Shearless barriers in the conservative Ikeda map

TL;DR

The paper addresses transport barriers in the conservative Ikeda map, a two-dimensional area-preserving system with parameters $\theta$ and $\phi$. It combines perturbative rotation-number analysis for $\phi\ll1$ with numerical bifurcation tracking to identify and characterize shearless curves, linking their existence to fixed-point bifurcations and resonance reconnections. The main contributions include an analytical expression for the rotation-number profile in the weak-perturbation regime, identification of distorted pitchfork and subcritical period-doubling bifurcations associated with shearless curves, and a demonstration that shearless barriers act as robust transport inhibitors whose breakup enables slow chaotic mixing. The results advance the understanding of transport phenomena in globally degenerate conservative maps and provide a general framework for locating and analyzing shearless barriers in similar systems, with potential applications to laser cavity dynamics and other Hamiltonian-like settings.

Abstract

We investigate the dynamics of the Ikeda map in the conservative limit, where it is represented as a two-dimensional area-preserving map governed by two control parameters, $θ$ and $φ$. We demonstrate that the map can be interpreted as a composition of a rotation and a translation of the state vector. In the integrable case ($φ= 0$), the map reduces to a uniform rotation by angle $θ$ about a fixed point, independent of initial conditions. For $φ\ne 0$, the system becomes nonintegrable, and the rotation angle acquires a coordinate dependence. The resulting rotation number profile exhibits extrema as a function of position, indicating the formation of shearless barriers. We analyze the emergence, persistence, and breakup of these barriers as the control parameters vary.

Figures (9)

• Figure 1: For $\phi=0.01$, (a) extreme value of the rotation number as a function of $\theta$. In the interval $\theta\in\left(\pi/2,3\pi/4\right)$, the rotation number profile is monotonic. (b) Nonmonotonic rotation number profile obtained for $\theta=1$. (c) Monotonic rotation number profile obtained for $\theta=3.5$.
• Figure 2: For $\theta=1$, (a) bifurcation diagram of the shearless curve and (b) bifurcation diagram of the fixed points, where blue denotes stability and red denotes instability. The saddle-center bifurcation around $\phi=3.8$ (marked by the vertical black line) is associated with the disappearance of the shearless curve. The vertical green lines in the inset indicate the two consecutive $\phi$ values considered in Fig. \ref{['fig:fig3']}.
• Figure 3: Phase space and rotation number profile for $\theta=1$: (a) and (b) show $\phi=3.8245$, before the saddle-center bifurcation, and (c) and (d) show $\phi=3.8456$, after the saddle-center bifurcation.
• Figure 4: Reconnection-collision sequence of the $\omega=1/7$ twin resonances. Phase spaces (upper row) and rotation number profiles (lower row) for $\theta=1$ and decreasing values of $\phi$; the dashed line segment at $x=0.5$ in the phase spaces represents the initial conditions used to compute the rotation number profile. In (a) and (b), $\phi=0.4833$; (c) and (d), $\phi=0.46745$; (e) and (f), $\phi=0.4623$; (g) and (h), $\phi=0.455$.
• Figure 5: For $\theta=3.5$, (a) bifurcation diagram of the shearless curve and (b) bifurcation diagram of the fixed point, where blue denotes stability and red denotes instability. A subcritical period-doubling bifurcation around $\phi=0.7$ (marked by the vertical black line) is associated with the appearance of the shearless curve. The vertical green lines in the inset indicate the three consecutive $\phi$ values considered in Fig. \ref{['fig:fig6']}.