Characterizing chaos in systems subjected to parameter drift

To characterize chaos in systems subjected to parameter drift, where a number of traditional methods do not apply, we propose viable alternative approaches, both in the qualitative and quantitative sense. Qualitatively, following stable and unstable foliations is shown to be efficient, which are easy to approximate numerically, without relying on the need for the existence of an analog of hyperbolic periodic orbits. Chaos originates from a Smale horseshoe-like pattern of the foliations, the transverse intersections of which indicate a chaotic set changing in time. In dissipative cases, the unstable foliation is found to be part of the so-called snapshot attractor, but the chaotic set is not dense on it if regular time-dependent attractors also exist. In Hamiltonian cases stable and unstable foliations turn out to be not equivalent due to the lack of time-reversal symmetry. It is the unstable foliation, which is found to correlate with the so-called snapshot chaotic sea. The chaotic set appears to be locally dense in this sea, while tori with originally quasiperiodic character might break up, their motion becoming chaotic as time goes on. A quantity called ensemble-averaged pairwise distance evaluated in relation to unstable foliations is shown to be an appropriate tool to provide the instantaneous strength of time-dependent chaos.

Figure 1

Snapshot unstable foliation and its relation to the snapshot attractor in the scenario ${\varepsilon }_0=0.08$, $\alpha =0.0005$ ($\beta =0.01$, $\omega =1$) at time $n=25$. The filaments are generated from $N=50000$ points on a vertical segment of length $dl=0.2$ centered at (a) $(-1,0.8)$ and (b) $(0.85,-0.5)$, marked by purple lines in both panels. The (a) green and (b) brown dots are the $k=10\mathrm{th}$ images of the segments initiated at instant $n-k=15$. The small light blue intervals are obtained from subintervals of length (a) $dl=4\times {10}^{-4}$ and (b) $dl=2\times {10}^{-3}$. (c) The snapshot unstable manifold (yellow) of the SHP (red cross), overlaid with the foliations of (a) and (b) at the same time instant. The unstable manifold was generated using $N=50000$ points and with $dl=5\times {10}^{-7}$. (d) The snapshot attractor (blue dots) at $n=25$, evolved from an initially uniform distribution of $N=40000$ points on a square of edge length 0.8 centered at the origin at time 0. Two disks of radius 0.2 are cut out from the shape of the ensemble about $(\pm 1,0)$.

Figure 2

The snapshot stable foliation, its relation to regular snapshot attractors, and a snapshot chaotic horseshoe at time $n=25$ for the scenario used in Fig. 1. (a) The foliation is generated from three vertical segments of length $dl=0.2$, each containing $N=50000$ points, centered at $(-1,0.8)$, $(0.25,0)$, $(0.85,-0.5)$, initiated at time $n+k=35$, obtained after $k=10$ backward iterations, and marked by pink dots. The “basins of attraction” in green and yellow of the left and right SNPs (red dots), respectively, are overlaid with the stable foliation at time $n=25$. The basins are obtained by generating the $k_b=15\mathrm{th}$ inverse image of two circles of radius 0.2 about the SNPs, taken at the $n+k_b=40\mathrm{th}$ iterate. (b) Snapshot chaotic horseshoe obtained by overlaying the unstable (stable) foliation of Fig. 1 and [Fig. 2] marked with green (pink) initiated $k=10$ steps earlier (later).

Figure 3

Hamiltonian snapshot foliations ($\omega =1$). (a) Unstable and (b) stable foliation in the same scenario as in Fig. 1 (with $\beta =0$) belonging to $n=25$ with $k=20$. The initial segments are the same three as in Fig. 2 (but with $N=10000$) for both foliations. (c) Snapshot chaotic horseshoe at time $n=25$ with $k=20$ obtained by overlaying the foliations in (a) and (b). The intersection points correspond to the snapshot chaotic saddle. (d) Snapshot phase portrait at $n=25$ containing the same four snapshot tori as panel (c), as well as a snapshot chaotic sea initiated from the stationary phase portrait at $n=0$. These are generated from the initial conditions $x_0=-0.7,0.95,1.56,1.65$, $v_0=0$ for the tori, and $x_0=0.1$, $v_0=0$ for the chaotic sea, respectively.

Figure 4

EAPD curves of unstable foliations in the scenario ${\varepsilon }_0=0.08$, $\alpha =0.0001$. The location of the initial vertical segment in both images is the same as in Fig. 1. The displayed (rounded-up) $\lambda$ values are a few instantaneous Lyapunov exponents. (a) Dissipative case with $\beta =0.01$. The dynamics is chaotic up to $n\approx 50$ with a positive but ever decreasing instantaneous Lyapunov exponent, followed by a phase of negative slope. (b) Hamiltonian case: The upper curve represents the unstable foliation. For completeness, on the lower curve we show the case of a snapshot torus initiated from the KAM torus of initial conditions $x_0=-0.53,v_0=0$ started from $n=15$.