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

Here we consider the anharmonic oscillator that is a dynamical system givenby $y_{xx}+\delta y^{n}=0$. We demonstrate that to this equation corresponds anew example of a superintegrable two-dimensional metric with a linear and atranscendental first integrals. Moreover, we show that for particular values of$n$ the transcendental first integral degenerates into a polynomial one, whichprovides an example of a superintegrable metric with additional polynomialfirst integral of an arbitrary even degree. We also discuss a general procedureof how to construct a superintegrable metric with one linear first integralfrom an autonomous nonlinear oscillator that is cubic with respect to the firstderivative. We classify all cubic oscillators that can be used in thisconstruction. Furthermore, we study the Li\'enard equations that are equivalentto the anharmonic oscillator with respect to the point transformations. We showthat there are nontrivial examples of the Li\'enard equations that belong tothis equivalence class, like the generalized Duffing oscillator or thegeneralized Duffing--Van der Pol oscillator.

On the geometric and analytical properties of the anharmonic oscillator