Weak tracking in nonautonomous chaotic systems

Previous studies have shown that rate-induced transitions can occur in pullback attractors of systems subject to “parameter shifts” between two asymptotically steady values of a system parameter. For cases where the attractors limit to equilibrium or periodic orbit in past and future limits of such an nonautonomous systems, these can occur as the parameter change passes through a critical rate. Such rate-induced transitions for attractors that limit to chaotic attractors in past or future limits has been less examined. In this paper, we identify a new phenomenon is associated with more complex attractors in the future limit: weak tracking, where a pullback attractor of the system limits to a proper subset of an attractor of the future limit system. We demonstrate weak tracking in a nonautonomous Rössler system, and argue there are infinitely many critical rates at each of which the pullback attracting solution of the system tracks an embedded unstable periodic orbit of the future chaotic attractor. We also state some necessary conditions that are needed for weak tracking.

Figure 1

In (a), the Rössler attractor for parameter values $a=b=0.2$ and $c=5.7$. This also shows the period-one unstable periodic orbit ${\mathrm{\Gamma }}_{+}$, and Poincaré section $\mathrm{\Sigma }$ defined as $x\left(t\right)-y\left(t\right)=0$. In panel (b) we plot the the projection of the $x$-component of the return map of Rössler system. Assuming that a trajectory $\left[x\right(t),y(t),z(t\left)\right]$ intersects with $\mathrm{\Sigma }$ at $t=t_n$ for $n=1,2,...$, we define $x_n=x\left(t_n\right)$. $({\gamma }_x,{\gamma }_z)$ represents the intersection of the periodic orbit ${\mathrm{\Gamma }}_{+}$ with the section $\mathrm{\Sigma }$.

Figure 2

The parameter shift $a\left(s\right)$ and the piecewise linear approximation $\widehat{a}\left(s\right)$ vs. time, for $a_{+}=-a_{-}=0.2$ and $\delta =0.001$.

Figure 3

Two examples of weak tracking (EtoP connection) for Eq. (5). The parameters are $b=a_{max}=-a_{min}=0.2, c=5.7$ and $T=150$. (a) and (c) show the EtoP connection at $r=0.9202212159423$, (b) and (d) show the connection at $r=0.995651959127$.

Figure 4

A schematic diagram showing the shooting method we use to find the the connection between $Z_{-}$ and ${\mathrm{\Gamma }}_{+}$ for Eq. (5), see the GitHub repository [31] for animated version of this figure.

Figure 5

Graphs of $\eta$ for increasing integration time $T$ to add additional intersections of $\mathrm{\Sigma }$: roots correspond to connections from $A_{-}$ to the periodic orbit ${\mathrm{\Gamma }}_{+}$. (a) $T=125$, (b) $T=135$, (c) $T=145$, and (d) $T=155$. It can be seen that additional zeros of $\eta$ (corresponding to critical rates that give weak tracking) appear as $T$ increases. The parameter values are $b=a_{max}=-a_{min}=0.2$ and $c=5.7$.