So far we have mainly focused on the synchronous stability of coupled R¨ossler sy-stems with an x-x coupling. The synchronous stability of two coupled spiral-type R¨ossler attractors with the other coupling cases (x-y coupling, y-x coupling and y-y coupling) have been studied in Ref. [7]. There, the authors showed that the simple core approximation can be used to successfully estimate the synchronous stability because the steady outward spiralling motion near the (x, y) plane dominates the full trajectory. For the two coupled funnel-type R¨ossler attractors with an x-x coupling, we have shown that the synchronous stability must be reconstructed using the weig-hted average of MTLEs of the trigger part and the planar part. In this section we would like to show that other coupling possibilities also exhibit similar results. For definiteness, the parameters of the funnel-type R¨ossler attractor are fixed as follows:
a = 0.375, b = 0.2 and r = 7.
According to Ref. [7], the approximate MTLEs of the simple core analysis with the other coupling cases can be calculated as follows.
(i) x-y coupling:
The coupling matrix E is expressed as
E =
and the variational equation of two coupled simple cores is
which corresponds to the Eq. (13) of Ref. [7]. (But do notice that the original notation of the MTLE, λ⊥of Ref. [7], has been changed to λmax in the present work, and the parameter ω of Ref. [7] has also been set to 1.)
(ii) y-x coupling:
The coupling matrix E is expressed as
E =
and the variational equation of two coupled simple cores is
which corresponds to Eq. (15) of Ref. [7].
(iii) y-y coupling:
The coupling matrix E is expressed as
E =
and the variational equation of two coupled simple cores is
which corresponds to Eq. (17) of Ref. [7].
Fig. 6.5(a)(b)(c) show the MTLE versus ε of the complete trajectory (black square), the spiral in the (x, y) plane (red circle), the consecutive trigger (blue up-triangle), the weighted average (green downup-triangle), and the simple core (magenta line) when the coupling is x-y, y-x, and y-y, respectively. The magenta lines of Fig.
6.5(a)(b)(c) are calculated from Eq. (6.17), Eq. (6.19) and Eq. (6.21), respectively.
The initial separation vectors are directly copied from the complete trajectory when we do the weighted average. The three figures show that the simple core approxima-tion (magenta line) no longer gives a satisfactory result if a significant porapproxima-tion of the complete trajectory rises high above the (x, y) plane, though it does provide partial agreement when one just focus on the orbital motion near the (x, y) plane. The ε-dependence of MTLE can be successfully reconstructed using the weighted average method, which is improved from the simple core analysis by taking into account the fact that chaos in the system under consideration is derived from two main sources:
the almost planar spiralling motion, and the z-triggering events.
6.4 Possible future work
It seems that this idea should be equally applicable to other simple chaotic systems such as the Lorenz attractors [5]. Indeed, for the Lorenz system described by
dx
where, σ, ρ and β are parameters with suitably chosen values, the Lorenz attrac-tor looks loosely like two thin leaves symmetrically placed and tilted in the phase space. For this geometry, we may conveniently divide the complete trajectory of the Lorenz attractor into two parts: The spiral part and the transition part. The spiral part is composed of two approximate planar spiral motions around the fixed points (±pβ(ρ − 1), ±pβ(ρ − 1), ρ − 1), each sitting at the center of its associated leaf.
When the trajectory switches from one leaf to another, the path it takes falls into the transition part. Using idea similar to the simple core analysis of the two coupled R¨ossler attractors, the MTLEs of the spiral parts of the coupled Lorenz attractors can be approximately calculated by the linear stability analysis at the fixed point. For the spiral part, our preliminary calculations show that the distribution of the MTLE versus the coupling strength calculated by the linear stability analysis is consistent with the distribution of the numerical results, provided that a proper range of the spiral motion is defined. (But we are still working on how to define the range of the spiral motion in a most sensible way.) Our preliminary success with the current method prompts us into expecting that the MTLE of the complete trajectory can be reconstructed by the time-weighted average of the MTLEs of the spiral part and transition part if the proper initial separation vectors are chosen.
6.5 Brief summary
In this work, we have analyzed the synchronous stability of two coupled R¨ossler at-tractors, be they of spiral-type or funnel-type. For spiral-type attractor, the spiralling motion near the (x, y) plane dominates most part of the time evolution so that the simple core approximation is a good method for the investigation of the synchronous stability. In contrast, the funnel-type R¨ossler attractors spend quite some time exhi-biting consecutive triggering behavior, which does not bear any direct resemblance to the spiralling motion, and simple core cannot be used to quantify synchronous chaos.
To improve on this, we have divided the complete trajectory of the funnel-type R¨ossler attractor into two parts: the (planar) spiral part and the consecutive trigger part. The MTLEs for the two parts can be calculated separately, and their time-weighted average can be calculated, which is shown to faithfully reproduce the value computed from the complete trajectory.
