Boundary Surface and Stable Manifold in Sequential Circuits

Milan GUZAN

Department of Theoretical Electrical Engineering and Electrical Measurement, Technical University of Košice, Park Komenského 3, 040 01 Košice, Slovak Republic

milan.guzan@tuke.sk

Abstract. This item compares the boundary surface and stable manifold of a ternary memory. It points out to their consistency in the case of conventional saddle singularity. The other case, however, occurs at the second saddle – so called virtual. Then the boundary surface calculated using grid method differs from the stable manifold calculated by backward integration technique. The suggested procedure of calculating the stable manifold is hard for calculating and depicting a large number of trajectories - carriers of information about the stable manifold. After calculating the trajectories, stable manifold itself is depicted using the contour lines in the projection into u1, u2 plane.

Keywords Saddle point, boundary surface, stable manifold.

1. Introduction Every electrical circuit with at least two attractors

must contain regions that separate the attractors from each other. Therefore, in the field of mathematics, the term stable variety was introduced, in foreign literature - stable manifold (SM). Author of works [1] - [12] introduced the term, boundary surface (BS), independently on stable manifold term. Although the graphic representation of terms BS and SM is in most cases identical for sequential circuits, there are specific cases where this does not apply.

2. Stable Manifold and Boundary Surface The term SM was first introduced probably in [13].

There was just found the boundary, separating two stable states, by backwards integration technique in the plane - separatrix. Independently from the work [13], BS separating stable was graphically calculated and displayed, using the same technique of backward integration, or steady states each other not only in the plane (m = 2), but even for cases where m 3 [3] - [8]. Author of the mentioned works generalized the known, or proposed his

own new graphic methods for depicting trajectories or BS. Accuracy of these procedures was later verified by computer and positively commented in [14].

In [15], BS was calculated using PC by the grid method. It brought faster calculation and depiction of BS, compared with the graphic methods. This work gave the answer to question in [16]: "What boundary separates the domain of attraction for the chaotic attractor and the domain of attraction for the periodic attractor?" It was just BS, which could be visualized only by grid method with regard to existence of three unstable singularities in Chua`s circuit. But it was the first case, which did not reach the same results by backward integration and by grid method. Backward integration technique assumes that there is unstable singularity between stable singularities, which has the character of the saddle point. Saddle singularity is thus characterized (in system of the third order) with one positive eigenvalue and two negative eigenvalues, or negative real parts in the case of complex conjugate eigenvalues.

3. MVL Memory Circuit Multiple valued logic (MVL) memory circuit is

shown in Fig.1. Description of the circuit by differential equations and its parameters can be found in [17] in this proceedings, in the Introduction chapter. In the next section of this item we will consider supply voltage U=440mV, control pulse I=0, R=0, L=100pH and C=C1=C2 .

In [18], the case was shown where the circuit in Fig.1 was characterized for the parameters listed in Table 1 [17] and C=500fF, except the three stable singularities, also by other three undesirable stable limit cycles (SLC). Their presence caused a malfunction of the analyzed memory. Therefore, such values of C were searched, for which the circuit would be actually ternary. There were cross sections of BS made in u1,u2 plane for C=10fF to 500fF by grid method. In the range of C=10fF to 260fF the circuit was actually characterized only by three areas of attraction for stable singularities S1, S2 and S3, what the authors of [17] considered a condition for smooth memory control.

978-1-61284-324-7/11/$26.00 ©2011 IEEE

Fig. 1 MVL memory circuit

However, when cross sections were made through the unstable singularities N1, N2, it turned out that at C=10fF (Fig.2) BS does not pass through the saddle point N1 (BS does not separate attractors S1 and S2), as it is for C=20fF (Fig.3) to 260fF. IV characteristics of the element and the load in Fig.2a or Fig.3a in i,u2 plane correspond to parameters listed in Table 1 [17]. IV characteristics in plane u1,u2 when R = 0 are projected as the line, Fig.2b or Fig.3b. Fig.2 and Fig.3 present the Monge projection of the cross section of the BS through singularity N1. Singularities S1, S2 and S3 are stable and N1, N2 are unstable – according to Table 1, the conventional saddle points. Green color corresponds to the area of attractivity for the attractor S1 (in gray scale it is labeled as big number 1) and gray or red color illustrates the area of attractivity for the attractor S2 (big number 2) or S3 (big number 3). The question therefore is whether the element of the BS is a functional like in [18], where BS was divided due to the presence of SLC and N1, N2 were not conventional saddles.

C=C1=C2=10fF

N1 N2

1 5.5329285459354e+12 4.7971008544295e+12

2 -2.8645631623059e+12 -2.4183230407659e+12

3 -1.58365383629462e+11 -1.8877781366355e+11

C=C1=C2=20fF

1 2.6759750941224e+12 2.2941834990518e+12

2 -1.2303900314649e+12 -9.4724570137984e+11

3 -1.9058506265746e+11 -2.5193779767196e+11

Tab. 1 Eigenvalues of saddle points N1, N2.

The answer was given in work [19], where the Fig.4 showed that the element of the BS is a functional, even if initial conditions chosen for one or the other part of the element led to the same attractor S1. More detailed commentary on the Fig.4 is given in the already mentioned work [19]. It was very interesting that the trajectory was joined into single stream, denoted by large arrow. Therefore, the question arose, what the area looks like, which is invisible for the grid technique - Fig.2, but it causes the trajectories from Fig.4 bypass it?

