Coupled relaxation oscillators and circle maps

Physica D 30 (1988) 61-82 North-Holland, Amsterdam

C O U P L E D RELAXATION OSCII J ,ATORS AND CIRCLE MAPS

R. PEREZ PASCUAL and J. L O M N I T Z - A D L E R [nstituto de Fisica, U.N.A.M., Apartado Postal 20-364, MJxico 01000, D.F., Mexico

Received 29 December 1986 Revised manuscript received 23 September 1987

We discuss the problem of two non-linear oscillators of the saw-tooth type coupled with each other at either the threshold or base in such a way that the threshold (base) of the first oscillator is equal to a constant plus a term proportional to the value of the state of the second, and similarly for the other oscillator. The solution of such problems involves the decoupling of the component subsystems through the use of geometrical techniques to reduce the problem to that of the iteration of two independent maps of the unit segment onto itself. Depending on the sign of the proportionality constant and whether the couplings takes place at the threshold or base, there are ten different systems of this type, resulting in twelve different maps. The dependence of the associated maps on the parameters of the initial system, and bifurcation spaces of a number of these maps are obtained analytically. This yields the bifurcation spaces of the original coupled non-linear oscillators.

1. Introduction

In recent years much effort has been dedicated to the study of forced non-linear oscillators. This interest arises from two motivations. The first is that these systems often provide a reasonable rep- resentation of physical [1], biological systems such as nervous or cardiac tissue subjected to applied periodic stimuli [2], geophysical systems such as seimic faults [3] and electronic circuits [4]. The

second aspect of these problems which has excited interest is the fact that certain universal properties have been observed for non-linear systems as they

approach the transition to chaos [5]. Periodically driven non-linear oscillators, being some of the simplest systems for which such behaviour may be observed, are important because of their tractabil- ity, in many cases allowing analytical treatment.

In fig. 1 we show a non-linear oscillator of the saw-tooth type. In this system we have a single variable y which increases with time at a constant rate A, until a threshold h is reached. At this point y drops discontinuously to the base value b. If the system is unforced, that is if A, b and h are constant, then the system has simple clock-like

behaviour with period • = ( h - b ) / A . One may force these oscillators by changing the values of the threshold or base periodically with time; these have been studied in ref. [6].

In this paper we consider the more complicated case of two such systems which force each other. In fig. 2 we show the case of two oscillators, the state of the first oscillator determines the threshold

and base values h 2, b2 of the second and similarly the state of the second oscillator determines the

threshold and base values hi, b 1 of the first. In this way the coupling of the two oscillators takes place. Such a system may be considered to be an extension of the original forced non-linear oscillator to the case of a two-oscillator system, and it was shown in ref. [7] that these systems can be solved by transforming the coupled two-oscillator system into two uncoupled maps of the unit segment onto itself. One specific system having simi-

lar characteristics as the oscillators under study has been used in the study of coupled neurons [8].

In this paper we shall discuss the class of systems in which each of the component subsystems is coupled either at the threshold or base by introducing a linear dependence of them on the

0167-2789/88/$03.50 © Elsevier Science Publishers B.V. (North-Hol land Physics Publishing Division)

62 R. P~rez Pascual and J. Lomnitz-A dler / Coupled relaxation oscillators

T I M E -

Fig. 1. The evolution of an unforced saw-tooth type osciUator.

i

TIME

Fig. 2. A pair of saw-tooth oscillators which are forced at the threshold and base. In this case there are periodic orbits.

1

state of the other oscillator,

Z l ( y ) = h I "4- oqy or B l ( y ) = fllY; (1)

r 2 ( x ) = h2 + ~2x or B 2 ( x ) = & x .

If each subsystem is coupled at either the threshold or the base, and considering that the parameters a and fl may have either a positive or negative sign we find that there are ten different cases of coupling in the oscillators.

This paper is organized as follows. In section 2 we shall show how the original system is decou- pied into two interval maps. We shall discuss one case in detail and state the results for the other ten. We also show how the parameters of the original two-oscillator system are related to those of the maps in the decoupled system, which maps may appear for each coupled system and at which values of their parameters.

Section 3 is devoted to the discussion and solution of the iterations of the resulting maps. In this

section we shall obtain the full bifurcation spaces of seven of the maps and we discuss the expected solutions of the remaining five. This section is technical. To aid the reader we have included for each map a figure of its bifurcation space. The casual reader may skip this section and only look at the figures.

In section 4 we discuss the relation between the coupled system and its corresponding maps, the results are summarized in table I.

2. Decoupling the two-oscillator system

The way in which one may decouple two non- linear oscillators which are coupled at the threshold and base has been shown in ref [7]. As we mentioned in the introduction, there are ten different types of coupling which are discussed in this paper. Each of these is determined by whether the oscillator's coupling is at the threshold or base, and by the sign of this coupling. Thus we may denote the systems of interest by the symbols (FS, Gs), where F, G stand for either threshold or base and where S and s are the associated signs. For example, the system (T + , B - ) is one in which

Tl(y)=ha +oqy, B I = 0 ; (2)

7"2 = h2, B 2 ( x ) = - & x ,

where /'1, T 2 are the thresholds and B 1, B 2 the base values of the x and y systems, respectively, and the coupling is obtained by their dependence on the value of the other oscillator. The a 's and fl 's will always have a positive sign.

R. P~rez Pascual and J. Lomnitz-Adler / Coupled relaxation oscillators 63

I I

I

/ ! I

I I I

, /

/ / /

TIME *'

Fig. 3. The time development of two oscillators coupled at the threshold.

To illustrate the technique, we shall discuss the (T + , T + ) case shown in fig. 3, which is coupled at both thresholds with a positive sign. Without losing generality we have normalized the original threshold values to one, so

T l ( y ) = l + a i y ; T2(x ) = l + a 2 x . (3)

As a first step in the analysis of this coupled system, we represent it in phase space, that is the R 2 space whose coordinates are (x, y), the values

of the state variables of the two oscillators. We first note that the domain on which the system

may be found is constrained to a polygonal re-

gion: when the y system is in the state y = 0, x may only take on values ranging from zero to one. As y increases, the maximum value that x can reach, that is the x threshold, increases according to the rule

x = 1 + a ly . (4)

The same argument holds for the y variable, so that all possible states of the system are to be found in the interior of a region bounded by four lines:

O < x < T l ( y ) ; O < y < T 2 ( x ). (5)

yt

a b c x

Fig. 4. Allowed phase space for the system of oscillators shown in fig. 3. Those trajectories which intersect the perimeter at the double line in the upper right-hand corner have both variables reset to zero.

