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Domains of attraction for multiple limit cycles of coupled van der pol equations by simple cell mapping
ht. J. Non-Linear Mechanics, Printed in Great Britain.
Vol. 20, No. S/6, pp. 507-517.
OOZO-7462/85 Pergamon
1985
$3.00 + .oO Prss Ltd.
DOMAINS OF ATTRACTION FOR MULTIPLE LIMIT CYCLES OF COUPLED VAN DER POL EQUATIONS BY SIMPLE CELL MAPPING JIANXUE Xu Department of Engineering Mechanics, Xi’an Jiaotong University, Xi’an, Shaanxi Province, The People’s Republic of China R. S. GUTTALU Department of Mechanical Engineering, University of Southern California, Los Angeles, CA 90089, U.S.A.
and C. S. Hsu Department of Mechanical Engineering, University of California, Berkeley, CA 94720, U.S.A. Ahstrati-One of the most ditIIcult tasks in non-linear analysis is to determine globally various domains of attraction in the state space when there exist more than one asymptotically stable equilibrium states and/or periodic motions. The task is even more demanding if the order of the system is higher than two. In this paper we consider two coupled van der Pol oscillators which admit two asymptotically stable limit cycles. For systems of this kind we show how the method of cell-to-cell mapping can be used to determine the two four-dimensional domains of attraction of the two stable limit cycles in a very effective way. The final results are shown in this paper in the form of a series of graphs which are various two-dimensional sections of the four-dimensional state space.
1. INTRODUCTION
In the classical analysis of non-linear analysis [l-4], once the equations of motion for a system have been formulated, one usually tries first (i) to locate all the possible equilibrium states and periodic motions of the system. Thereafter, one would (ii) study the stability characteristics of these solutions. Logically, the next task is (iii) to find the global domain of attraction in the state space for each asymptotically stable equilibrium state or periodic motion. In some sense one could say that to find and to delineate these domains of attraction is really the ultimate aim of any analysis of a non-linear system. However, when one examines the literature, one finds that while the tasks (i) and (ii) are subjects of a great number of investigations, the task (iii) has not been dealt with as often, particularly when the order of the system is higher than two. This lack of intensity of activity in this direction is probably due to the fact that there is no generally valid analytical method available for this task when the system is other than weakly non-linear. A direct numerical integration approach seems to be the only viable one. However, such an approach is usually prohibitively time-consuming even with the present-day high power computers and, therefore, is not a practical or attractive one. In the last few years, as an attempt to find more efficient and practical ways of determining the global behavior of strongly non-linear systems, methods of cell-to-cell mapping [5-91 have been proposed. These methods are currently being developed in various directions. Up to now two types of cell mappings have been investigated. One is the simple cell mapping [5,6] and the other the generalized cell mapping [7,8]. Both types can be used effectively to determine the global behavior of non-linear systems [6,8,10,11]. Moreover, as new tools of analysis, cell mappings have led us to new methodologies in several other directions. For instance, the generalized cell mapping has been found to be a very efficient method for locating and determining the statistical properties of strange attractors [12]. In the field of automatic control, the cell mapping concept has allowed us to devise a new discrete method of optimal control [13]. The cell mapping method has also implications on the numerical methods of locating all the zeros of a vector function [14]. Up to now, as a method to determine domains of attraction the simple cell mapping has only been applied to second order systems. In this paper we extend the usage to fourth order systems. Specitically, we study a system of two coupled van der Pol oscillators which possess 507
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J. Xu et al.
two stable limit cycles and we use the simple cell mapping to determine the two domains of attraction. 2.
COUPLED
VAN
DER
POL
EQUATIONS
Consider a system of two coupled van der Pol equations.
lY11 j;l
“2
-p
1-y:
0
0
1-y;j,
91 10
-v
l+v
+
l+q+v
1 -v
y1
IO y2
=
0
00’
P-1)
Here v is the coupling coefficient and rl may be called a detuning coefficient. If v = q = 0 the system is reduced to two identical uncoupled van der Pol oscillators. If v = 0 it is reduced to two uncoupled but not identical oscillators, one with its undamped natural circular frequency equal to 1 and the other (1 + q) ‘j2. This system with additional linear velocity coupling terms has been considered by Pavlidis [ 151 and Rand and Holmes [ 161. An excellent and detailed perturbation analysis of this system when ,u,q and v are small is given in [16]. The analysis gives us the conditions under which there exist 4,2 or 0 limit cycles. They have also studied stability of these limit cycles. However, the third task mentioned in the Introduction of determining the domains of attraction when there are two stable limit cycles is not considered in [ 161. As far as the notation is concerned, p, q and v used in this paper are connected with E, A and u used in [16] through v = Ea.
rj = &A,
p = 6,
(2.2)
For the subsequent discussion, it is more convenient to write (2.1) as a system of four first order equations. Letting Xl = Yl,
x2 = $1,
x3 = Y2,
x4 = 32,
(2.3)
we have i = 1, 2, 3, 4,
kj = E(x),
(2.4)
where Fl =
x2,
Fz =
p(l - x:)x, - (1 + v)xr +
F3 =
x4,
F4 =
~(1 -
vx3,
(2.5) x:,x,
+
vxl
-
(1 + u + v)xs.
