Resonance and Weak Chaos in Quasiperiodically-Forced Circle Maps

TL;DR

The paper studies dynamics of quasiperiodically forced circle maps and classifies orbits into resonant, incommensurate, weakly chaotic (strange nonchaotic attractors, SNAs), and strongly chaotic regimes. It introduces and applies the weighted Birkhoff average (WBA) to compute rotation vectors with near machine precision for regular orbits and to diagnose chaos via convergence digits and Lyapunov diagnostics. Regular behavior is distinguished by resonance order using proximity to resonance planes ${\cal R}_{m,n}$ with distance $\Delta_{m,n}(\omega)=\frac{|m\cdot\omega-n|}{\|m\|_2}$, revealing rich resonance tongues and pinching phenomena. SNAs are identified as weakly chaotic (nonpositive Lyapunov exponent) with slow WBA convergence, and the authors demonstrate WBA-based detection of SNAs and improved Lyapunov estimates. Collectively, the work enables fast, quantitative classification and parameter-dependent statistics of orbit types in quasiperiodically forced circle maps, with implications for understanding resonance structures in higher-dimensional torus maps.

Abstract

In this paper, we focus on a numerical technique, the weighted Birkhoff average (WBA) to distinguish between four categories of dynamics for quasiperiodically-forced circle maps. Regular dynamics can be classified by rotation vectors, and these can be rapidly computed to machine precision using the WBA. Regular orbits can be resonant or incommensurate and we distinguish between these by computing their "resonance order." When the dynamics is chaotic the WBA converges slowly. Such orbits can be strongly chaotic, when they have a positive Lyapunov exponent or weakly chaotic, when the maximal Lyapunov exponent is zero. The latter correspond to the strange nonchaotic attractors (SNA) that have been observed in quasiperiodically-forced circle maps beginning with the models introduced by Ding, Grebogi, and Ott. The WBA provides a new technique to find SNAs, and allows us to accurately compute the proportions of each of the four orbit types as a function of map parameters.

Figures (8)

• Figure 1: Poincaré slices of regular orbits of the quasiperiodically-forced circle map with $a_1=0.8$ and $\Omega_2 =\gamma$, for (a) $a_2 = 0.6$, (b) $a_2 = 2.49$ and (c) $a_2 = 5$ (these are the first, fourth, and eighth values shown in Fig. \ref{['fig:qpPropa2']}). The plot shows a grid of 200 for $\Omega_1 \in [0,1]$, with the orbits colored as in Fig. \ref{['fig:qpMapa1']} using the value of $\omega_1$ computed with $T = 10^6$. Resonant orbits are plotted with larger dots. Each orbit is iterated $10^5$ times to remove transients, and the next 1000 points on the Poincaré slice $|x_2| < 0.0005$ are shown.
• Figure 2: Nonresonant (panels (a) and (c)) and resonant (panels (b) and (c)) orbits for the quasiperiodically-forced circle map (\ref{['eq:QPForce']}) as a function of $(a_1,\Omega_1)$ for $a_2 = 0.6$ (top panels) and $a_2 = 1$ (bottom panels), with $\Omega_2 = \gamma$ (\ref{['eq:GoldenMean']}). These orbits are distinguished using (\ref{['eq:2DCriterion']}) and (\ref{['eq:Incommensurate']}). The orbits are colored using $\omega_1$ as shown in the color bars with black indicating no orbits of the given type.
• Figure 3: Resonant regions for the quasiperiodically-forced circle map (\ref{['eq:QPForce']}) for $a_1 = 1.2$. The larger regions are labeled by $(m_1,m_2,n)$, Each resonance is colored by $\omega_1$ and black regions correspond to either chaotic or incommensurate orbits.
• Figure 4: Chaotic orbits of the quasiperiodically-forced circle map (\ref{['eq:QPForce']}) for a $1000 \times 1000$ grid in the $(\Omega_1,a_1)$ plane for (a) $a_2 = 0.6$ and (b) $a_2 = 1$. The color bar corresponds to $\lambda_T$ (\ref{['eq:WBLyap']}). Parameters with $\lambda_T \le 0$ are gray; they correspond to weak chaos, or strange nonchaotic attractors. Parameters with nonchaotic orbits are colored dark blue, and strongly chaotic orbits have colors that vary with $\lambda_T$.
• Figure 5: Quasiperiodically-forced circle map (\ref{['eq:QPForce']}) for $a_1 = 1.2$ for a $1000\times1000$ grid in $(\Omega_1,a_2)$ showing orbits with sensitive dependence using Criterion (\ref{['eq:ChaosThreshold']}), colored by Lyapunov exponent (\ref{['eq:WBLyap']}). The grayscale indicates strange nonchaotic attractors. Blue indicates nonchaotic parameters, most of which correspond to the resonance regions in Fig. \ref{['fig:qpResonant']}.