Example of simplest bifurcation diagram for a monotone family of vector fields on a torus

Abstract

We present an example of a monotone two-parameter family of vector fields on a torus whose bifurcation diagram we demonstrate to be in the class of "simplest" diagrams proposed by Baesens & MacKay (2018 Nonlinearity 31 2928--81). This shows that the proposed class is realisable.

Figures (8)

• Figure 1: Two simplest bifurcation diagrams for monotone families of vector fields on a torus (slightly edited version of Baesens_2018). In this work we will focus on (a). The infinitely many rhc curves and tongues emanating from the $H$ point have not been fully depicted.
• Figure 2: (a) The resonance region $\mathcal{R}$; (b) Bifurcation diagram for the equilibria. To depict these figures, as well as the upcoming ones, we have chosen $\phi=1.1/24$ for aesthetic reasons.
• Figure 3: Phase portraits in the five regions of parameter space into which the $\textnormal{snic}$ curves divide the strip $|\Omega_y|<1-\delta$. Attractive periodic orbits are depicted in red and repelling ones in blue. The signs $\pm$ indicate that the periodic orbit created by $\textnormal{snic}$ is right-going/left-going. The transitions to $\textnormal{snic}$ along the curves of $\textnormal{sne}$ require $Z$ points that are not depicted here.
• Figure 4: Sketch of the bifurcations diagram found so far in the top of the resonance region, $\mathcal{Q}$, excluding the trace-zero curves. We also depict the $\delta$-neighbourhood of $\boldsymbol{\mathbf{\Omega}}= (S,1)$ in dashed lines. The sign $\pm$ indicates right-going/left-going and a/b indicates that the homoclinic connection is above/below the saddle.
• Figure 5: Sketch of the phase portraits for the reversible family of vector fields \ref{['eq:reversible_approx']} at selected values of $\rho$. To increase visibility, the $\rho$-axis and the phase portraits have been slightly distorted.