Many cases with different sets of values of ,u, rl and v have been studied. Here we present three cases for which there exist two stable limit cycles. First consider the case with p = 0.1, q = 0.04, and v = 0.1. To exhibit the two limit cycles which reside in a 4-D state space, we plot their projections on the xl - x2, xl - x3, xl - x4, x3 - x2, x2 - x4, and x3 - x4 planes as shown in Fig. 1. These results as well as the limit cycle results for the other cases to be shown later were obtained by using the fourth-order Runge-Kutta method of numerical integration. From Fig. l(b) it is readily seen that the limit cycle labelled “the jirst” is essentially an “inphase coupled” limit cycle while the one labelled as “the second” is essentially an “out-of-phase coupled” limit cycle. This is even better seen in Fig. 2 where the time histories of the two limit cycles for this case are shown. The periods of these two limit cycles are, respectively, 6.246 and 5.668 units. In Fig. 3 we show the two stable limit cycles for the second case p = 0.25, q = 0.04, and v = 0.1. Here, the departure from 0” or 180” of the phase difference of the phase-locked coupled motions is more pronounced than that of the first case. Also, there is more distorsion from a circular shape in Figs 3(a) and (f), indicating substantial higher harmonics in the limit cycles.
Multiple limit cycles of coupled van der Pal equations
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x” 0
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-2 .:Q -2
0
2
(f)
2nd \ 1st
x”
0
-2 21Q -2
0
2
x3
Fig. 1. Projections of the two limit cycles for the case p = 0.1, 1 = 0.04 and Y = 0.1. (a) 2-
0
I
I
I
3.123
6.246
Time I.
B
2.634 Time
Fig. 2. Time histories of the two limit cycles for the case p = 0.1, q = 0.04, and Y = 0.1.
J. Xv et al.
510
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, 2$$ (cl
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1st
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0
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2
x3
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x"
o-
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0
2
5
@ -2
0
2
x3
Fig. 3. Projections of the two limit cycles for the case p = 0.25, q = 0.04, and v = 0.1.
In Fig. 4 we show the two limit cycles for the third case with /A= 0.1, q = 0.04, and v = 0.02. Here, the coupling is stronger than in the first case. The results indicate that the phase difference departure from 0” or 180” is fairly small. Before leaving this section we wish to make one remark about the conditions derived in [ 161 concerning the number of limit cycles a system may possess. These conditions are based upon an asymptotic perturbation analysis with ~1as the smallness parameter. In general, they may be no longer adequate when ~1is not very small. For example, when p = 0.35, rl = 0.04, and v = 0.1, one stable limit cycle is found while, according to the criteria derived in [16], no phase-locked periodic solution is expected. 3. METHOD
OF SIMPLE
CELL MAPPING
Before taking up the matter of determining the domains of attraction for the two limit cycles, we shall give in this section a brief description of the method of simple cell mapping to be used for this purpose. For a more detailed discussion of the method thelreader is referred to [5,6 I. Consider a dynamical system governed by (2.4). The basic idea of cell mapping is to consider the state space not as a continuum of uncountable number of points but rather as a collection of a large number of state cells, with each cell being a state entity. The cell structure over the state space may be introduced in a variety of ways. The simplest one is the following.
511
Multiple limit cycles of coupled van der Pol equations
x”
o-
-2
-2
0
2
x3
Fig. 4. Projections of the two limit cycles for the case p = 0.1, f = 0.04, and v = 0.2.
A four dimensional state cell is designated by a cell vector Z which has integer-valued components Z1 , Zz, Z3, and Zq. A point x of the state space with components x1, x2, x3 and x4 belongs to a cell Z if and only if (,i-_)hisxi<
(Zi+f)hi,
i=1,2,3,4,
(3.1)
where hi is the interval size associated with xi. Having constructed the cell state space, we can create a cell to cell mapping for the dynamical system (2.4) in the following manner. For an autonomous system (2.4) we first choose an appropriate time interval T.Apart from a requirement that T should not be too small, its precise magnitude is not crucial. For any state cell, to be denoted here by Z(n), we locate its center point xc(n). Next, using xc(n) as an initial state, we integrate (2.4) over a time interval T to obtain the state of the system x’(n + 1) at the end of that time interval. This process of integration can be done either analytically if possible or numerically if necessary. The cell in which x’(n + 1) lies is then taken to be the image cell of Z(n) after one mapping step and is denoted by Z(n + 1). This is to be done for each state cell of interest. In this manner a simple cell-to-cell mapping C is created for the system (2.4). Z(n + 1) = C[Z(n)]. NM20:5/6-L
(3.2)
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J. Xv et al.