The phase space corresponding to the system of fig. 3 is shown in fig. 4. In phase space the evolution of the system is represented in a simple fashion. In the absence of a relaxation to base value both x and y increase linearly with time, and as time passes the system traces a straight line in phase space whose slope is

m = A 2 / A x, (6)

where At, A 2 are the rates at which x and y increase in time. When this line intercepts one of

6 4 R. P~rez Pascual and J. Lomnitz-Adler / Coupled relaxation oscillators

the thresholds a relaxation will occur. Let us assume that the y threshold is reached. If x < 1, then only the y variable is reset to base value and the system jumps to a state whose x value remains

unchanged while y relaxes to zero. If x is greater than one when the y threshold is reached the relaxation in y stimulates a reset in the x subsystem, leading to a state where both x and y are equal to zero. If the x variable were the first to reach threshold the behaviour would be analo-

gous. What we wish is to predict the structure of the

orbits of the two subsystems, in particular to be

able to predict the periodicity of the two subsystems when given the parameters which determine the system. Let us look at the time interval ~'y, which is the elapsed time between the n th and the (n + 1)st relaxations of the y variable. In general, after the n th reset in y we have x = x , and y = O,

and at the ( n + l ) s t reset x = x , + 1 and y = T2(x,+l) . Since y increases linearly with time, at a

rate d y / d t = A 2, we find

V = T 2 ( X n + l ) / A 2 ' ( 7 )

so that a knowledge of the sequence x , of values of x when y relaxations occur is sufficient to determine the time intervals between relaxations in y. In our case the intervals are given by

1"y = (1 + a2x.+l)/A2. (8)

I t is clear that the point (x , ,0 ) determines uniquely the next point (x ,+ l ,0 ) (see fig. 4) according to a rule x ,+ x = f ( x , ) , then the sequence of x ' s will be an orbit of the discrete dynamical system generated by the map x ,÷ 1 = f ( x , ) . If one of the y relaxations occurs at an x value which is greater than the x threshold which ap- plies after the reset, then both variables will relax and either the x or the y interval will have to be

determined more explicitly. This is not a great

f(×) C

. . . . . . . . I . . . . . . i . . . . . . . . . . . . . . . I I I I I I I I

I t I

a b x

Fig. 5. The x map corresponding to the case shown in fig. 3. This is map II of section 3.

difficulty since the subsystems can have only three

types of asymptotic behaviour: (i) stable orbits of period N, (ii) quasiperiodic orbits, or (iii) chaotic orbits. In the first case the undetermined interval needs to be determined only once since the orbit repeats itself (the periodicity of the orbit may be determined without the explicit knowledge of the missing interval). In the second and third cases the double reset for which both subsystems slip may occur only once in the entire time series, so that the indeterminate time interval is irrelevant.

The map x ,+ 1 = f ( x , ) for the special case under discussion is shown in fig. 5. It is a map of the unit segment onto itself, and it has three parameters: the value c = f (0 ) , and the values a, b which delimit the range over which double relaxations will occur (see fig. 4). The time evolution of subsystem x is obtained from a similar map

Y,,+I = g(Y,,), where y, is the value of y at the nth relaxation of the x variable and which only differs f rom the one of the variable y in the values of the three parameters. This approach resembles the Poincar6 map technique.

The maps constructed above are for a special form of coupling, but the "decoupling" technique is applicable to any pair of oscillators which are coupled at the base a n d / o r threshold. Specifically,

R. Pdrez Pascual and J. Lomnitz-Adler / Coupled relaxation oscillators 65

r / I T + , T-I-) ( T + , T - ) ( T - , T - }

( e + , 8 + ) LB+.B- ) ( B - , B - I

S (T+oB+) (T+, B - ) I T - , B+)

( T - , B - |

Fig. 6. The phase spaces associated to the ten forms of coupling dealt with in this paper. Those trajectories that intersect double lines have both variables reset to zero.

the ten forms of coupling which we discuss in this paper are associated to the ten phase space diagrams shown in fig. 6. These in turn give rise to the twelve kinds of maps shown in fig. 7, and which of these maps appear for each variable given the values of the original system's parameters is tabulated in table I.

In many situations of interest the states vary non-linearly with time, and the coupled threshold and base of one oscillator have a non-linear dependence on the state of the other. While the analysis of such systems is much more complex, and may prove impossible, the decoupling technique discussed above may still be applied and will lead to great simplifications in numerical treatment.

In the case of n coupled oscillators the same technique will lead to mappings of R n-1 onto itself. Such work is already in progress.

I

b ~ . . . . . .

c

. . . . 4 - - - - - -

o

/ i V I o b

d . . . . . . .

c . . . . I ' - - - -

o

X

dt . . . .

o

II

o b

i i i I I I

d - - d

. . . . 4-'--

o b

"~rrr

I/ I : ~ . . . .

o b

-,rr

i ! I i i I I I I I I I i I

c

0

xTr

a b

c ~ . . . .

o

-xm-

I I '

~ ~ I I t I I I iI t i I i i i I I I I i I I i i

Fig. 7. The twelve maps of the unit segment onto itself associated to the ten diagrams shown in fig. 6.

3. M a p p i n g s

In this section we shall discuss the orbits generated by iterating the twelve maps obtained upon decoupling the original ten coupled systems. Each of these maps is characterized by a number of parameters which determine the nature of the orbits. These may be stable or unstable periodic orbits with period N, quasiperiodic, chaotic or orbits approaching asymptotically one of these. In all of the maps shown in fig. 7 we can identify zero with one and study them as maps of a circle onto itself. In this way we can use the concept of rotation number defined by

• 1 (9)

66 R. Pdrez Pascual and J. Lomnitz-Adler/ Coupled relaxation oscillators

where O is a function that adds one each time that the map turns once around the circle. If R is well-defined and is a rational number p/q, then there is a periodic orbit of period N, an integer multiple of q. If R is irrational the orbit is either quasiperiodic or chaotic. There exist different kinds of chaos [9], several of them may occur in the class of systems which we are studying. The dependence of the type of asymptotic orbit on map's parameters is presented by means of a bifurcation space, in which we distinguish those regions of the parameter space which correspond to a given type of orbit. The map's parameters may be obtained from those of the original two-oscillator system as will be shown in section 4.

Map I

This is the first map which we discuss. It consists of two straight segments, the first starting at f (0) = b and the second ending at f (1) = a < b (with a jump at x=c such that f _ ( c ) = l , f÷(c) -- 0, as shown in fig. 8). It is of a form which has been treated by Keener [9], who found that such maps may only give rise to periodic and quasiperiodic behaviour.

To construct the bifurcation space we look for the preimages of the point c: c o = c, c 1 = f - l ( c ) . . . . . CN=f-N(C), where N is such that

f (x) / b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

/ C x

Fig. 8. Illustration of map type I.

c N lies in the range (a, b) and has no preimage, thus ending the sequence (N may be infinite). We state that if n < N + 2, fn has no fixed points, but if n = N + 2 , f n will have N + 2 stable fixed points corresponding to a stable orbit of period N + 2 o f f .

