In Section 3.1 we replaced x of Eq. (3.1b) with a fixed value hxi = 0 and analytically explained the behavior of y as we vary a in R¨ossler attractor. The solution of y of Eq. (3.4b) is
where D = r/a − y(0) is a parameter whose value depends on the origin of time we have chosen. In this section we will find the approximate analytical forms of x and z of Eq. (3.4) in the different regions of consecutive triggering period, and the relation between different periods will be connected by a sort of connection formula [40].
Fig. 3.4(a) shows z versus t for Eq. (3.4) and we have divided the certain period which includes two peaks and two troughs into four regions. In region I we may
neglect z in Eq. (3.4a), for z is small enough in comparison to y. The approximate equation in region I is
dx In the above, C1 is a constant of integration, and T1 is the length of time between the origin and the point when z climbs out of the previous trough. Fig. 3.4(b) shows z versus t and Eq. (3.15) with the appropriate value of C1 in region I. Similarly the solutions of z in region III are
z(t) = C20e2ar(t−T10)2−D1a2ea(t−T 01)+D1a (t−T10). (3.16) In region II the value of z is much larger than the value of y, so this time we neglect y in Eq. (3.4a). The approximate equation in region II is
dx
dt = −z, dz
dt = zx.
(3.17)
The solutions of Eq. (3.17) in region II are x(t) =p
where E1is a constant of integration. Fig. 3.4(c) shows z versus t and Eq. (3.19) with the appropriate value of E1 in region II. From Fig. 3.4(a) we observe the duration
short (near 0.12), which is much smaller than T1. The form of Eq. (3.15) in the matching area can be approximated as
z(t) = C1e2ar (t+T1)2−D1a2ea(t+T1)+D1a (t+T1) Because the time of the matching area is about -0.12 and the value of E1 is near 1000, which render e−
√E1t quite large when t is substituted by this value, the form of Eq.
(3.19) in this region can be approximated as z(t) = E1
Comparing Eq. (3.20) and Eq. (3.21) we get the relation
C1e2arT12−D1a2eaT1+D1a T1 = 2E1, (3.22a)
Similarly we can use the same procedure to connect region II and region III and get another relation
Comparing Eq. (3.22b) and Eq. (3.23b) and expanding eaT1 and e−aT10 to the first order, we get T1 = T10 and
The value of D depends on the selection of origin of time. Now we choose t = T10+ T2 as a new origin of time to get D2 where T2 is the length of time between the
next trough and the peak following it. We can get the relation between D1 and D2 as
D2 = D1ea(T10+T2)
= D1ea(T1+T2). (3.25)
From Eq. (3.24) the relation between E1 and E2 can be rewritten as E2 =hr
a − D2 T2i2
= E1 r − aD1ea(T1+T2) r − aD1
T2 T1
2
. (3.26)
Now we will solve for the relation between C1 and C20. From Eq. (3.22a) and Eq.
(3.23a) we get
C20 = C1e−2D1a2 (sinh aT1−aT1). (3.27) Note that the value of C20 depends on the choice of the initial moment of time, and it will change to C2 when the origin of time is taken to be T1+ T2. The relation between C20 and C1 is then
C20e−D1a2 = C2e−D2a2
= C2e−D1a2ea(T1+T2) (3.28)
and we can combine Eq. (3.27) and Eq. (3.28) to get
C2 = C1e−2D1a2 {sinh aT1−aT1+12[1−ea(T1+T2)]}. (3.29) The last step is to connect T1 and T2. Using Eq. (3.22a), Eq. (3.23a) and Eq.
(3.28) we get
2E1e−2arT12+D1a2e−aT1+D1a T1 · eD1a2[ea(T1+T2)−1] = 2E2e−2arT22+D2a2eaT2−D2a T2, (3.30) where E2 =T2r/a − D1ea(T1+T2) 2
. The recursion relations of the four parameters
described above can be rewritten as Having obtained the recursion relations for the parameters, we may derive an approximate expression for the original variables x and z in terms of the auxiliary parameters introduced above. For example, x and z in the region with z << y can be written as where i is the index of the region of interest. The analytical forms of x and z in the region with z >> y can be written as
x(t) =p
Finally, the cumulative times τi and τi0 are defined respectively as
Figure 3.1: (a) Three variables x,y and z of Eq. (3.1) versus t with a = 0.25, r = 14.97 and the initial conditions (0, 51.9, 500). (b) x versus ln z for Eq. (3.3) with ¯y1 = 50,
¯
y2 = 25, ¯y3 = 0 and ¯y4 = −50. Here the integrated constant C = 378.54.
Figure 3.2: (a) x,y and z of Eq. (3.4) versus t with a = 0.25, r = 14.97 and the initial conditions (0, 57.88, 500). (b) The numerical values y of Eq. (3.1) versus t with the initial condition y0 = 51.9 and the approximate form Eq. (3.5) with y(0) = 57.88 for different values of a. The approximate form successfully captures the decreasing trend of the numerical results. (c) ∂y∂a versus t with different a.
Figure 3.3: (a) The (x, y) projections of three trajectories of Eq. (1.1) with a = 0.1, a = 0.23 and a = 0.36. (b) The (x, y) projections of three short segments of the trajectories with the maximal value of y0 which are captured from (a) . Here we have changed the sign of y and set r = 14, b = 0.1.