Within the context of approximation by discretizing the original state space into a cell state space, this cell-to-cell mapping C contains all the dynamics information of the system, including its global properties. It is then a matter of extracting these properties from this cell mapping C. Let C” denote the cell mapping C applied m times with Co understood to be the identity mapping. Periodic motions and periodic cells
A sequence of K distinct cells Z*(j), j = 1,2,. . . , K, which satisfy Z*(m + 1) = C?[Z*(l)],
m=
1,2 ,..., K-
1, (3.3)
z*(l) = Ck[Z*(l)],
is said to form a periodic motion of period K. For convenience we call such a motion a P - K motion. We call each of its elements Z*(j) a periodic cell of period Kor simply a P - Kcell. A P - 1 cell may also be regarded as an equilibrium cell under the cell mapping C. Domains of attraction
A cell Z is said to be “r-steps removed from a P - K motion” if r is the minimum positive integer such that C’(Z) = Z*(j) where Z*(j) is one of the P - Kcells of that P - Kmotion. In other words, Z is mapped after r steps into one of the P - Kcells of the P - Kmotion and any further mapping will lock the evolution of the system in this P - Kmotion. The set of all cells which are r steps or less removed from a P - K motion is called the “r-step domain of attraction” for that P - K motion. The total domain of attraction, or simply the domain of attraction of a P - K motion is its r-step domain of attraction with r + co. Sink cell
A crucial concept in the method of cell mapping is the introduction of a cell called the sink cell. For practically all physical problems, once a state variable exceeds a certain positive or negative magnitude, we are no longer interested in the further evolution of the variable. This implies that there is only a finite region of the cell state space which is of interest to us. Taking advantage of this, all the cells outside this finite region can be lumped into a huge cell, a sink cell, which may be assumed to map into itself under the mapping C. The finite region of interest of the cell state space may be large, but the key point is that the total number of cells of interest will be finite. This means that the cell mapping (3.2) may be taken as an array of finite size. Because of this feature we were able to devise an algorithm [6] which allows us to determine the periodic cells and their domains of attraction in a very effective manner. For an explanation of this algorithm the reader is referred to [6]. 4.
LIMIT
CYCLES
AS PERIODIC
MOTIONS
IN SIMPLE
CELL
MAPPING
To demonstrate how the global properties of the coupled van der Pol equations (2.4) may be determined by using simple cell mapping, we shall take the case p = 0.1, q = 0.04, and v = 0.1. A variety of cell structures using different cell sizes have been investigated. In this paper we present the results from one of such investigations. To set up the cell structure we take the region of interest of the state space to be -2.5 6 x1 < 2.5,
-3.0 $ x2 < 3.0,
-2.5 5 x3 < 2.5,
- 3.0 5
x4
<
3.0.
(4.1)
In each coordinate direction 59 intervals are used resulting in hi = h3 = 0.084746,
h2 = h4 = 0.101695.
(4.2)
The cells involved in the simple cell mapping are 12,117,361 regular cells plus a sink cell. The time interval T used to create the simple cell mapping is 2.2 units. In simple cell mapping, a stable limit cycle of the coupled van der Pol equations will show up as one or more periodic motions. An unstable equilibrium state of the system may also
513
Multiple limit cycles of coupled van der Pol equations
show up in the form of one or more periodic motions, [17]. For the present case, the first limit cycle which is the in-phase coupled one is represented by two periodic motions consisting of 34 periodic cells. The second limit cycle which is essentially out-of-phase coupled is represented by three periodic motions consisting of 158 periodic cells. The unstable equilibrium state of the system at the origin of the state space is represented by three periodic motions consisting of seven periodic cells. We display these periodic cells in Fig. 5 by plotting the projections of their center points on the x1 - x2, x1 - x3, x1 - x4, x3 - x2, x2 - x4, x3 - x4 planes. The cells associated with the first liiit cycle are designated by the symbol “ x “, those ofthe second limit cycle by “e”,and those associated with the unstable equilibrium state at’the origin by “+“. For comparison we have also plotted in Fig. 5 the projections of the limit cycles on the six planes. _ 3 (a) e
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514
J. Xv et al. 5.
DOMAINS
OF ATTRACTION
OF THE
LIMIT
CYCLES
The
simple cell mapping computer run which yields the periodic cells representing the limit cycles also gives us the domains of attraction of these two limit cycles. The global behavior of the 12 117 361 regular cells may be summarized as follows. (i) There are 4 772 425 regular cells which are attracted to the 34 periodic cells representing the in-phase coupled limit cycle. The total number of cells in this domain of attraction including the periodic cells themselves is 4 772 459. (ii) There are 4568420 regular cells which are attracted to the 158 periodic cells representing the out-of-phase coupled limit cycle. The total number of cells in this domain of attraction including the periodic cells themselves is 4 568 578. (iii) There are 2 776 317 regular cells which are mapped into the sink cell according to the present version ofsimple cell mapping. These are the cells located near the corners of the fourdimensional block of (4.1). For these cells this particular simple cell mapping cannot tell us which domains of attraction they belong to. (iv) The remaining seven regular cells are periodic cells representing the unstable equilibrium state at the origin of the state space. There are no cells which are attracted to these periodic cells. This is a consistent result as this equilibrium state is an unstable one. Of course, it is impractical to describe the global behavior of all the cells. Even to display the two 4-D domains of attraction is difficult. In this paper we show in Figs 6 and 7 the domains by twelve 2-D sections of the 4-D cell state space. They are respectively: Figure 6(a) Zr (b) Z, (c) Zr (d) Z1 (e) Z, (f) Z1 Figure7(a) Z3 (b) Z3 (c) Z3 (d) Z3 (e) Z3 (1’) Z3
-
Zz Z2 Z2 Zz Zz Z2 Z4 Z4 Z4 Z4 Z4 Z4
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at at at at at at at at at at at at
Z3 Z3 Z3 ZJ Z3 Z3 Z1 Zr Z1 Z1 Z1 Z1
= = = = = = = = = = = =
0 and Zq = 0, 15 at Z4 = 0, - 15 and Z4 = 0, 0 and Z4 = 15, 0 and Z4 = - 15. 15 and Z4 = 15. 0 and Zz = 0, 15 and Zz = 0, - 15 and Z2 = 0, 0 and Zz = 15, 0 and Zz = - 15, 15 and Zz = 15.