This statement is proved in the following way. (1) When n < N + 2, fn consists of a number of linear branches with positive slopes with discontinuities at the points c k as shown in fig. 9. These discontinuities are such that at all Ck:~C~_I, f~_(Ck)>fg_(Ck), and at cn_ 1 the function f " jumps from one to zero. (2) If f " has a fixed point, then it must have n fixed points, that is all of the branches but one must cross the diagonal line x ' = x, and the slopes of the branches which cross the diagonal must be equal. (3) There are no fixed points on f~ whose slope is greater than one. Let us assume that one of the branches of fo crosses the identity. We can find by inspection that neither the first not the last branch of f " crosses the identity with slope greater than one. This means that we cannot have n segments crossing the diagonal with slope greater than one. (4) If n < N + 2 then f has no fixed points. At x = cn_ t we have a discontinuity which takes us from f " = 1 to f~ = 0, and neither of the two branches leading up to the discontinuity may cross the identity, so that f " has no fixed points at all. (5) fN+2 has N + 2 fixed points which make up a globally stable orbit of period N + 2. Since CN+ 1 does not exist, fN+2 never attains the value one, and so never presents the jump from zero to one which was the feature responsible for the nonexistence of fixed points in the previous case. Now all of the discontinuities are upwards, but since the function fN+2 covers less than the full range [0,1] this produces at least one crossing (and thus N + 2) with the diagonal. Note that fN+2 consists of only N + 2 branches all of which intersect the diagonal (see fig. 10). All segments have a slope which is less than one since there are no downward jumps, proving the stability of the orbit.

We now compute the inverse images of c as a function of the parameters, until we find one

R. P~rez Pascual and J. Lomnitz-Adler / Coupled relaxation oscillators 67

f"(x}

/ / I /

/ /

/ /

/ /

/ /

/ - - /

/ /

/ /

/ - - /

fi/ r ~

/ i r.....- --'~ /

/ 1 / t

/ I

x

f""(x)

f

/

#

Fig. 9. The n th iterate of a type I map for n less than N + 2. There are n + 1 branches, and none may cross the diagonal (see text).

Fig. 10. The ( N + 2 ) n d iterate of map I. There are N + 2 branches whose image spans less than the full range and which intersect the identity at N + 2 points.

which lies between a and b, thus ending the sequence and determining the period of the stable

orbit. If a < c < b then N = 0 and the stable orbit has period two. Let c not lie within the range and, without loss of generality, we may assume that a < b < c. Then c 1 will be given by

c (c - b) (10) cl = 1 - b

and there will be a stable orbit of period three if

a < c 1 < b. This equation defines a region limited by the curves

a ( 1 - b ) = c ( c - b ) ; b ( 1 - b ) = c ( c - b ) . (11)

Given a value for c the first equation defines a hyperbola which joins the point a = 0, b = c, with

the point on the line a = b given by

a = b 1 = ½( c + 1 + X/1 + 2 C - 3C 2 ), (12)

where we have to take the negative sign, because for the positive choice b is greater than c; the

second equation gives the straight line b = b 1.

b

O=C

Fig. 11. The bifurcation space of map I shown in fig. 8, for a < b. The numbers in the white regions are the periodicity of the stable orbits of the maps corresponding to these parameters. The dashed regions correspond to maps whose orbits have higher periods or whose orbits are quasiperiodic.

These two curves define the period three region shown in the lower left part of fig. 11.

I f c 1 does not lie within the range (a, b) we must calculate c2, which satisfies

C ( C 1 - - b) c2 1 - b (13)

68 R. P~rez Pascual and J. Lomnitz-Adler / Coupled relaxation oscillators

Again we have a situation similar to that discussed above. There will be a stable period four orbit if a < c= < b. The limits of the period four region will be a vertical line at the b value given by

b(1 - b 2) = c( c 2 - c b - b + b 2) (14)

and another curve which joins the point a = b = b 2 on the line a = b with the point (a = 0, b = bl) as shown in fig. 11. Following this process to find a

region of period k + 2 we calculate c k and use the condition a < c k < b; in general there will be more than one possibility and one of them will have the toothlike shape which we have already found for

c 1 and c 2. The structure of the other regions of period

N + 2 which we have left out in the above discus-

sion may be obtained from a theorem by Keener [9], which states that the rotation number of maps having a j u m p singularity is independent of the

initial condition, and that if we alter the map by means of a continuous parameter ~, the rotation number R is a Cantor function of h which takes on rational values in non-empty intervals and irrational values on a Cantor set of measure zero. When the rotation number is rational there exists a periodic orbit, and when it is irrational, the i n v a r i a n t s e t f') f n ( [ 0 , 1 ) ) is a

Cantor set. As we can see from above, the regions in (a , b) space which we constructed explicitly

correspond to rotation numbers of the form l / p , where p is the period. From Keener's theorem we find that there exist regions of any rotation number between 1 / p and 1 / ( p + 1) along any arc spanning f rom a region having R = 1 /p to another one with R = 1 / ( p + 1). We can thus construct (in a topological sense) the full bifurcation space of a and b; for a fixed value of c it is shown in

fig. 11. The class of systems which lead to map I can

never be such that b < a. However we can have the situation where there is no jump discontinuity, which has a unique stable orbit of period one. In

the bifurcation space of fig. 11, we have included this orbit in the region b < a.

Map H

This map has the following structure. It consists of three linear segments, the first of which begins

at f ( 0 ) = c, and continues up to the point x = a, where f ( a ) = O = l . At this point we have a discontinuity in the derivative and the second segment has slope zero, mapping the entire region

(a , b) onto the point x = 0. At x = b we begin the third segment which extends from f ( b ) = 0 to

f (1 ) = c. We shall call the first and third segments the first and second branches, and the intervening

segment the flat region. Map II is shown in fig. 12. The orbital structure of the iterations of this

map is straightforward to obtain once it is observed that the map is the inverse of map I. That is to say that, where the periodic solutions of map I were obtained by searching for the preimage of c which lies in the range between a and b, the periodic solutions of map II are obtained by iterating this

map forward until one of the iterates of c falls within the range (a, b). The next iterate of c will have the value zero, and the one after that will

again be equal to c.

The point to note is that, although the maps themselves are very different, the equations which define regions of a specific periodicity in the

f {×)

¢

b x

Fig. 12. Illustration of map type II.

R. Pdrez Pascual and J. Lomnitz-Adler / Coupled relaxation oscillators 69

bifurcation space are identical, and so the bifurcation space of map II is the same as shown in fig. 11.