Fig. 4 Trajectories of the conventional saddle N2 and virtual

saddle N1 for C=10fF. Some trajectories join into one bold trajectory denoted by the large arrow. More detailed commentary on the figure is in the work [19].

4. Calculation of the Stable Manifold Element of the BS (EBS) was first described in [6].

EBS is a tangent plane to BS in a saddle singularity, linear replacement of the BS in the vicinity of the saddle and the source of trajectories calculated by backward integration, which are part of SM. Thus it was possible to depict trajectories, which were part of BS or SM using graphic methods (in the absence of PC and later using grid

Fig.2 Cross - section through virtual saddle point N1 in Monge projection.

Fig.3 Cross-section through conventional saddle point N1 in Monge projection. Description of axes is the same as in Fig.2.2.

method). This finding was used independently on [6] also in work [20] using an analogue computer.

Based on [6], [21], the plane of the EBS is defined by linear transformation of system (1) as follows

1 11 12 1 13 2 0y i u u (5)

where 1k are the eigenvectors and the appropriate u1, u2, i corresponds to incremental quantities of state

variables. Values of eigenvectors and coordinates of singularities for N1 and N2 are listed in Table 2.

N1 N2 N1 N2

i [mA] 3,5 2,92 17,707 -760,71

u1 [mV] 354 91 -0,0202 -37,492

u2 [mV] 86 348 1 1

Tab. 2 Coordinates and eigenvectors of saddle points N1, N2.

In Fig. 2b and Fig. 3b the main line of the EBS is indicated by symbol o. On EBS - Fig. 2b was chosen a circle (yellow) of initial conditions (IC) with a radius of 26mV, not to exceed the breakpoints IV characteristic of the element. In the plane i,u2 in Fig. 2, the circle of Fig. 3b was projected as a line coinciding with the IV characteristic. This projection of IC is also marked by yellow. The circle in Fig. 2b was divided into 360 points / degrees. These are the initial conditions for trajectories calculated using the Runge-Kutta method by backward integration. All 360 trajectories in Monge projection are shown in Fig. 5. From this figure it is clear that there does exist some boundary dividing the trajectories converging to 1. from the trajectories converging to 2. . This boundary was found between 117° 118° and 297° 298°. It should be noted that each calculated trajectory in Fig. 5 and the following is a part of SM. The next calculation procedure was following: At first, the segment separating 1. from 2. was found and this segment was consequently divided again into 360 points. Calculation of IC (i, u1, u2) continued in this way on EBS until the result became stabilized up to the 15-th decimal place. Segment of 1° between 117° and 118° and between 297° and 298° has been divided in such way four times. The results of the calculation of trajectories and their depiction are shown in Fig. 6.

Since there is still unfilled space between the trajectories in Fig. 6, such intervals were subsequently re-searched, which would fill the unfilled space by the other trajectories. Example of a re-search of IC to fill space between the trajectories shown in Fig. 6 is illustrated in Fig.7, but only for a range of 117° to 118°. Particular colored segments indicate the interval in which the trajectories were calculated, whereas some sections were overlaid by previous trajectories. When we compare Fig.6b with Fig.7b, we could see that by further manual dividing we obtained new trajectories - carriers of information about the shape of SM. In Fig.5b and 6b, similarly as in Fig.3b

are also main lines of EBS plotted for N2 passing through the saddle N2.

Symbol + denotes the main line of the EBS for current level enlarged against current level of the saddle N2 by +1mA, i.e. 3.9mA, and - denotes the main line of EBS for N2 for the corresponding current level of 1.9mA. The main line o corresponds to the current level of saddle N2 - iN2=2.9mA. These designations were retained also because of the Fig.8, because the shape of SM is obvious from Fig.8 for both N1 (red points), and for N2 (green points).

Fig. 7 The example of next fine dividing to fill unfilled (white)

space from Fig.6 between 117°and 118°

Fig. 5 360 depicted trajectiories

Fig. 6 Other trajectories obtained by successive dividing between 1. a 2.

Fig. 8 Stable manifold depicted by contour lines for virtual

saddle N1 and conventional saddle N2

Although N2 is conventional saddle, there was similar calculation of SM made for N2 to compare the accuracy of calculating SM for N1. There was also the circle chosen on the EBS passing through conventional saddle N2. Boundaries found on this circle were located between 153°

154° and 333° to 334°. In the case of the saddle N2, BS is identical to the SM. Points delineated in Fig.8 representing SM of both singularities N1 and N2 are the results of intersection of trajectories shown in Fig.5 7 with current levels from 0.5mA to 5mA with 0.5mA step. From Fig.8 it is clear that the SM in the neighborhood of N1 is steeper than in the neighborhood of N2. More detailed calculation of the SM should be carried out also for the region u1x u2, i.e. for u1 <0.22 to 0.44 V> x u2<0.22 - 0.44 V>. This will be the subject of other activities of the author.

Conclusion The procedure for calculating the depiction surfaces

shows that although there is no uniform method for calculation and depiction of surfaces separating attractors from each other in the state space, combination of grid method and backward integration step technique makes possible the depiction of such new phenomena as the double-arm stable manifold, published in work [21]. Mentioned analysis and circuit simulation can help to understand the behavior of sequential or chaos-generating circuits in case of their failure or other problems.

Acknowledgements We support research activities in Slovakia / Project is co-financed from EU funds. This paper was developed within the Project "Centre of

Excellence of the Integrated Research & Exploitation the Advanced Materials and Technologies in the Automotive Electronics", ITMS 26220120055

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