In these figures a cell belonging to the domain of attraction of the first limit cycle is indicated by the symbol “ x “. A cell belonging to the domain of attraction of the second limit cycle is indicated by “.“. A blank spot means the location ofa cell which is mapped into the sink cell in its evolution. To check the validity of the domain of attraction results obtained by simple cell mapping, we have also computed figures of the type shown in Figs 6 and 7 by direct numerical integration. The center point of each cell is taken as an initial state. The subsequent motion and the limit cycle it eventually approaches are determined by numerical integration. In Fig. 8 we show a typical example of comparison between the simple cell mapping result and the direct numerical integration result. The figure is the same as Fig. 7(f) but with the addition of a zig-zag line which represents the separation of the two domains of attraction determined by direct numerical integration. One sees that the results from the two methods agree remarkably well. There are a few cells near the boundary separating the two domains of attraction which are not predicted correctly by the simple cell mapping method. This is to be expected because the simple cell mapping was created by using only the center point of each cell. Next, we shall make some comments about the computer needs to do a problem of this nature. For this system of couple van der Pol equations, several sets of values of p, q and v have been investigated. For each set several cell sizes are used. For example, for the case p = 0.1, q = 0.04, and v = 0.1, the total number of regular cells used to cover the region of interest (4.1) is 274, 314, 514 or 594. The global domain of attraction results obtained from these separate investigations are very much the same. In this paper we have presented the case 594 which involves the largest number of cells used by us up to now. This will allow us to convey to the reader an appreciation of the memory capacity, computer speed, and computer
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....
515
. ................... :....::::::::::::::::::::::::: ............................
r
0
Z,
(f) at Z,=l5,Z,=l5
Fig. 6. Z1 - Z2 sections of the domains of attraction for p = 0.1, q = 0.04 and v = 0.1. “ x “-for first limit cycle, “.“-for the second limit cycle.
the
time needs involved. All the computations reported here were done on VAX 11/750. With intelligent programming 1Zmillion cells can be used but this is probably the limit if one is restricted to a machine of VAX 1l/750 capacity. The total computer time used to carry out a complete 4-D simple cell mapping analysis using 594 cells, including the creation of the simple cell mapping C of (3.2), is about 19 h. This is a computing job of substantial size but it is not an unusually large one in present-day academic and industrial laboratories. On the other hand, if we were to generate the domains of attraction by direct numerical integration to cover the same region of interest (4.1) and to use the same interval sizes, the computer time required would be of the order 3800 h, based upon our experience. Thus, the algorithm of simple cell mapping achieves approximately a 200 folds improvement in computing efficiency for this problem. 6. CONCLUDING
REMARKS
The task of determining the domains of attraction for strongly non-linear systems is usually so time-consuming computationally that it is often put aside by scientists and engineers as
516
J. XV et al.
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Z,=O,
(c)at
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Z,=-15,Z,=O
L
at Z,=l5,
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_
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Z,=O,
Z,=-15
.
0
Z3 (f)
Z,
at Z,=l5,2,=15
Fig. 7. ZJ - Z4 sections of the domains of attraction for p = 0.1, q = 0.04 and Y = 0.1. “ x “-for first limit cycle, “.‘‘-for the second limit cycle.
the
something difficult to achieve. The advance of computer capability is, of course, constantly changing this picture. But even now, for a 4-D problem the front attack of this task by direct numerical integration is still prohibitively time-consuming. In this paper we show, however, that by using the method of simple cell mapping the task can be accomplished with a reasonable amount ofeffort. We hope and envision that as the methodology of cell mapping is developed further and as the computer technology is advanced further, the computation requirements could be drastically reduced and that a complete global analysis would be possible for 6-D systems in a not-too-distant future. Acknowledgements-This
No. MEA-8217471.
material is based upon work supported by the National Science Foundation under Grant
Multiple limit cycles of coupled van der Pol equations
517
::
N”
07: :: :: :: ::
I
‘:““..“’
Fig. 8. A comparison between the domains of attraction determined by simple cell mapping and by direct numerical integration. For the case p = 0.1, n = 0.04 and Y= 0.1. At Zr = 15 and Zz = 15.
1. 2. 3. 4. 5. 6. 7. 8.
REFERENCES C. Hayashi, Non-linear Oscillations in Physical Systems. McGraw-Hill, New York (1964). A. H. Nayfeh and D. T. Mook, Non-linear Oscillations. Wiley-Interscience, New York (1979). N. Minorsky, Non-linear Oscillations. Van Nostrand, Princeton, NJ (1962). N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-linear Oscillations. Gordon & Breach, New York (1961). C. S. Hsu, A theory of cell-to-cell mapping dynamical systems. J. appl. Mech. 47,931 (1980). C. S. Hsu and R. S. Guttalu, An unravelling algorithm for global analysis of dynamical systems: an application of cell-to-cell mappings, J. appl. Mech. 47, 940 (1980). C. S. Hsu, A generalized theory of cell-to-cell mapping for non-linear dynamical systems. J. appl. Mech. 48,634 (1981). C. S. Hsu, R. S. Guttalu and W. H. Zhy A method of analyzing generalized cell mappings. J. appl. Mech. 49,885 (1982).
9. C. S. Hsu, A probabilistic theory of non-linear dynamical systems based on the cell state space concept. J. appl. Mech. 49,895 (1982). 10. R. S. Guttalu and C. S. Hsu, A global analysis of a non-linear system under parametric excitation. .I. Sound Vibrat. 97, 399 (1984).