We also investigated the distribution of the separation vector ~dn at the onset of each type of motion, and found that its distribution exhibits very simple features.
Specifically, there is a critical value ε∗ of the coupling strength ε above which ~dn
at the onset of a triggering event is restricted to a particular direction. We can analytically determine this direction using simple core analysis. Surprisingly, ~dn at the onset of a spiralling motion also tends to align itself to that same direction when ε is large. We provided a theoretical analysis showing why this must be the case using the large ε expansion. For ε < ε∗, one may approximately assume that the direction of ~dn is uniformly distributed along a circle lying in the (x, y) plane. Our conclusion is not restricted to the x-x coupling only. We showed that other coupling schemes also satisfy the conclusions we have drawn from our study.
Figure 6.1: Different sampling strategies affect the computed ε-dependence of the MTLE. Shown in all figures are: complete trajectory (black square), planar spiral (red circle), consecutive trigger (blue uptriangle), the weighted average (green do-wntriangle), and the simple core (magenta line). The initial separation vectors are sampled differently: (a) Directly copied from the true trajectory. (b) Sampled from the uniformly distributed points on the surface of a sphere of radius d0. (c) Sampled
6.4
Figure 6.2: (a) Eq. (6.4) (thick black line) and the values of δy/δx at the onset of the consecutive triggering region (green line) and the planar spiral region (blue line). (b) The trajectories of x, z and the values of δy/δx within a certain consecutive triggering period when ε = 10.
Figure 6.3: (a) The comparison between the numerical trajectories of x and y with those predicted using Eq. (6.11) during one triggering event. (b) The comparison between the numerical values of δz and Eq. (6.12). (c) The comparison between the numerical values of δx, δx and the results of perturbation calculation up to the second order. (d) The comparison between the numerical values of δy/δx and Eq. (6.13).
Figure 6.4: The ε-dependence of MTLE for the complete trajectory (black square), planar spiral (red circle), consecutive trigger (blue uptriangle), the weighted average (green downtriangle), and the simple core (magenta line). The orange vertical line signifies ε = ε∗. For ε > ε∗, the initial separation vector at the onset of either the spiralling motion or the consecutive triggering event is sampled using Eq. (6.4) and the renormalized condition: pδx2+ δy2 = d. When ε < ε∗ the initial separations are sampled from the uniformly distributed points on a circle with the radius d0 that lies in the (x, y) plane.
Figure 6.5: The ε-dependence of MTLE for different coupling schemes: (a) x-y cou-pling, (b) y-x coupling and (c) y-y coupling. The initial separation vectors for either the spiral region or the consecutive triggering region are sampled from the complete trajectory (black square), planar spiral (red circle), consecutive trigger (blue
uptri-Chapter 7 Conclusion
In this thesis, certain interesting properties of funnel-type R¨ossler attractor are in-vestigated, including the spike dynamics and the synchronous stability. First, we analytically study the dynamics of a typical trajectory in a funnel-type R¨ossler at-tractor. The necessary condition for the R¨ossler system to exhibit the consecutive triggering behavior is derived. By performing a suitable averaging method we get the approximate form of variable y in the consecutive triggering period. The trend of y versus time with different values of a is qualitatively explained, and its quan-titative agreement with numerical simulations is very good. Furthermore, we use a sort of connection formula to connect the approximate forms of variable z between the different regions in the consecutive triggering period and describe the originally more difficult time-continuous chaotic behavior of the system using an approximate four-parameter surrogate iteration scheme.
The surrogate iteration scheme has the merit of helping one understand more easily, and analytically, why the peak-to-peak durations and the heights of the peaks of the z variable behave the way they do. All the analytical results are well consistent with those obtained through the numerical integration of the system. Our approach can explain some important properties which are observed in the R¨ossler attractor.
The numerical results show the phase shift of the variables x and z between the equation with the average approximation and without the averaging approximation.
Finding the reason behind the phase shift should be an interesting endeavour, which
I plan to explore in the future.
We also investigate the synchronous stability between two coupled R¨ossler attrac-tors with a unidirectional diffusive coupling. A previous research proposed the simple core equation which approximates the trajectory of the R¨ossler attractor as a planar outward motion and used it to effectively describe the synchronous stability of two coupled R¨ossler systems. We show that this is a good approximation only when one is dealing with the spiral-type attractor. But when the funnel-type attractor exhibiting repeated motion of trigger-and-reinject becomes equally important as the spiral mo-tion, the (consecutive) triggering part must also be explicitly included in the analysis.
We show the synchronous stability of the two coupled funnel-type R¨ossler attractors can be reconstructed using the time-weighted average of MTLEs of the two parts:
spiral part and trigger part.
