Bifurcations of a nonlinear spherical pendulum with vibrating suspension point

Yan Luo; Kaicheng Sheng

doi:10.1142/s0218127425501482

Home/Archive/arXiv

arXiv · 1708.01313

Bifurcations of a nonlinear spherical pendulum with vibrating suspension point

Yan Luo·Kaicheng Sheng

View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper considers a nonlinear spherical pendulum whose suspension point performs high-frequency spatial vibrations. The dynamics of this pendulum can be described by averaging its Hamiltonian over phases of vibrations. Rotationally symmetric conditions on vibrations are assumed in the averaged Hamiltonian. Under these conditions, a bifurcation diagram for the phase portraits of the averaged system is presented. Numerical simulations of different examples of vibrations are performed. The case of proper degeneration in KAM theory guarantees the coherence of dynamical characteristics between the averaged and exact systems.

Explore related subjects

math.GM

Keep this discovery

BibTeX RIS

Yan Luo, Kaicheng Sheng. 2017-08-03. Bifurcations of a nonlinear spherical pendulum with vibrating suspension point. https://doi.org/10.1142/s0218127425501482

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗

arXiv2026

The decrease in sectional curvature of the Fisher--Rao manifold of beta distributions

In the paper, the author develops a general convexity lemma and applies it to demonstrate the monotonic decrease of a function involving polygamma functions, arising from the sectional curvature of the Fisher--Rao manifold of beta distributions.

math.GM↗

arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