In the case of map I we first constructed the toothlike sections associated to rotation number l / N , and then used Keener's theorem to obtain the structure of the bifurcation space between them. These toothlike regions correspond to orbits in which all of the points but one fall on only one of the two branches of the original map, and the last point falls on the other branch. If c > b they fall on the second branch while if c < a they fall on the first. In map II we have the same toothlike structure along the a = 0 and the b = 1 margins of the bifurcation space, and these regions correspond to orbits in which all of the points but one fall on one of the two branches, with the last point being in the range (a, b) where f (x ) has its flat portion. Again, if c > b the points fall on the second branch, and if c < a they fall on the first.

When b < a, the origin is included in the flat region and we have a stable orbit of period one.

Map III

feature of a number of maps to be studied in this paper: the flat region which maps a range of finite measure onto the origin.

Since this map corresponds to b = 1, for c > a we have a period two orbit. When c < a the bifurcation space, in terms of c and a, corresponds to varying these two parameters while remaining on the fight edge of fig. 11. In this edge, we pass directly from regions of rotation number R --- 1/N to ones with R = 1 / ( N + 1), and of course all steps fall on the first branch.

For a given value of c, we may define AN(C ) as the value of a which separates those regions having periods (N + 2) and (N + 3). The equation for A N is obtained by iterating f(c) N times and equating this value to a, resulting in the equation

aN÷l(1--a) = c ( 1 - - c ) N+I. (15)

The following statements are easy to prove. (1) Eq. (15) has at most two real solutions for a, one is trivial and gives A N - - ( 1 - c). (2) The other

Map III, shown in fig. 13, is a special case of map II, in which the parameter b is equal to one. We felt that it is worthwhile to exhibit this case explicitly since it highlights the most prominent

f (x) ~.________________I . . . . . . . . . . . . . .

Fig. 13. Illustration of map type III.

/ / / /-,,, -2\\\\\\\\ \

I I I I

o

Fig. 14. The bifurcation space of map type IlL All orbits have rotation number l /N, where N is the number inscribed in the corresponding region.

70 R. Pdrez Pascual and J. Lornnitz-A dler / Coupled relaxation oscillators

solution is such that at c = 0 , A s = 0 , and at c = 1, A s = 1. (3) When there is only one solution, A s = (1 - c) and dAs /dc = 1. With this informa- tion we are able to plot the bifurcation space shown in fig. 14.

Map IV

This map, shown in fig. 15, contains features of maps I and II. We shall not solve this map with the same degree of rigour, but present the principal structures of the bifurcation space. As in map II this map consists of three linear segments, the second of which is a "fiat region" which maps an entire range of values a < x < b onto the value x = 0. As in the above-mentioned case this region will be associated to double resets in the coupled

system. This map is distinguished from map II by the

fact that there exists a discontinuity at x = 0 and x = 1, that is f ( 0 ) = c is greater than f ( 1 ) = d. Since a discontinuity of this form is responsible for the appearance of periodic orbits in map I, we may expect the regions of low periods in the bifurcation space to be broadened, as in fact does happen.

We shall divide the bifurcation space up into the six regions shown in fig. 16, according to the

c

f ix )

I . . . . . . . -I . . . . . . . . . . . . . . . . . . . . . ! I I I

. . . . . . . . -4 . . . . . . . . . . . . . . / d

a b x

Fig. 15. Illustration of map type IV.

o=I

/

b=d b=c b

iii

O=C

o=d

b=1

Fig. 16. The different regions of map type IV, according to eq. (16).

following combinations of parameters:

(i) a < b < d < c , (ii) a < d < b < c , (iii) a < d < c < b , (iv) d < a < b < c , (v) d < a < c < b , (vi) d < c < a < b .

(16)

f ' t x~

I - 1 1

/ ! / I / i / I - ~ - - .V" . . . . . -~'c'

t , ~' / /

, / / f f

/ . / /

C' X'

Fig. 17. Case (vi) of map IV after carrying out the transformation given in eq. (17) which shifts the discontinuity of the function f ( x ) away from the end point.

R. Pdrez Pascual and J. Lomnitz-Adler/ Coupled relaxation oscillators 71

Cases (iii) and (v) clearly support a stable orbit of period two, since f ( 0 ) = c and f ( c ) = 0. To determine the structure of the remainder of the bifurcation space, we note that a range of the values of the variable x are inaccessible to the system: after the first iteration of map IV it is impossible for the variable x to take on values between x = d and x = c. Let us redefine the variable x by cutting out the range ( d , c ) ,

identifying these two values and renormalizing the new range:

X t = x / ( 1 - (c - d) ) ,

( x - ( c - d ) ) / ( 1 - ( c - d ) ) ,

i f x < d ,

i f x > c .

(17)

This new variable may take on values ranging from zero to one, and the transformed map f ' is shown in fig. 17 for cases (vi). We see that, as in the case of the original function f , there is a range of x ' which is again forbidden. However, this discontinuity has now been moved over to the point x ' ( d ) = x ' ( c ) = d / (1 - (c - d ) ) so that f ' (0) =f ' (1) . We note that the slopes of the two branches of f , before and after the flat region, are left unchanged by this transformation.

In case (i) the discontinuity in f ' occurs at x ' ( d ) , to the right of the flat region. By changing

a, l

4

b-d

a,c

a-d

b-c b,l b

Fig. 18. Bifurcation space of map type IV.

the variable

y = l - x ' ; g ( y ) = l - f ' ( x ) ,

we may observe that map (i) transforms into the map (vi) shown in fig. 17. We shall discuss that case below.

In a similar fashion, case (ii) transforms into case (vi) In this case the lower part of the pro- hibited region of map IV coincides with the upper range of the flat region. This means that a section of the flat region will be cut off when construction (17) is carried out and that the discontinuity will take place at x ' = d ' = b'. As in the previous case, we see that there is a range in x ' which is inaccessible to the successive iterations of f ' . If we redefine the range of the variable x ' in the same way as in eq. (17) we observe that the new map f " ( x " )

is again of the same form as the original map f ( x ) , but with the new parameters a", b " = d " and c" such that this map is a limiting case of case (i), whose results we can apply directly.

As we mentioned before, cases (iii) and (v) have a period two orbit which may be obtained triv- ially. Case (iv) also has a stable two-orbit. When we carry out the transformation (17) in case (iv) the resulting map has no flat part, and there are two forbidden regions at the lower and upper ranges of x ' , that is the iterates of f ' ( x ' ) may not take on values in the ranges (0, ( b - c)/(1 - ( c - d))) or (d(1 + a - c ) / a ( 1 - (c - d)), l) . Repeat- ing the transformation (17) the resulting map is of type I with its parameters in the 2-orbit range.

