the $q = 1$ case, while both transformations tend to increase Lyapunov exponents for regular orbits. Some exceptions to this trend remain unexplained regarding Kolmogorov-Arnold-Moser (KAM) tori stability. |
| title | Lyapunov stability under $q$-dilatation and $q$-contraction of coordinates |
| topic | Chaotic Dynamics 34C28, 37D45 |
| url | https://arxiv.org/abs/2411.18691 |