11. Apiwon Polchai and C. S. Hsu, Domain of stability of synchronous generators by a cell mapping approach. Int. .I. Control. (in press). 12. C. S. Hsu and Myun C. Rim, Statistics of strange attractors by generalized cell mapping. J. stat. Phys. 38,735 (1985). 13. C. S. Hsu, A discrete method of optimal control based upon the cell state space concept. .I. Optim. Theory Appl. (in press). 14. C. S. Hsu and W. H. Zhu, A simplical mapping method for locating the zeros of a function. Q. J. appl. Math. 42, 41 (1984).
15. T. Pavlidis, Biological Oscillators: Their Mathematical Analysis. Academic Press, New York (1973). 16. R. H. Rand and P. J. Holmes, Bifurcation of periodic motions in two weakly coupled van der Pol oscillators, Int. J. Non-Linear Mech. 15, 387 (1980). 17. C. S. Hsu and Apiwon Polchai, Characteristics of singular entities of simple cell mappings. Jnt. J. Non-Linear Mech. 19, 19 (1984).
Vol. 20, No. S/6, pp. 507-517.
OOZO-7462/85 Pergamon
1985
$3.00 + .oO Prss Ltd.
DOMAINS OF ATTRACTION FOR MULTIPLE LIMIT CYCLES OF COUPLED VAN DER POL EQUATIONS BY SIMPLE CELL MAPPING JIANXUE Xu Department of Engineering Mechanics, Xi’an Jiaotong University, Xi’an, Shaanxi Province, The People’s Republic of China R. S. GUTTALU Department of Mechanical Engineering, University of Southern California, Los Angeles, CA 90089, U.S.A.
and C. S. Hsu Department of Mechanical Engineering, University of California, Berkeley, CA 94720, U.S.A. Ahstrati-One of the most ditIIcult tasks in non-linear analysis is to determine globally various domains of attraction in the state space when there exist more than one asymptotically stable equilibrium states and/or periodic motions. The task is even more demanding if the order of the system is higher than two. In this paper we consider two coupled van der Pol oscillators which admit two asymptotically stable limit cycles. For systems of this kind we show how the method of cell-to-cell mapping can be used to determine the two four-dimensional domains of attraction of the two stable limit cycles in a very effective way. The final results are shown in this paper in the form of a series of graphs which are various two-dimensional sections of the four-dimensional state space.
1. INTRODUCTION
In the classical analysis of non-linear analysis [l-4], once the equations of motion for a system have been formulated, one usually tries first (i) to locate all the possible equilibrium states and periodic motions of the system. Thereafter, one would (ii) study the stability characteristics of these solutions. Logically, the next task is (iii) to find the global domain of attraction in the state space for each asymptotically stable equilibrium state or periodic motion. In some sense one could say that to find and to delineate these domains of attraction is really the ultimate aim of any analysis of a non-linear system. However, when one examines the literature, one finds that while the tasks (i) and (ii) are subjects of a great number of investigations, the task (iii) has not been dealt with as often, particularly when the order of the system is higher than two. This lack of intensity of activity in this direction is probably due to the fact that there is no generally valid analytical method available for this task when the system is other than weakly non-linear. A direct numerical integration approach seems to be the only viable one. However, such an approach is usually prohibitively time-consuming even with the present-day high power computers and, therefore, is not a practical or attractive one. In the last few years, as an attempt to find more efficient and practical ways of determining the global behavior of strongly non-linear systems, methods of cell-to-cell mapping [5-91 have been proposed. These methods are currently being developed in various directions. Up to now two types of cell mappings have been investigated. One is the simple cell mapping [5,6] and the other the generalized cell mapping [7,8]. Both types can be used effectively to determine the global behavior of non-linear systems [6,8,10,11]. Moreover, as new tools of analysis, cell mappings have led us to new methodologies in several other directions. For instance, the generalized cell mapping has been found to be a very efficient method for locating and determining the statistical properties of strange attractors [12]. In the field of automatic control, the cell mapping concept has allowed us to devise a new discrete method of optimal control [13]. The cell mapping method has also implications on the numerical methods of locating all the zeros of a vector function [14]. Up to now, as a method to determine domains of attraction the simple cell mapping has only been applied to second order systems. In this paper we extend the usage to fourth order systems. Specitically, we study a system of two coupled van der Pol oscillators which possess 507
508
J. Xu et al.
two stable limit cycles and we use the simple cell mapping to determine the two domains of attraction. 2.
COUPLED
VAN
DER
POL
EQUATIONS
Consider a system of two coupled van der Pol equations.
lY11 j;l
“2
-p
1-y:
0
0
1-y;j,
91 10
-v
l+v
+
l+q+v
1 -v
y1
IO y2
=
0
00’
P-1)
Here v is the coupling coefficient and rl may be called a detuning coefficient. If v = q = 0 the system is reduced to two identical uncoupled van der Pol oscillators. If v = 0 it is reduced to two uncoupled but not identical oscillators, one with its undamped natural circular frequency equal to 1 and the other (1 + q) ‘j2. This system with additional linear velocity coupling terms has been considered by Pavlidis [ 151 and Rand and Holmes [ 161. An excellent and detailed perturbation analysis of this system when ,u,q and v are small is given in [16]. The analysis gives us the conditions under which there exist 4,2 or 0 limit cycles. They have also studied stability of these limit cycles. However, the third task mentioned in the Introduction of determining the domains of attraction when there are two stable limit cycles is not considered in [ 161. As far as the notation is concerned, p, q and v used in this paper are connected with E, A and u used in [16] through v = Ea.
rj = &A,
p = 6,
(2.2)
For the subsequent discussion, it is more convenient to write (2.1) as a system of four first order equations. Letting Xl = Yl,
x2 = $1,
x3 = Y2,
x4 = 32,
(2.3)
we have i = 1, 2, 3, 4,
kj = E(x),
(2.4)
where Fl =
x2,
Fz =
p(l - x:)x, - (1 + v)xr +
F3 =
x4,
F4 =
~(1 -
vx3,
(2.5) x:,x,
+
vxl
-
(1 + u + v)xs.