The remaining region of parameter space is case (vi), where d < c < a < b . We know from the discussion in the previous sections that there are a number of regions in the bifurcation space with rotation number 1 / N corresponding to orbits which consist of ( N - 1) steps on the first branch of f and one step in the flat region. These orbits are independent of the specific value of d and coincide exactly with those obtained for map II. On the other hand, when a = b there is no flat region and map IV coincides with map I. This means that along the diagonal of fig. 16 the regions

72 R. P~rez Pascual and J. Lomnitz-A dler/ Coupled relaxation oscillators

of rotation number p / q are broadened, and rather than ending at a point they have a certain width. Although we have not proven it, it is not unreasonable to expect that as we vary the parameters of the map away from a = b by means of a parameter which we vary continuously, the resulting bifurcation space will have a similar structure as the previous maps discussed in this paper. Thus we expect the regions of lower periodicity to be broadened at the expense of those of higher period. This is observed in numerical simulations of the map.

The resulting bifurcation space is shown in fig. 18. In this diagram we have fixed the parameters c and d and present the regions of a and b corresponding to a given period. The boundaries of the period three region is that part of the diagram associated to cases (i) and (ii) is obtained analytically in a straightforward manner. They are given by the two inequalities

a < d ( c - b ) / ( 1 - b) < b case (i),

a < d ( c - b ) / ( 1 - b ) < d case(ii) .

The most important point of map IV remaining to be discussed refers to the question of bistability. In case (ii) one may show that bistability does not occur because the condition that f ( 0 ) = f ( 1 ) only allows the existence of stable periodic orbits which contain the point zero as one of the fixed points. As this condition is no longer satisfied, we may encounter the situation where two stable orbits exist for a given set of parameters, the first, of type A has zero as a fixed point, while the second (type B) has orbits similar to those encountered in map I.

It is not easy to determine whether this bistability does in fact exist. One may prove that the only region in parameter space where a 2-orbit of type B exists is (iv), where no orbits of type A are allowed, indicating that such orbits are never encountered in situations of bistability. Similarly one may prove that if map IV is such that it supports an orbit of type A whose winding number

R is equal to 1/N, that is the parameters lie within one of the toothiike sections of the bifurcation space, then the map will not support stable orbits of type B.

It appears that, if there are regions in the bifurcation space which support more than one stable orbit, then these will be such that both orbits will be rather complicated in form. At present no example has been found of a map of type IV which presents such orbits, but the possibility of its occurrence has not been elimin- ated.

Map V

This map (shown in fig. 19) again consists of two branches of positive slope interpolated by a fiat part which maps a region of non-zero measure onto the value one. As in the case of maps II, III and IV this map will lead to stable periodic orbits of rotation number 1 / N when c < a and a < f ( c ) < b. However, the features of the bifurcation space in the intermediate regions is far from clear since this map has no inverse, making it impossible for us to use Keener's theorem. It is quite likely that the bifurcation space will contain regions corresponding to stable orbits of all rotation numbers. In addition it is possible that those regions of the bifurcation space which correspond to quasiperiodic orbits will have finite measure.

f(x)

d

c

,

. . . . . . . . . . . . . . . i . . . . . . . . . . .

! p

a b x

Fig. 19. Illustration of map type V.

R. P$rez Pascual and J. Lomnitz-Adler / Coupled relaxation oscillators 73

Chaotic situations where the rotation number is not well-defined [9] may exist.

Map VI

Map VI is shown in fig. 20. It is another variant of the map with two branches and a fiat region. As for map IV we may separate the bifurcation space into six regions:

(i) a < b < e < c ,

(ii) a < e < b < c , (iii) a < e < c < b ,

(iv) e < a < b < c ,

(v) e < a < c < b ,

(vi) e < c < a < b .

Case (i) has a single stable orbit of period one with a fixed point at

e - c b x 1 - b - c + e" (18)

Cases (iii) and (v) correspond to stable orbits of period two where the variable x takes on the values zero and c. For case (iv) we may truncate the range of the variable x as we did in map IV, cutting out the range (0, e) and identifying x = 0 with x = e to transform case (iv) of map VI into cases (i) of map IV. Cases (ii) and (vi) will have toothlike structures corresponding to rotation

numbers 1 / N of the kind with which we are already familiar from previous sections. The regions lying between these will probably have a structure similar to that encountered in maps II and IV.

Map VII

Map VII is shown in fig. 21. It consists of two segments whose slopes are of opposite signs and with magnitudes

1 - c d - c (19) m l = a ' m 2 = l - a "

Rather than discussing the bifurcation space of this diagram in terms of the parameters a, c, and d, we shall construct it in terms of the two magnitudes m 1 and m 2 for a given value of a. We see that

1 1 m 2 a O < m l < a ' O < m 2 < 1 - a ' m 1 l - a "

(20)

and the two parameters c and d are given in terms of m 1, m 2 and a b y

c = 1 - aml;

d = m2(1 - a ) + c = 1 + m 2 - a ( m 1 + m2). (21)

f(x) c

[- . . . .

a • b

Fig. 20. Illustration of map type VI.

f(x)

o

Fig. 21. Illustration of map type VII.


When the absolute values of the two slopes are greater than one no stable orbits are allowed. Also, when m 2 is less than one, and d is greater than a, there exists a stable orbit of period one. This condition is satisfied when

a ( m 1 + 1) -- 1 m 2 > 1 - a (22)

If m 2 is less than one, and the above condition is not satisfied, all orbits are of the following form: there can never be two consecutive points which fall upon the second branch. For a given set of parameters we define the two integers N and M by

f N - l ( c ) < a ; f N ( c ) > a , (23)

f m - l ( d ) < a ; f M ( d ) > a , (24)

and let us consider the iterated function fk , with k < M. This function consists of k + 1 branches, the first of which has a positive slope equal to mkx, while all of the following branches have a negative

slope whose absolute value is equal to m2m~ -1. The discontinuities between the different branches occur at the points

a 0, a 1 . . . . . ak_l ; a 0 = a ; f k ( a k ) = a , (25)

We now evaluate fk(c) and f k (d ) for k < M. For both of these functions, all of the points along the sequence are on the first branch of the initial function, and they are

f k ( c ) = c + mlc + m2c + . . . +mfc

_ c ( 1 - m l ~ + 1 )

1 - - m 1

f k ( d) = c + mlc + m2c + . . . +mkl d

_ C(1 - - m k) + m l k d .