We also analytically investigate the time evolution of the separation vector. When the coupling strength is large enough, the separation vector tends to a specific di-rection whether the system has spent a long enough period of time spiralling outward near the (x, y) plane or exhibiting the consecutive triggering motion. For the spi-ral motion, we perform the simple core analysis to show that the separation vector evolves in time so that the onsets of the trigger part can be determined. But what is more interesting is: Even for the triggering motion which exhibits the very rapid up and down motion, the separation vectors still retain the regularity after the trig-gering motion. The onsets of the spiral part can be perturbatively calculated. The fact actually serves as a reminder that chaos does not necessarily imply randomness.
Besides the R¨ossler system, the synchronous stability of the coupled Lorenz systems has been partially analyzed using our proposed method. Work along this line is under way.
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Appendix A Appendix
In this appendix some supplementary materials to the research contents described in the main text are included here for completeness. They include the detailed descrip-tion of the steps leading to the approximadescrip-tion of Ti in the recurrence relation, and the refined treatment of the simplified equations of the R¨ossler attractor in the triggering stage.
A.1 The detail of the approximation leading to the recurrence relation
In Chapter 4 we compare the numerical results of Eq. (3.4) with our analytical ap-proximate forms Eqs. (3.32)–(3.35). If the recursion relations of parameters are based on Eq. (3.31) ( Eq. (4.6) ), we call it the first (second) set recurrence relation. Fig.
4.5 shows the comparison of the calculated parameters for the numerical results, using the first recurrence relation, and using the second recurrence relation, respectively.
Besides, there are several intermediate steps involving approximations which connect Eq. (3.31a) and Eq. (4.6a).
Eq. (4.1) is the equation we obtain after we omit the first term of Eq. (3.31a), which is all right because it is much smaller than the other terms. We also expand the three exponential functions to the second order in the coefficient a to get the
approximate cubic equation Eq. (4.2). Next, in our interested range of parameters some terms of Eq. (4.2) can be ignored, for their numerical values are much smaller than the values of the terms that have been retained. After all these have been done, the approximate quadratic equation Eq. (4.4) is finally arrived at.
Fig. A.1(a) shows Ti + Ti0 versus i in the numerical result and 2Ti versus i for the five approximate steps. Fig. A.1(b) shows the normalized results Zi/Z1 versus i in the numerical result and Ei/E1 versus i for the five approximate steps. In both Fig. A.1(a) and Fig. A.1(b), we see that the results of the first set of the recurrence relation, no first term approximation and the approximate cubic equation are closer to the numerical result than that provided by the other two approximations. We have traced back the source of the major error, and it is caused by our approximating the cubic equation by a simpler quadratic equation.
A.2 The refined treatment of the R¨ ossler equation in the triggering stage
In the main text of this article we showed that, under the approximation of Eq. (6.10), one can derive an approximate form for the time dependence of x and z, which in turn allows us to derive an approximate form of δz0. This information on δz0 then makes it possible for us to get the perturbed solutions of δx and δy to the first two leading orders. Here in this Appendix we show that actually a refinement on the time dependence of x and z is possible.
From Eq. (5.8) or Fig. 3.1(b) we conclude that ¯y < 0 is a necessary condition for the curve relating x and z to form a closed loop. And this also corresponds to having the possibility of exhibiting consecutive triggers. For such a situation, a first order approximation can be made by substituting the y on the right hand side of Eq. (1.1a) with an effective constant ¯y which is negative-valued. Within a triggering event, the
approximate equations of x and z can be written as Eq. (5.7)
These coupled equations can be solved to yield x(t) =√
where E is a constant of integration. Fig. A.2(a) shows the comparison between the numerical trajectories of x, y and z with those predicted by Eq. (A.2). The agreement between Eq. (A.2) and the numerical results is seen to be excellent. We can further plug Eq. (A.2) into Eq. (6.8c) to solve for the approximate form of δz0, and the result is
where K is yet another constant of integration. Eq. (A.3) and the numerical values of δz are shown in Fig. A.2(b). By substituting Eq. (A.3) into Eq. (6.9) the solutions of δx and δy to the first two leading orders and their numerical counterparts are compared in Fig. A.2(c). Once again, the approximate solution successfully captures the characteristic dip in δx. The approximate expression for δy/δx can be derived from
Fig. A.2(d) shows the comparison of the numerical values of δy/δx and Eq. (A.4) within one triggering event. The approximate solution of δy/δx is seen to fit well with numerical results.
Figure A.1: (a) Ti+ Ti0 versus i in the numerical result and 2Ti versus i for the five approximate cases. (b) The normalized results Zi/Z1 versus i in the numerical result and Ei/E1 versus i for the five approximate cases.
(a) (b)
(c) (d)
Figure A.2: (a) The comparison between the numerical trajectories of x, y and z and Eq. (A.2) during a triggering event. (b) The comparison between the numerical values of δz and Eq. (A.3). (c) The comparison between the numerical values of δx, δx and the results of perturbation calculation up to the second order. (d) The comparison between the numerical values of δy/δx and Eq. (A.4).