Many cases with different sets of values of ,u, rl and v have been studied. Here we present three cases for which there exist two stable limit cycles. First consider the case with p = 0.1, q = 0.04, and v = 0.1. To exhibit the two limit cycles which reside in a 4-D state space, we plot their projections on the xl - x2, xl - x3, xl - x4, x3 - x2, x2 - x4, and x3 - x4 planes as shown in Fig. 1. These results as well as the limit cycle results for the other cases to be shown later were obtained by using the fourth-order Runge-Kutta method of numerical integration. From Fig. l(b) it is readily seen that the limit cycle labelled “the jirst” is essentially an “inphase coupled” limit cycle while the one labelled as “the second” is essentially an “out-of-phase coupled” limit cycle. This is even better seen in Fig. 2 where the time histories of the two limit cycles for this case are shown. The periods of these two limit cycles are, respectively, 6.246 and 5.668 units. In Fig. 3 we show the two stable limit cycles for the second case p = 0.25, q = 0.04, and v = 0.1. Here, the departure from 0” or 180” of the phase difference of the phase-locked coupled motions is more pronounced than that of the first case. Also, there is more distorsion from a circular shape in Figs 3(a) and (f), indicating substantial higher harmonics in the limit cycles.
Multiple limit cycles of coupled van der Pal equations
509
2 2nd
x” 0
1st
x” 0
-2
-2
k-e-
lx
k--F+-
2
Xl
X,
cc) 2 1st’ ‘2nd 0
-2 .:Q -2
0
2
(f)
2nd \ 1st
x”
0
-2 21Q -2
0
2
x3
Fig. 1. Projections of the two limit cycles for the case p = 0.1, 1 = 0.04 and Y = 0.1. (a) 2-
0
I
I
I
3.123
6.246
Time I.
B
2.634 Time
Fig. 2. Time histories of the two limit cycles for the case p = 0.1, q = 0.04, and Y = 0.1.
J. Xv et al.
510
X”
, 2$$ (cl
2
1st
x"
0
'2nd
0@
x”
2nd -
-
1st
-2
-2 -2
0
-2
2
0
2
x3
Xl (f) 2--
2nd \ y 1st
x"
o-
-2-2
0
2
5
@ -2
0
2
x3
Fig. 3. Projections of the two limit cycles for the case p = 0.25, q = 0.04, and v = 0.1.
In Fig. 4 we show the two limit cycles for the third case with /A= 0.1, q = 0.04, and v = 0.02. Here, the coupling is stronger than in the first case. The results indicate that the phase difference departure from 0” or 180” is fairly small. Before leaving this section we wish to make one remark about the conditions derived in [ 161 concerning the number of limit cycles a system may possess. These conditions are based upon an asymptotic perturbation analysis with ~1as the smallness parameter. In general, they may be no longer adequate when ~1is not very small. For example, when p = 0.35, rl = 0.04, and v = 0.1, one stable limit cycle is found while, according to the criteria derived in [16], no phase-locked periodic solution is expected. 3. METHOD
OF SIMPLE
CELL MAPPING
Before taking up the matter of determining the domains of attraction for the two limit cycles, we shall give in this section a brief description of the method of simple cell mapping to be used for this purpose. For a more detailed discussion of the method thelreader is referred to [5,6 I. Consider a dynamical system governed by (2.4). The basic idea of cell mapping is to consider the state space not as a continuum of uncountable number of points but rather as a collection of a large number of state cells, with each cell being a state entity. The cell structure over the state space may be introduced in a variety of ways. The simplest one is the following.
511
Multiple limit cycles of coupled van der Pol equations
x”
o-
-2
-2
0
2
x3
Fig. 4. Projections of the two limit cycles for the case p = 0.1, f = 0.04, and v = 0.2.
A four dimensional state cell is designated by a cell vector Z which has integer-valued components Z1 , Zz, Z3, and Zq. A point x of the state space with components x1, x2, x3 and x4 belongs to a cell Z if and only if (,i-_)hisxi<
(Zi+f)hi,
i=1,2,3,4,
(3.1)
where hi is the interval size associated with xi. Having constructed the cell state space, we can create a cell to cell mapping for the dynamical system (2.4) in the following manner. For an autonomous system (2.4) we first choose an appropriate time interval T.Apart from a requirement that T should not be too small, its precise magnitude is not crucial. For any state cell, to be denoted here by Z(n), we locate its center point xc(n). Next, using xc(n) as an initial state, we integrate (2.4) over a time interval T to obtain the state of the system x’(n + 1) at the end of that time interval. This process of integration can be done either analytically if possible or numerically if necessary. The cell in which x’(n + 1) lies is then taken to be the image cell of Z(n) after one mapping step and is denoted by Z(n + 1). This is to be done for each state cell of interest. In this manner a simple cell-to-cell mapping C is created for the system (2.4). Z(n + 1) = C[Z(n)]. NM20:5/6-L
(3.2)
