English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Consider the ensemble of Gaussian random potentials $\{V^L(q)\}_{L=1}^\infty$on the $d$-dimensional torus where, essentially, $V^L(q)$ is a real-valuedtrigonometric polynomial of degree $L$ whose coefficients are independentstandard normal variables. Our main result ensures that, with a probabilitytending to 1 as $L\to\infty$, the dynamical system associated with the naturalHamiltonian function defined by this random potential, $H^L:=\frac12|p|^2+V^L(q)$, exhibits a number of chaotic regions which coexist with apositive-volume set of invariant tori. In particular, these systems aretypically neither integrable with non-degenerate first integrals nor ergodic.An analogous result for random natural Hamiltonian systems defined on thecotangent bundle of an arbitrary compact Riemannian manifold is presented too.

Non-integrability and chaos for natural Hamiltonian systems with a random potential