Figure 3.4: (a) z versus t for Eq. (3.4), and we have divided the certain period which includes two peaks and two troughs into four regions. (b) z versus t and Eq. (3.15) with the appropriate value of C in region I. (c) z versus t and Eq. (3.19) with the
Chapter 4
Results and discussion of the analytical form
In Chapter 3 we derived the analytical form of the funnel structure of an approx-imate R¨ossler system. The originally different time-continuous chaotic behavior of system can be reduced into an approximate four-parameter recursion relations. In this chapter we will compare the approximate analytical forms obtained by the ite-ration scheme with the numerical solutions. The analysis about some properties of our approximate model will be described and how they might help us understand the actual R¨ossler attractor will also be explained. Besides, the numerical integration shows the obvious phase shift of the variables (x and z) between the equation without the average approximation and the equation with the average approximation hxi = 0.
We plan to find out the mechanism behind this phenomenon in our future work. In the last section we briefly summarize this study.
4.1 Comparison to numerical solutions
Fig. 4.1 and Fig. 4.2 show the comparison between the numerical solutions of Eq.
(3.4) and the approximate forms Eq. (3.32)(3.33)(3.34)(3.35). The recursion relations of parameters Ti, Di, Ei and Ci are based on Eq. (3.31). Here we have chosen the value of T and D in such a way that the computed values of x and z correspond
precisely to the initial values of the system which are described by Eq. (3.4). We then calculate E1 and C1 via Eq. (3.24) and Eq. (3.22a ) and start our recursion scheme. The computed time dependence of the variables are then checked against the numerically integrated results of the system.
Fig. 4.1 shows the case with a = 0.25 and r = 14.97 and the initial conditions (0, 57.88, 500). From Fig. 4.1(a)(b) we see that the trends of the heights of the peaks and troughs of z in the approximate form are quite consistent with the numerical results. Furthermore, the parameter T of Eq. (3.31a) will increase with each iteration, which implies that the duration between two consecutive peaks and the duration between two consecutive troughs both increase with time. This trend is also consistent with the numerical results. Not only that, when the number of iterations of the recursion relation Eq. (3.31a) exceeds a certain value, no positive solution of Ti+1
that can meaningfully correspond to consecutive trigger will be found. That is, there is an upper bound to the allowed number of iterations. Put differently, the R¨ossler system can not “fire” indefinitely, and our theory can be used to predict the number of triggers allowed for a given initial conditions on the attractor. From Fig. 4.1(c) we also observe that the approximate form of x is well fit to the numerical solution. Fig.
4.2 shows the case with a = 0.375, r = 14.97 and the initial condition (0, 38.92, 500).
Again, we see that the trend of Ti and the trends of height of peaks and troughs of our approximate formulas compare favorably with numerical solutions.
Because the dependence of Ti+1on the parameters cannot be solved exactly via Eq.
(3.31a) with each iteration i , we now try to develop further approximations to get a more definite expression. To proceed, we note that the term ln
Ti+1[ra−Diea(Ti+Ti+1)]
Ti(ra−Di)
2
is much smaller than the other terms so we may regard it as a perturbation. The form of the remaining terms is
r
We now expand the three exponential functions and ignore all terms which are third order in the coefficient a or higher. Because we can easily check that Ti+1 = −Ti is
one solution of Eq. (3.31a) and Eq. (4.1), which is trivial and of no use to us, we may write the finished equation as
(Ti+ Ti+1)
We can rewrite Eq. (4.2) as
8
In our interested range of parameters 6r/5a3Di and 6/5a2 are much larger than 2Ti/5a, and 8/5a is much larger than Ti, and so we will ignore these terms. The The solution of Eq. (4.4) is
Ti+1 =3(r − aDi)
inside the radical sign is indeed less than unity so that the approximation makes sense. Next, we retain up to the second order term in this expansion and combine Eqs. (3.31b)(3.31c)(3.31d) into a new set of even more simplified recursion relations
involving four parameters as Fig. 4.3 and Fig. 4.4 show the comparison between the numerical solutions and this further simplified approximate forms. Specifically, Fig. 4.3 is the case with a = 0.25, r = 14.97 while Fig. 4.4 is for the case with a = 0.375, r = 14.97. The initial conditions for these two cases are the same as in Fig. 4.1 and Fig. 4.2, respectively.
Although the peaks of z of this further simplified form in Fig. 4.3 decays faster than the case in Fig. 4.1, the decreasing trend is still intact. Also, the cumulative time τi and τi0 in Fig. 4.3 are smaller than those in Fig. 4.1 for the same i.
The comparison of parameters using the first (Eq. (3.31)) and the second (Eq.
(4.6)) set of recursion relation with the numerical results is shown in Fig. 4.5. In particular, Fig. 4.5(a) shows Ti+ Ti0 versus i in the numerical result of Eq. (3.4) and 2Ti versus i for the two sets of recursion formulas. Ti + Ti0 in the numerical result is simply the duration between two troughs of z and 2Ti in the recursion formulas correspond to the same thing because we had assumed Ti = Ti0 in our approximation.
In all the three cases, the durations between two consecutive peaks are observed to increase with each trigger. However, for the same index i, the predicted value of 2Ti from the first set of recursion formula is slightly less than the value of Ti+ Ti0 of the numerical result, and the predicted value of 2Ti using the second set of (ever more crude) recursion formula is seen to yield a result off by more, which is understandable.
Fig. 4.5(b) shows Zi versus i in the numerical result and Ei/2 versus i for the two sets of recursion formulas. Zi and Ei/2 represent the height of the peak of z.
Fig. 4.5(c) shows the normalized results Zi/Z1 and Ei/E1 versus i. We can also observe the normalized value Ei/E1 of the first recurrence case is slightly less than the normalized value Zi/Z1 of numerical result. All in all, it is encouraging to see
that the computed trends do agree with numerical integrations of the system which is described by Eq. (3.4).