512
J. Xv et al.
Within the context of approximation by discretizing the original state space into a cell state space, this cell-to-cell mapping C contains all the dynamics information of the system, including its global properties. It is then a matter of extracting these properties from this cell mapping C. Let C” denote the cell mapping C applied m times with Co understood to be the identity mapping. Periodic motions and periodic cells
A sequence of K distinct cells Z*(j), j = 1,2,. . . , K, which satisfy Z*(m + 1) = C?[Z*(l)],
m=
1,2 ,..., K-
1, (3.3)
z*(l) = Ck[Z*(l)],
is said to form a periodic motion of period K. For convenience we call such a motion a P - K motion. We call each of its elements Z*(j) a periodic cell of period Kor simply a P - Kcell. A P - 1 cell may also be regarded as an equilibrium cell under the cell mapping C. Domains of attraction
A cell Z is said to be “r-steps removed from a P - K motion” if r is the minimum positive integer such that C’(Z) = Z*(j) where Z*(j) is one of the P - Kcells of that P - Kmotion. In other words, Z is mapped after r steps into one of the P - Kcells of the P - Kmotion and any further mapping will lock the evolution of the system in this P - Kmotion. The set of all cells which are r steps or less removed from a P - K motion is called the “r-step domain of attraction” for that P - K motion. The total domain of attraction, or simply the domain of attraction of a P - K motion is its r-step domain of attraction with r + co. Sink cell
A crucial concept in the method of cell mapping is the introduction of a cell called the sink cell. For practically all physical problems, once a state variable exceeds a certain positive or negative magnitude, we are no longer interested in the further evolution of the variable. This implies that there is only a finite region of the cell state space which is of interest to us. Taking advantage of this, all the cells outside this finite region can be lumped into a huge cell, a sink cell, which may be assumed to map into itself under the mapping C. The finite region of interest of the cell state space may be large, but the key point is that the total number of cells of interest will be finite. This means that the cell mapping (3.2) may be taken as an array of finite size. Because of this feature we were able to devise an algorithm [6] which allows us to determine the periodic cells and their domains of attraction in a very effective manner. For an explanation of this algorithm the reader is referred to [6]. 4.
LIMIT
CYCLES
AS PERIODIC
MOTIONS
IN SIMPLE
CELL
MAPPING
To demonstrate how the global properties of the coupled van der Pol equations (2.4) may be determined by using simple cell mapping, we shall take the case p = 0.1, q = 0.04, and v = 0.1. A variety of cell structures using different cell sizes have been investigated. In this paper we present the results from one of such investigations. To set up the cell structure we take the region of interest of the state space to be -2.5 6 x1 < 2.5,
-3.0 $ x2 < 3.0,
-2.5 5 x3 < 2.5,
- 3.0 5
x4
<
3.0.
(4.1)
In each coordinate direction 59 intervals are used resulting in hi = h3 = 0.084746,
h2 = h4 = 0.101695.
(4.2)
The cells involved in the simple cell mapping are 12,117,361 regular cells plus a sink cell. The time interval T used to create the simple cell mapping is 2.2 units. In simple cell mapping, a stable limit cycle of the coupled van der Pol equations will show up as one or more periodic motions. An unstable equilibrium state of the system may also
513
Multiple limit cycles of coupled van der Pol equations
show up in the form of one or more periodic motions, [17]. For the present case, the first limit cycle which is the in-phase coupled one is represented by two periodic motions consisting of 34 periodic cells. The second limit cycle which is essentially out-of-phase coupled is represented by three periodic motions consisting of 158 periodic cells. The unstable equilibrium state of the system at the origin of the state space is represented by three periodic motions consisting of seven periodic cells. We display these periodic cells in Fig. 5 by plotting the projections of their center points on the x1 - x2, x1 - x3, x1 - x4, x3 - x2, x2 - x4, x3 - x4 planes. The cells associated with the first liiit cycle are designated by the symbol “ x “, those ofthe second limit cycle by “e”,and those associated with the unstable equilibrium state at’the origin by “+“. For comparison we have also plotted in Fig. 5 the projections of the limit cycles on the six planes. _ 3 (a) e
.
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I:0 . ._: c3 ‘.
.
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-I
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:.,
-3c
-3t -2
-I
0
,
-2
2
-I
.’
0
1
2
X,
X,
Fig. 5. Periodic cells representing the limit cycles and equilibrium state for the case p = 0.1,~ = 0.04 and v = 0.1. “ x “-the first limit cycle, “.” -the second limit cycle, u + “-the equilibrium state at the Origin.