1 - m 1

(27)

(28)

Given a we may section out the space (ml, m2) into regions associated to different values of M. The equation for the boundary separating the regions corresponding to M and M + 1 is

( 1 - a m x ) ( 1 - m ~ ) a= 1 - m 1

+ m ~ [ 1 + m 2 - a ( m 1 + m2)] ,

m 2 ( m l ) = m ~ + l ( 1 - am1)- (1 - a ) (1 - ma)(1 - a)ml M

(29)

(30)

for a given value of m l , we subtract m~+l ( rn l ) f rom m~(ml) to find

that is a i is the i th preimage of a. Now the first branch of f goes from fk(o) to f k (ak_ l )= 1; it cannot cross the identity and thus a stable k-orbit can only involve the remaining k segments. All of these branches have negative slope, since they only involve one step on the second branch of the original map. As long as k < M, the last segment

of f k does not cross the identity since f~'(a)= f k - l ( d ) < a and thus no k-orbit exists. This simply reflects the fact that it requires at least M steps to return to the second branch and so all orbits must have at the most a rotation number of 1 / ( M + 1). Similarly, there cannot be more than N steps along the first branch of f , and thus the rotation number must be greater than 1 / ( N + 1),

1 / ( N + 1) < R < 1 / ( M + 1). (26)

m ~ t ( m x ) - m ~ + l ( m l ) = - - 1

(31) m M + l •

On the other hand, the sections in (ml, m2) space which correspond to different values of N have boundaries which only depend on m~, since the equation for N is independent of d. These boundaries appear as vertical fines in the bifurcation space, and the boundaries for M and for N = M must coincide at m I such that m2(ml) = 0. The equation satisfied by m I is

m ? + l ( 1 - aml) = (1 - a ) . (32)

There are two solutions to this equation, one of which occurs at m 1 = 1 and is meaningless. Thus we have determined the range which is allowed for

m 2 the rotation number at a given point in parameter space.

Given the parameter ml, m 2 we determine what is the value of M which corresponds to this orbit. One may prove that there exists an (M + 1)-orbit, since the ( M + 2)nd branch begins at fM(a)= fM+l(d) > a, and this branch has a negative slope, thus by necessity crossing the identity. Of course, if fM+l crosses the identity once, then it crosses the identity M + 1 times, since there are no shorter period orbits. (This may beseen in the following way: for k < M + 1, the first branch of fk starts at fk(c) and ends at fk___ 1, and thus does not cross the identity; but neither does the last branch cross the identity since fk(a) < a. Since there are only k + 1 branches, and two of them cannot cross the identity, none of them can.)

Now if N is not equal to M, then in addition to ( M + 1)-orbits we can have higher period orbits, up to ( N + 1). Let us see what happens when N = M + I . In this case we have fM+l(a)= fM(d) > a, and fM+l(1) =fM(c) < a, and there is an intermediate value of x which we denote by z, for which fM+l(z)=a. For all of the iterates fk(x), with k < ( M + 1), we have found that the last branch of f covered the range (a, 1). When N = k = ( M + 1), however, this last branch will cover the range between z and one, and will begin at fk+l(z)= 1 to end at fM+I(C), crossing the identity and having a slope of mlm 2. We see then that when N ~: M we may have bistability, provided that both types of orbits are stable.

Now we approach the question of stability. The orbits which we have discussed all consist of k steps on the first branch with one step on the second branch. For an orbit to be stable we must satisfy

m~m 2 < 1. (33)

In fig. 22 we show the resulting bifurcation space. In it we have plotted three different curves: those separating sections corresponding to different M's , straight vertical lines corresponding to different N 's, which coincide for N = M at m 2 = 0, and the

1 nq.

R. P~rez Pascual and J. Lomnitz-A dler / Coupled relaxation oscillators 75

m 2

1 m 1

Fig. 22. Bifurcation space of map type VII.

curves satisfying eq. (33). The regions for which eq. (33) is not satisfied for the corresponding M are unstable, and the regions where N = M and condition (33) is satisfied for M, M + 1, M + 2 . . . are bistable, with permitted orbits having the periods M + 1, M + 2, M + 3, and so forth.

Map VIII

In fig. 23 we show map VIII consisting of three segments which has some of the features of maps II and VII. The first branch has positive slope, with an initial value f(0) -- c which terminates at x = a, f (a)= 1 = 0. The second segment is a flat region which maps the entire range (a, b) onto x = 0, and finally we have a second branch of negative slope such that f(b) = d and f(1) = c.

The presence of the first segment and the flat region indicates that the bifurcation space in the variables (a , b) for a given pair (c, d) will have the same toothlike structure which we have already observed in all maps containing this feature. We

76 R. P#rez Pascual and J. Lomnitz-Adler / Coupled relaxation oscillators

f ( x )

I I i I I I I I I I I I

. . . . . . . . . _!I . . . . . . I . . . . . . . . . .

I-

a t

f(x)

. . . . . . . . . . . . . . . . . . . . . . . . . . . C

o x

Fig. 23. Illustration of map type VIII. Fig. 24. Illustration of map type IX.

also expect to have orbits of the kind obtained for map VII. In general, for a given set of parameters it is possible to have stable orbits of both kinds, meaning both those which pass through the origin every so many steps and those which do not. For example, let us look at the case where d > b, b > c > a and ( d - c ) < ( 1 - b ) . In this case we may have a stable orbit of period 2 which passes through the origin and the point x = c as well as a one orbit at the stable fixed point

d - cb x = 1 + d - b - c " (34)

In general, we may expect maps of type VIII to support both types of orbits. To determine whether orbits of the kind exhibited in map VII may exist, we extend the second branch into the flat region, to obtain a map of type VII with

d ' = d + m 2 ( b - a ) (35)

so long as d ' is less than or equal to one. Given this map we obtain the range of periodicities M and N from the previous case. To determine whether these orbits may exist, we iterate this transformed map M, M + 1, M + 2 . . . . . N times to obtain explicitly the points of the orbits of the corresponding periods. If any of the fixed points falls within the range (a, b), then this orbit is forbidden in map VIII.

M a p I X

Map IX is a generalization of map VII, differing from the previous map in the fact that f(1) is now greater than f(0) . This difference is not fundamental: as before, if d > a a stable one-orbit may exist if the magnitude of the slope of the second branch is less than one. When d < a all stable orbits will consist of groups of steps on the first branch separated by single steps on the second branch, and the structure of that part of the orbit which lies on the second branch is identical to what was found for map VII. Because we have an additional essential parameter which may be used to decrease the slope of the second branch, we will find that when e = c the bifurcation space of map IX is exactly that of VII, but that as e increases m 2 decreases, thus increasing the range of parameters for which stable orbits exist. The bistability which was observed in the previous case will persist. Map IX is shown in fig. 24.

