Why is it that the direction of ~d0,n at the onset of either the spiral or the trigger part cannot be assumed random despite the chaotic nature of the system? For the spiral part, we can use the simple core approximation to analyze this behavior after the system has spent a long enough period of time spiralling outward near the (x, y) plane and is ready to take off. Specifically, from Eq. (5.5) the differential equation for the variable δy is written as
dδy2
The coefficients Bi and Di can be calculated from the initial condition of δx, δy and their derivatives.
First we focus on the case with η > 1. After the system has a spent a relatively long period of time spiralling outward near the (x, y) plane, the value of δy/δx can be approximated as
which depends only on η but not the initial value δx(0) and δy(0)! Because the end of the spiralling motion corresponds exactly to the beginning of the next phase,
that is, consecutive triggering regime, Eq. (6.4) effectively gives us the approximate directional distribution of the onset separation vector for the trigger part.
Fig. 6.2(a) compares Eq. (6.4) (thick black line) with the values of δy/δx collected for the true trajectory at the onset of the (consecutive) trigger part (green line) and of the spiral part (blue line). In this numerical experiment, the complete trajectory of funnel-type R¨ossler attractor is consisted of 185 consecutive triggering periods and 185 planar spiral periods. Thus, inside each of the 10 horizontal grids (corresponding to ε = 2, 3, ..., up to 11), there are 185 samples. From the figure we see that Eq.
(6.4) gives a fairly good prediction for the values of δy/δx at the onset of consecutive triggers.
The previous theory shows that the constancy of δy/δx at the onset of each trigger part is due to the fact that the planar spiralling motion prior to the onset is just the combination of a pure rotation and a simple radial expansion. However, a surprise is in store here: Taking a look at the blue curves of Fig. 6.2(b), which shows δy/δx in the consecutive triggering region, we immediately see that it also tends to converge to the value it started with at the end of each triggering event (the “teeth” near the bottom of the figure). Since the system is doing a sort of reinjection when trigger occurs, whose dynamics is rather different from that of the simple spiralling motion we encountered before, this begs for explanation.
To understand this in a quantitative way, we decided to analyze the situation using perturbation theory. Specifically, we will derive the solution to the variational equation of the coupled R¨ossler systems using the large ε expansion.
The variational equation of two R¨ossler systems with an x-x coupling is dδx
(6.4), the ratio of δy to δx at the onset of consecutive trigger can be approximated pη2− 1) with η = (a + ε)/2, provided that the system has spent some time
spiralling outward near the (x, y) plane. When the value of ε is very large, this ratio can be approximated as −2η = −ε. So we expand the solution of Eq. (6.5) as the series
where ε is the (large) coupling strength. Substituting Eq. (6.6) into Eq. (6.5) and collecting terms of the same order, we obtain
O(ε): Eq. (6.7) and Eq. (6.8) together allows one to solve for δx and δy up to the first two leading orders, and we obtain
δx '
where C2 is the initial value of δy and δz0 can be calculated via Eq. (6.8c).
Though x and z of R¨ossler equation cannot be solved analytically, we can still work something out when it is exhibiting triggering motion. To proceed, we note that we may neglect −y in Eq. (1.1b), because z typically is much larger than y at this stage. (For a refined treatment, please see the Appendix.) We can also safely
ignore b in Eq. (1.1c), because z(x − r) is much larger than b except near the moment when x = r. The simplified approximate equations become
dx
dt = −z, dz
dt = z(x − r).
(6.10)
The solutions of Eq. (6.10) are
x(t) =√
where E is the constant of integration [24]. Here, the origin of time has been chosen to correspond to the moment when z has reached its peak in a triggering event. (Thus, E/2 is simply the peak of z). Fig. 6.3(a) shows the comparison between the numerical trajectories of x and z with those predicted by Eq. (6.11) for a certain time span when z is triggered. Clearly, Eq. (6.11) is consistent with the numerical trajectories when z is large. We can plug Eq. (6.11) into Eq. (6.8c) to solve for the approximate form of δz0. The answer is
where K is some constant of integration. Predictions of Eq. (6.12) and the numerical values of δz when z is triggered are shown in Fig. 6.3(b). Substituting Eq. (6.12) into Eq. (6.9), we compare the solutions of δx and δy to the first two leading orders with their numerical values in Fig. 6.3(c). The approximate solution of δx successfully captures the characteristic sudden dip caused by the triggering motion one sees in the numerical simulation. To determine the direction of the separation vector, we
note that δy/δx of our approximate solution can be expressed as
Fig. 6.3(d) shows the comparison of the numerical values of δy/δx with that predicted by Eq. (6.13) when z is triggered. As can be seen, our approximate solution of δy/δx agrees rather well with the numerical results. From Fig. 6.3(b) and Eq. (6.12) we know the value of δz0 is very small at the moment when triggering begins or ends.
According to Eq. (6.13), the value of δy/δx at the end of a triggering motion can come close back to its starting value (≈ −ε), provided that ε is large enough compared to a and t, which happens to hold for our system. Indeed, this also explains why the scatter in the blue data points of Fig. 6.2(a) and their deviation from the theoretical prediction appear to be smaller for larger ε: Our perturbation theory employs a large ε expansion, and it works better when ε is large.
Next, we consider the case when η < 1. From Eq. (6.2), and after the system has spent some time spiralling out near the (x, y) plane, the value of δy/δx can be expressed as
From Eq. (6.14) and Eq. (6.15) we see that the value of δy/δx is dependent on the time t and the initial values of δx(0) and δy(0). Unlike the case with η > 1, the value of δy/δx of the case with η < 1 oscillates and does not converge toward some fixed value.
Because the system exhibits two different trends for δy/δx, depending on whether η > 1 or η < 1, we should accordingly adopt two different sampling strategies for the initial separation vectors. When ε > ε∗(η > 1), we should only sample the values of (δx, δy) for the initial separation vector based on that dictated by Eq. (6.4) and the renormalized condition: pδx2+ δy2 = d. But for ε < ε∗(η < 1), we should sample the values of (δx, δy) for the initial separation vector from the uniformly distributed points on a circle with the radius d0 lying in the (x, y) plane. Fig. 6.4 shows the MTLE versus ε for five cases: complete trajectory (black square), spiral in the (x, y) plane alone (red circle), consecutive trigger alone (blue uptriangle), the weighted average (green downtriangle), and the simple core (magenta line). The vertical orange line signifies where ε = ε∗. For the region where ε > ε∗, the ε-dependence of the weighted average is consistent with that of the complete trajectory when the appropriate sampling strategy has been adopted. Not surprisingly, the ε-dependence calculated from pure planar spiral motion agrees well with that of the simple core calculated from Eq. (5.6), but it alone cannot accurately reproduce the MTLE for any ε. For the region of ε < ε∗, the ε-dependence of the weighted average is still consistent with that of the complete trajectory (but again only if the appropriate sampling strategy has been adopted), and the simple core calculation is seen to agree with the true solution when ε is small. In all the above, the simple core approximation does not perform as well as it was reported in the previous work, simply because the attractor we are looking at has been chosen so that pure, planar spiralling motion does not tell the whole story of how the system gets its chaotic behavior. But since the motion in a funnel-type attractor can still be approximately treated in an analytically satisfactory way, one can still compute MTLE with success.