arXiv subjects

Charles Pugh

Publications and source records attributed to Charles Pugh.

5 recordsLinked to original sources

arXiv2026

Global analysis of the Kuramoto flow

Kuramoto's differential equation describes a synchronization process between several harmonic oscillators. It has been used to model biological phenomena such as the synchronization of heart cells, the circadian rhythm, or brain waves. It is also used in power system control. The simplest possible model assumes that all oscillators are identical and connected to each other with equal pairwise attraction. In this paper, we give a full geometric description of its global dynamics in terms of Morse theory and dynamical systems. Most of this description is stable in the sense that it is topologically preserved under small perturbations of the parameters.

math.DS↗

arXiv2012

Periodic Points on the 2-sphere

For a $C^{1}$ degree two latitude preserving endomorphism $f$ of the 2-sphere, we show that $f$ has $2^{n}$ periodic points.

math.DS↗

arXiv2012

Hölder foliations, revisited

We investigate transverse Hölder regularity of some canonical leaf conjugacies in partially hyperbolic dynamical systems and transverse Hölder regularity of some invariant foliations. Our results validate claims made elsewhere in the literature.

math.DS↗

arXiv2011

Jordan curves and funnel sections

We study the case when solution of an ODE at a given initial condition fail to be unique and investigate what are the possible time-1 sections of the `solution funnel'. Along the way we give construction of a natural complete metric on the space of Jordan curves and prove that the generic Jordan curve is nowhere pierceable by arcs of finite length.

math.DS↗

arXiv2002

Partial differentiability of invariant splittings

A key feature of a general nonlinear partially hyperbolic dynamical system is the absence of differentiability of its invariant splitting. In this paper, we show that often partial derivatives of the splitting exist and the splitting depends smoothly on the dynamical system itself.

math.DS↗