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

We consider reversible non-conservative perturbations of the conservativecubic H\'enon maps $H_3^{\pm}: \bar x = y, \bar y = -x + M_1 + M_2 y \pm y^3$and study their influence on the 1:3 resonance, i.e. bifurcations of fixedpoints with eigenvalues $e^{\pm i 2\pi/3}$. It follows from the work by Dullinand Meiss, this resonance is degenerate for $M_1=0, M_2=-1$ when thecorresponding fixed point is elliptic. We show that bifurcations of this pointunder reversible perturbations give rise to four 3-periodic orbits, two of themare symmetric and conservative (saddles in the case of map $H_3^+$ and ellipticorbits in the case of map $H_3^-$), the other two orbits are nonsymmetric andthey compose symmetric couples of dissipative orbits (attracting and repellingorbits in the case of map $H_3^+$ and saddles with the Jacobians less than 1and greater than 1 in the case of map $H_3^-$). We show that these localsymmetry-breaking bifurcations can lead to mixed dynamics due to accompanyingglobal reversible bifurcations of symmetric non-transversal homo- andheteroclinic cycles. We also generalize the results of Dullin and Meiss to thecase of the $p:q$ resonances with odd $q$ and show that all of them are alsodegenerate for the maps $H_3^{\pm}$ with $M_1=0$.

On 1:3 resonance under reversible perturbations of conservative cubic H\'enon maps