Slim Fractals: The Geometry of Doubly Transient Chaos

Traditional studies of chaos in conservative and driven dissipative systems have established a correspondence between sensitive dependence on initial conditions and fractal basin boundaries, but much less is known about the relation between geometry and dynamics in undriven dissipative systems. These systems can exhibit a prevalent form of complex dynamics, dubbed doubly transient chaos because not only typical trajectories but also the (otherwise invariant) chaotic saddles are transient. This property, along with a manifest lack of scale invariance, has hindered the study of the geometric properties of basin boundaries in these systems—most remarkably, the very question of whether they are fractal across all scales has yet to be answered. Here, we derive a general dynamical condition that answers this question, which we use to demonstrate that the basin boundaries can indeed form a true fractal; in fact, they do so generically in a broad class of transiently chaotic undriven dissipative systems. Using physical examples, we demonstrate that the boundaries typically form a slim fractal, which we define as a set whose dimension at a given resolution decreases when the resolution is increased. To properly characterize such sets, we introduce the notion of equivalent dimension for quantifying their relation with sensitive dependence on initial conditions at all scales. We show that slim fractal boundaries can exhibit complex geometry even when they do not form a true fractal and fractal scaling is observed only above a certain length scale at each boundary point. Thus, our results reveal slim fractals as a geometrical hallmark of transient chaos in undriven dissipative systems.

Popular Summary

Small changes can sometimes have big effects. That is the essence of chaos, where the final state of a process depends sensitively on the initial condition—a phenomenon popularly known as the “butterfly effect” (where, the metaphor goes, a butterfly flapping its wings might affect storm formation miles away). Visualizations of how these states interact can create mesmerizing images known as fractals, where boundaries between regions that correspond to different final states intermingle at finer and finer scales. This is well established in scenarios where energy is conserved, as well as in energy-dissipating systems that are driven. But this geometrical behavior is not well understood in scenarios where no energy is injected to balance dissipated energy. We develop a new conceptual framework for studying the geometry of such systems and demonstrate, for the first time, that their boundaries often exhibit genuine fractals.

Specifically, we derive a general dynamical condition under which the boundaries are fractal at all length scales. We use this condition to show that the boundaries form a new type of fractal, which we call “slim fractals,” as they tend to appear sparser at smaller length scales. To analyze such structures, we introduce a novel notion of fractal dimension that quantifies the resulting scale-varying sensitive dependence on initial conditions.

Given the ubiquity of undriven dissipative systems, these results have profound implications in many contexts, such as the detection of gravitational waves from merging binary systems of stars and/or black holes, the transition to equilibrium in chemical reactions, and the dynamics of interacting vortices in viscous fluids.

Figure 1

Geometry and dynamics in the scattering region of undriven dissipative systems. (a) Fractal basin boundaries in the two-dimensional cross section $\dot{x}=\dot{y}=0$ for potential $U_2(r,\theta )$ and $\mu =0.2$. The basins of the exits $E_1$, $E_2$, and $E_3$ are colored red, green, and beige, respectively. Trajectories starting on the one-dimensional cross section at $x=-0.5$ (vertical line segment A) are shown with arrows indicating the direction of flow and colors indicating the basin to which they belong. (b) Successive magnifications of the one-dimensional cross section at $x=-0.5$ in (a), showing the fractal nature of the basin boundaries. (c) Finite-scale fractal basin boundaries for the potential $U_3(r,\theta )$ and $\mu =1$. Trajectories with initial conditions on the cross sections at $x=-0.2$ (segment ${\mathrm{A}}^{\prime }$), $x=-0.5$ (segment ${\mathrm{B}}^{\prime }$), and $x=-1$ (segment ${\mathrm{C}}^{\prime }$) are shown. (d) Successive magnifications of the cross section at $x=-1$ in (c), revealing a simple boundary.

Figure 2

Center manifold reduction of the dynamics at the origin for Eq. (1) with $U_3(r,\theta )$. (a) Three-dimensional projection of the approximate center manifold given by Eq. (5). (b) Approximate dynamics on the center manifold, given by Eq. (6). The blue curves are trajectories initiated at the (randomly chosen) initial conditions indicated by red dots.