M a p X

Map X is shown in fig. 25. It consists of two linear segments of negative slope with a discontinuity at x = a. The bifurcation space of this map is trivial: There will always exist two fixed points at

c . 1 + m2a (36) x l = l+m----~' x 2 = l + m - - ~ '

R. Pirez Pascual and J. Lomnitz-Adler / Coupled relaxation oscillators 77

f(x)

. . . . . . . . . . . L . . . . . . . . . .

r

Fig. 25. Illustration of map type X.

m2

BISTABLE ~ ~

ml

Fig. 26. Bifurcation space of map type X.

where the absolute value of the slopes are given by

1 - d c and m 2 = - - . (37) m l = a 1 - a

When m t is less than one and m 2 is greater than one only the first fixed point is stable, and conversely when m I > 1, m 2 < 1. If the absolute values of the two slopes are less than one the system will present bistability; whereas if both of them are greater than one, that is c > d, there are no stable orbits, the rotation number is not well- defined and chaotic orbits may exist [9]. The bifurcation space is shown in fig. 26.

/, A I I

I I I I I

f ( x ) i i

1 1 , , ,,

fCx)

Fig. 27. Illustration of map type XI.

'//, 'ltil/i/!/i;il , ?,',, JAt I I I I I

, I ' I I i I I i I I I I I I I i i

I I I I I I i ' I I , I I t l I I I i l I I I i I I I I I I I I I I I

X

Fig. 28. Illustration of map type XII.

separated by discontinuities which take the function f ( x ) from f = 1 to f = 0. The orbits of these maps are in general chaotic and unstable, having large sensitivity to initial conditions.

Map XII is a special case of map XI which was included as a separate diagram because of an interesting feature. If we denote as x , the nth point at which f has a discontinuity, map XII consists of an infinite number of branches such that there exists an accumulation point of { x , } as shown in fig. 28. The interesting feature is that the memory of initial conditions is forgotten within one iteration of the map.

Maps XI and )(11 4. Behaviour of the coupled system

Map XI is shown in fig. 27. It consists of a number of linear branches of positive slope

The bifurcation spaces of the dynamical systems generated by the maps discussed in the previous


Table I The maps associated to each diagram as a function of the parameters of the two oscillator system. Unless stated otherwise, the parameters h 1 and h 2 of eq. (1) are normalized to one. The first column states the class of diagram, the second defines the range of parameters in question, and the third and fourth columns contain the associated type of x and y maps, respectively.

Class Parameters x map y map

(T+ ,T+) A2/Al > Otl; A1/A2 > I /a 2 II II (T+ ,T+) A1/Z2<a 2 III origin ( T + , T - ) A1/A2>a 1 VI IV ( T + , T - ) A1/A2<a t III origin ( T + , T - ) A 2 / A I < a 2 origin III ( T - , T - ) 1/a l > a 2 I I (B+,B+) f l2<A2/Al<l/ f l l I I (B+,B+) A2/AI>I/ f l l IX X ( B + , B - ) A2/A1<i/fll;fl2Al/(A2+fl2al)<fll I XI ( B + , B - ) f l2A1/(A2+fl2A1)>fll XI I ( B + , B - ) A 2 / A t > I / f l I IX X ( B - , B - ) h l o r h 2 ~ 0 XI XI ( B - , B - ) h i = h 2 = 0 XII XII ( B + , T + ) A 2 / A I > I ; 1 / f l l > A 2 / A a > e t 2 VI V (B+,T+) A2/Ax<I;1/fl l>A2/AI>et2 VI IV ( B + , T + ) A 2 / A I > I / f l l ; A2/AI>a2 VII IV (B+,T+) 1/ f l l>A2/AI>I VI origin ( B + , T - ) A2/A x < l / i l l I I (B + ,T - ) A2/A 1 > 1~ill VII X ( B - , T - ) fll <(Al +flla2Az)/(Az+Ala2) I I ( B - , T - ) fll>(Alq'-flla2AE)/(A2+AlOt2) I XI ( B - , T + ) A 2 / A I > a 2 II V ( B - , T + ) A z / A I < a 2 origin III

sec t ion descr ibe the na ture of the orbi ts of each of

the i nd iv idua l osci l la tors once the geometr ical

" d e c o u p l i n g " has been carr ied out. A full so lut ion

of the coup led system must also de te rmine which

m a p s a p p e a r for the ind iv idua l subsys tems once a

set of p a r a m e t e r s has been es tabl ished for the full

system.

As several types of coupl ing can resul t in the

same type of maps , and since for one specific type

of coup l ing different coupl ing pa rame te r s can lead

to different maps , it is not in general poss ible to

es tab l i sh a one to one cor respondence be tween the

coup l ed sys tem and its associa ted maps. In this

sec t ion we show in detai l how the "decoup l ing"

m a p s are re la ted to the coupled system in some

specia l cases, and we present the combina t ion of

m a p s which appea r for all cases under s tudy in

t ab le I, wi thou t de te rmin ing the exact dependence

of the m a p ' s pa rame te r s on the system parameters .

F i r s t l y we s tudy the example which we have

used t h roughou t our presenta t ion, the (T + , T + )

system. This par t i cu la r system is p r o b a b l y the

s imples t because the two associa ted maps are al-

ways of type II , no mat te r which coupl ing pa rame-

ters a re used. F o r this reason it is a good s tar t ing

p o i n t in any discuss ion of the re la t ion of the maps

to the full system.

In fig. 29 we show a system of this type in which

we have insc r ibed the t ra jec tory of a s table o rb i t

in phase space. In this par t i cu la r example the

o rb i t begins at x = y = 0 and re turns to the same

p o i n t af ter five resets of the y var iable and three

resets of the x variable. This means that the

ro t a t i on n u m b e r of the orbi t in the associa ted m a p

R. P$rez Pascual and J. Lomnitz-Adler / Coupled relaxation oscillators 79

V',/

I

/F T/ ) /Y/ X

Fig. 29. Trajectory of a stable orbit of the (T + ,T + ) system for which the periods of the x and y subsystems are five and three, respectively. The rotation number of the x and y variables are ~ and ~, respectively.

" y " is ~, and the rotation number for the " x " map is 3. These illustrate a general property of the asymptotic orbits of the "decoupling" maps:

Let p be the period of the asymptotic orbit that has the greater period, and R 1 = q / p its rotation number. Then the period of the asymptotic orbit of the other map will be q. In case the maps have winding number one, the rotation number of the

second map is given by

R 2 = 1 / R 1 ( m o d l ) . (38)

How do the essential parameters of the sub- maps depend on those of the full system? Let us begin with the (T + , T + ) case in which the ratio of the two slopes m = A z / A I is equal to one, and the two coupling parameters are a x and a 2. As long as the two slopes are equal the coupled system has period one, and the associated maps lie in the region a > b of fig. 11. Let us decrease m until

1 (39) m = 1 + o r 1

At this point the parameters of the y map are

a = 0 ; b = l + , v 2 - m ; c = l . (40)

If we decrease m by an amount ~, the new parameters of the y map are given by

m (41) c = 1 - e q m '

a = l - m ( l + a l ) ; b = l + a E - m . (42)

We see that if m is very close to 1/(1 + al), then c is very dose to one, while a is small and b is less than one. This means that once m decreases beyond a certain point, we go from period one to an orbit of a very high periodicity. For c << 1 the change may not be apparent from looking at a finite number of y (or x) intervals, because they are very similar to the period one orbit, but the change becomes apparent when we look at both variables together, since for m > 1/(1 + al) the two systems are reset at exactly the same time, whereas for m less than this value the two subsystems have lost their synchronization.

