Emerging attractors and the transition from dissipative to conservative dynamics

The topological structure of basin boundaries plays a fundamental role in the sensitivity to the final state in chaotic dynamical systems. Herewith we present a study on the dynamics of dissipative systems close to the Hamiltonian limit, emphasizing the increasing number of periodic attractors, and on the structural changes in their basin boundaries as the dissipation approaches zero. We show numerically that a power law with nontrivial exponent describes the growth of the total number of periodic attractors as the damping is decreased. We also establish that for small scales the dynamics is governed by effective dynamical invariants, whose measure depends not only on the region of the phase space but also on the scale under consideration. Therefore, our results show that the concept of effective invariants is also relevant for dissipative systems.

Figure 1

Total number of periodic attractors detected for a specific value of the dissipation $\nu$, as a function of the number of initial conditions randomly chosen in $\mathbf{\Lambda }$ and evolved according to the map. The parameters for map (1) were $f_0=4.0$, and $\nu =0.0035$.

Figure 2

(Color online) Total number of periodic attractors, which were detected numerically for the single rotor map (black circles), and for the Hénon map (red diamonds) for different values of $\nu$. The parameters were $f_0=4.0$, for the single rotor map, and $A=1.075$ for the Hénon map. It is also shown that the power laws fits, the black and the red solid lines for the single rotor map and for the Hénon map, respectively; compare to the number of period-1 attractors expected from Eq. (2) (black dashed line) for the single rotor map. The divergence from this curve (dashed black line) to the other (black solid line) is clear, showing the important contribution of high-period periodic attractors in the regime of small damping for the single rotor map.

Figure 3

(Color online) Basins of attraction for $f_0=4.0$ at different values of $\nu$. Each color represents a set of initial conditions, which converge to one of the attractors. In (a) $\nu =0.32$, in (b) $\nu =0.3$, in (c) $\nu =0.2415$, and (d) is the blow up of a region $x∊\left[4,5\right],y∊\left[-15,-10\right]$ for $\nu =0.2415$. We used $2\times {10}^6$ initial conditions for each one of the figures.

Figure 4

(Color online) Blow up of the region $x∊\left[4.662,4.679\right],y∊\left[-13.955,-13.94\right]$ for $\nu =0.2415$ in Fig. 3d, showing the fractal Cantor structure.

Figure 5

The largest basin of attraction for $\nu =0.02$ and $f_0=4.0$. The black points represent initial conditions, which converge to the attractor formed from the main island. The initial conditions were chosen such that $\left(x_0,y_0\right)∊\left[0,2\pi \right]\times \left[-\pi ,\pi \right]$. We used the same parameters as those used in Ref. [10].

Figure 6

Fraction of uncertain initial conditions $f\left(\epsilon \right)$ as a function of the perturbation size $\epsilon$ for two different regions of the phase space. The regions, ${\mathit{R}}_i\subset \mathbf{\Gamma }$, were defined as ${\mathit{R}}_1\equiv \left(x,y\right)∊\left[0,1.5\right]\times \left[-1.0,-2.5\right]$, and ${\mathit{R}}_3\equiv \left(x,y\right)∊\left[1.0,2.5\right]\times \left[198.5,200.0\right]$. The utilized parameters were $f_0=4.0$, and $\nu =0.08$.

Figure 7

Fraction of uncertain initial conditions $f\left(\epsilon \right)$ scaling with the perturbation $\epsilon$ for three different regions of the phase space. The regions, ${\mathit{R}}_i\subset \mathbf{\Gamma }$, were defined as ${\mathit{R}}_1\equiv \left(x,y\right)∊\left[0,1.5\right]\times \left[-1.0,-2.5\right]$, ${\mathit{R}}_2\equiv \left(x,y\right)∊\left[1.0,2.5\right]\times \left[18.5,20.0\right]$, and ${\mathit{R}}_3\equiv \left(x,y\right)∊\left[1.0,2.5\right]\times \left[198.5,200.0\right]$. The utilized parameters were $f_0=4.0$ and $\nu =0.3$.

Figure 8

Fraction of uncertain initial conditions $f\left(\epsilon \right)$ scaling with the perturbation $\epsilon$ for three different regions of the phase space. The regions are the same as in Fig. 7. The utilized parameters were $f_0=4.0$ and $\nu =0.07$.

Figure 9

Fraction of uncertain initial conditions $f\left(\epsilon \right)$ scaling with the perturbation $\epsilon$ for three different regions of the phase space. The regions are the same as in Fig. 7. The utilized parameters were $f_0=4.0$ and $\nu =0.02$.

Figure 10

Illustration of the dependence of the number of periodic attractors with the precision. The figure shows attractor $x_i$ and $y_i$, with $i=1$ or 2. Hence, the attractors are of period two. If on the detection of such attractors, $ϵ>\xi$ is used, only one attractor of period two is detected. On the other hand, for $ϵ<\xi$, we are able to distinguish them.