Figure 3

Three shapes of the roulette surface that we consider, given by the functions $S_1$, $S_2$, and $S_3$ defined in the text. In each panel, the white dot indicates the unstable fixed point at the origin, the green dots the attractors ($A_1$, $A_2$, and $A_3$), and the red dots the saddle points away from the origin (only present for $S_3$). The part of each surface corresponding to $S_i(r,\theta )<0.5$ is shown in the bottom row. Surface colors indicate the value of the function, and a common color scheme is used in all six panels.

Figure 4

Slim fractal boundaries of the attraction basins for the roulette system. (a)–(c) Attraction basins for the roulette shapes shown in Fig. 3 in the two-dimensional subspace parametrized by the initial condition $(v_0,{\theta }_0)$. The red, green, and beige regions indicate the basins of the attractors $A_1$, $A_2$, and $A_3$ (marked in Fig. 3), respectively. (d)–(f) Spatial distribution of the (color-coded) fractality length scale $\ell (v_0,{\theta }_0)$ on the boundaries of the basins shown in (a)–(c). Note that $v_0$ and ${\theta }_0$ are normalized by $v_{\max }$ and $2\pi /3$, respectively, only for the axes of the plots and not for the computation of $\ell (v_0,{\theta }_0)$. We compute $\ell (v_0,{\theta }_0)$ using double precision and the bisection resolution of ${10}^{-13}$ for each of the $1024\times 1024$ grid points [corresponding to $\mathrm{\Delta }=2^{-10}{v}_{\max }({\theta }_0)$, which ranges from $3.38\times {10}^{-3}$ to $6.48\times {10}^{-3}$ depending on ${\theta }_0$ and the roulette shape].

Figure 5

Successive magnification of the attraction basins on vertical line segments through the points $P_1$ (a) and $P_2$ (b) in Figs. 4 and 4, respectively. The numbers on the left indicate the length of the magnified intervals. The details on the computational procedure used to generate this figure can be found in Appendix pp4.

Figure 6

Dimension of slim fractals. (a)–(c) Uncertainty function $f(ϵ)$ estimated by sampling pairs of points $ϵ$ apart on a vertical line segment centered at $P_1$, $P_2$, and $P_3$ in Fig. 4 for the roulette system with surface shapes $S_1$, $S_2$, and $S_3$, respectively, for successively smaller segments (left to right). (d) Equivalent dimension $D_{\mathrm{eq}}$, computed using Eq. (12) with $L=2\times {10}^{-3}$ and $D_{\mathrm{eff}}$ estimated as the slope for each segment in (a)–(c) from the linear least-squares fit [lines in (a)–(c) are offset for clarity]. The vertical dashed lines indicate the fractality length scale $\ell$ for $S_2$ and $S_3$. (e,f) Illustration of the effective dimension (e) and the final-state uncertainty (f) as a function of $\ln (1/ϵ)$ for three types of fractals.

Figure 7

Length $r_s$ of the simple boundary segment on the line $y=0$ for systems with potential $U_{\alpha }$. (a) The circles indicate the numerically estimated $r_s$ as a function of $\alpha$. The solid curve is the best nonlinear fit to the data, as described in Appendix pp3. (b) The same quantities shown as a function of $1/(\alpha -b_2)$, with the best-fit value of $b_2\approx 1.986$.

Figure 8

Distribution of fractality measures for the roulette system computed from the $2^{20}\approx 1.05\times 10^6$ grid points used in Figs. 4. (a) Estimated probability density function for the fractality length scale $\ell$. (b) Estimated probability mass function for the construction level $N_{\text{level}}$.

Figure 9

Ratio ${\ell }_{i+1}/{\ell }_i$ of consecutive basin interval lengths for the roulette system as a function of ${\ell }_i$. The inset shows a magnification of the region shaded in gray. For each shape of the roulette surface ($S_1$, $S_2$, or $S_3$), ten realizations of the (random) process described in Appendix pp6 are superimposed. The curves are to guide the eyes.