514
J. Xv et al. 5.
DOMAINS
OF ATTRACTION
OF THE
LIMIT
CYCLES
The
simple cell mapping computer run which yields the periodic cells representing the limit cycles also gives us the domains of attraction of these two limit cycles. The global behavior of the 12 117 361 regular cells may be summarized as follows. (i) There are 4 772 425 regular cells which are attracted to the 34 periodic cells representing the in-phase coupled limit cycle. The total number of cells in this domain of attraction including the periodic cells themselves is 4 772 459. (ii) There are 4568420 regular cells which are attracted to the 158 periodic cells representing the out-of-phase coupled limit cycle. The total number of cells in this domain of attraction including the periodic cells themselves is 4 568 578. (iii) There are 2 776 317 regular cells which are mapped into the sink cell according to the present version ofsimple cell mapping. These are the cells located near the corners of the fourdimensional block of (4.1). For these cells this particular simple cell mapping cannot tell us which domains of attraction they belong to. (iv) The remaining seven regular cells are periodic cells representing the unstable equilibrium state at the origin of the state space. There are no cells which are attracted to these periodic cells. This is a consistent result as this equilibrium state is an unstable one. Of course, it is impractical to describe the global behavior of all the cells. Even to display the two 4-D domains of attraction is difficult. In this paper we show in Figs 6 and 7 the domains by twelve 2-D sections of the 4-D cell state space. They are respectively: Figure 6(a) Zr (b) Z, (c) Zr (d) Z1 (e) Z, (f) Z1 Figure7(a) Z3 (b) Z3 (c) Z3 (d) Z3 (e) Z3 (1’) Z3
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In these figures a cell belonging to the domain of attraction of the first limit cycle is indicated by the symbol “ x “. A cell belonging to the domain of attraction of the second limit cycle is indicated by “.“. A blank spot means the location ofa cell which is mapped into the sink cell in its evolution. To check the validity of the domain of attraction results obtained by simple cell mapping, we have also computed figures of the type shown in Figs 6 and 7 by direct numerical integration. The center point of each cell is taken as an initial state. The subsequent motion and the limit cycle it eventually approaches are determined by numerical integration. In Fig. 8 we show a typical example of comparison between the simple cell mapping result and the direct numerical integration result. The figure is the same as Fig. 7(f) but with the addition of a zig-zag line which represents the separation of the two domains of attraction determined by direct numerical integration. One sees that the results from the two methods agree remarkably well. There are a few cells near the boundary separating the two domains of attraction which are not predicted correctly by the simple cell mapping method. This is to be expected because the simple cell mapping was created by using only the center point of each cell. Next, we shall make some comments about the computer needs to do a problem of this nature. For this system of couple van der Pol equations, several sets of values of p, q and v have been investigated. For each set several cell sizes are used. For example, for the case p = 0.1, q = 0.04, and v = 0.1, the total number of regular cells used to cover the region of interest (4.1) is 274, 314, 514 or 594. The global domain of attraction results obtained from these separate investigations are very much the same. In this paper we have presented the case 594 which involves the largest number of cells used by us up to now. This will allow us to convey to the reader an appreciation of the memory capacity, computer speed, and computer
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time needs involved. All the computations reported here were done on VAX 11/750. With intelligent programming 1Zmillion cells can be used but this is probably the limit if one is restricted to a machine of VAX 1l/750 capacity. The total computer time used to carry out a complete 4-D simple cell mapping analysis using 594 cells, including the creation of the simple cell mapping C of (3.2), is about 19 h. This is a computing job of substantial size but it is not an unusually large one in present-day academic and industrial laboratories. On the other hand, if we were to generate the domains of attraction by direct numerical integration to cover the same region of interest (4.1) and to use the same interval sizes, the computer time required would be of the order 3800 h, based upon our experience. Thus, the algorithm of simple cell mapping achieves approximately a 200 folds improvement in computing efficiency for this problem. 6. CONCLUDING
REMARKS
The task of determining the domains of attraction for strongly non-linear systems is usually so time-consuming computationally that it is often put aside by scientists and engineers as
516
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something difficult to achieve. The advance of computer capability is, of course, constantly changing this picture. But even now, for a 4-D problem the front attack of this task by direct numerical integration is still prohibitively time-consuming. In this paper we show, however, that by using the method of simple cell mapping the task can be accomplished with a reasonable amount ofeffort. We hope and envision that as the methodology of cell mapping is developed further and as the computer technology is advanced further, the computation requirements could be drastically reduced and that a complete global analysis would be possible for 6-D systems in a not-too-distant future. Acknowledgements-This
No. MEA-8217471.
material is based upon work supported by the National Science Foundation under Grant
Multiple limit cycles of coupled van der Pol equations
517
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1. 2. 3. 4. 5. 6. 7. 8.
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9. C. S. Hsu, A probabilistic theory of non-linear dynamical systems based on the cell state space concept. J. appl. Mech. 49,895 (1982). 10. R. S. Guttalu and C. S. Hsu, A global analysis of a non-linear system under parametric excitation. .I. Sound Vibrat. 97, 399 (1984).
11. Apiwon Polchai and C. S. Hsu, Domain of stability of synchronous generators by a cell mapping approach. Int. .I. Control. (in press). 12. C. S. Hsu and Myun C. Rim, Statistics of strange attractors by generalized cell mapping. J. stat. Phys. 38,735 (1985). 13. C. S. Hsu, A discrete method of optimal control based upon the cell state space concept. .I. Optim. Theory Appl. (in press). 14. C. S. Hsu and W. H. Zhu, A simplical mapping method for locating the zeros of a function. Q. J. appl. Math. 42, 41 (1984).
15. T. Pavlidis, Biological Oscillators: Their Mathematical Analysis. Academic Press, New York (1973). 16. R. H. Rand and P. J. Holmes, Bifurcation of periodic motions in two weakly coupled van der Pol oscillators, Int. J. Non-Linear Mech. 15, 387 (1980). 17. C. S. Hsu and Apiwon Polchai, Characteristics of singular entities of simple cell mappings. Jnt. J. Non-Linear Mech. 19, 19 (1984).