This behaviour is quite general for this system. As we vary the parameters of the full system away from a given situation one or the other variable may seem to behave in virtually the same manner as before even if one has made a transition from one region to another in the bifurcation spaces of the "decoupling" maps. The change in periodicity will become immediately apparent if in addition to observing the time series of the individual subsystems we also note the correlations between them.

Let us continue decreasing the slope m. At some point we will find that c = b, which is the beginning of the period two region of the y map. This appears as part of the period one region of the x map since there are no resets in the y variable except the one which returns us to the origin. If we decrease m still further the low periodicity will again be lost to be recovered when one subsystem settles into a period three region and the other into a period one orbit.

80 R. POrez Paseual and J. Lomnitz-A dler/ Coupled relaxation oscillators

y=1

Y,

%

(0,0)

.,__, . . . . . . . . . . .

i t \ I ! i \ I ! i \ i ', ! \ t I I 1 I \ !

i t \ I

. . . . . . . . . . . . . . . i--?\i / \ t

V X o x Xq x-1

Fig. 30. Phase space diagram of the (T - , T - ) system. The x map will only map the region which lies between (Xo, X 1) onto itself, and similarly for the y map.

It is of little use to go into this example with

any more detail, and we now proceed to discuss

the case ( T - , T - ) . This case's orbits will be

complete ly different f rom those of the previous

ones due to the face that none of the associated

maps can have a flat region. In fig. 30 we show the

phase space when x and y are normalized such

that their greatest values are respectively one. Here

we see that the associated x ~ x ' map values x in

the range X o and Xt, with

Xo

Xa - m

42(1 - 4 1 ) (43) ( m + a 2 ) ( 1 - ~ 1 4 2 ) '

1 - - a I (44) 1 - - a l a 2 '

where m = A2/At, and similarly for the y vari-

able, where we have

max(1 - 42) (45) )1o = ( m a l + 1)(1 - axa2)'

1 - a 2 (46) I11 - 1 - a l a 2 "

Let us s tudy the x map. Since the intervals (0, X0) and ( )Ca, 1) are inaccessible, we shall only be inter-

ested in mapp ing the range (Xo,)(1) onto itself.

We begin with m = 1, eq = a 2. The trajectory

which begins at (X0,0) will first cross the x

threshold at ~ = (1 - X0)/(1 + at), and then con-

t inue to increase until the y threshold is reached

at b = (1 - ? ) / ( 1 + 42 ) . In general, we have

X "4- (11 (47) x ' = (1 + 41)(1 + 42)

and so the map x ~ x ' consists of a single line segment which covers less than the full range

(Xo, Xt). RenormaliTing this range so that its full

length is equal to one, the resulting map is of type

I, where a > b and no singularity exists. There is a

stable orbit, and it is of period one. If we vary the

two parameters 41, a2, the basic feature of the resulting map will be the same, since x'(Xt) < X t f rom eq. (47). The y map is identical.

We now vary the slope m =A2/A v Following the same procedure as above we find that, for m

greater than (1 + 4 2 ) / ( 1 + a t ) , but less than

r l / ( x l - t o ) ,

1 - ( 1 - al)m + mx x ' = (m + 42)(1 + oqrn) ' (48)

the cor responding x map will be the same as

before. However, once m increases beyond the

value

rl m X1 _ Xo, (49)

tbe x map will consist of two branches. The x

map will begin at b = x'(Xo) of the order of X 1, as x increases, there will occur a singularity in the x

map at the point c defined by the condit ion m ( X a - c ) = Yr At this point x ' jumps f rom the

value x ' = X 1 to x ' = X o, after which it proceeds to increase until a=x ' (X1) is reached. It is s t ra ightforward to determine that b > a, and thus the x map is of type I.

Should we decrease m rather than increasing it, a similar effect will take place when m is such that

R. P ~rez P ascual and J. Lomni tz-A dler / Coupled relaxation oscillators 81

y=l

\

/ \ \

. . . . . - T , r X X=I

Fig. 31. Phase space of the ( T - , T - ) map when one coupling constant is greater than one.

f(x)

/

/ i

/ / / t / 1 !

t / /

/ /

/ /

Fig. 32. Limiting case of map I arising from the system shown in fig. 31.

m = (Y1 -Y)/X1. At this value the x map makes a transition from being of type I with b < a to being of the same type with a < b. The singular point when this transition occurs is c - - X 1, so that the period of the orbit is infinite. If m is decreased further, then finite-period orbits appear.

The discussion above covers all cases of the (T - , T - ) diagram except one: If either one of the two coupling parameters becomes greater than one, the associated diagram has dosed triangular regions in phase space, as shown in fig. 31, where we show the trajectory taken by the system in phase space. The full two-oscillator system will always collapse onto a complex orbit in which one of the variables (y in fig. 31) is reset at an ever-increasing rate until it almost remains fixed at it's base value. This point is not a fixed point of the subsystem's orbit, and eventually the full system makes a transition that resets the other variable. If both coupling parameters are greater than one, a similar trajectory is followed in the case of the second variable's orbit. Since there is an accumulation point in the orbit, the orbit of such systems is unique, even if it is not periodic. The map for one of the variables is shown in fig. 32.

As in the case of the (T + , T + ) diagram, the product of the two rotation numbers must be the same. In this case the two oscillators are not in general synchronized, and thus it is more difficult to determine from a reduced number of points whether a transition in the periodicity of the orbits has taken place.

The other eight systems may be analyzed in the same manner. The maps which may appear for any given system as its parameters are varied are shown in table I.

To summarize, we have taken the ten systems which are obtained when we couple each of the individual oscillators at either the threshold or base. We have "decoupled" them by means of the techniques shown in ref. [7] and we have found that there are twelve classes of "decoupling" maps. These maps were discussed in section 3, where we obtained the full bifurcation spaces for a number of them, and presented arguments of plausibility about the structure of the bifurcation spaces of the remaining ones. Finally we have shown in detail how the parameters of the maps vary with the parameters of the full system for two of the coupled systems and we presented a table describing

82 R. P$rez Pascual and J. Lomnitz-.4 dler / Coupled relaxation oscillators

this dependence for all of the systems under discussion.

References

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[3] C.G. Bufe, P. Wharsh and R.O. Burford, Geophys. Res. Lett. 4 (1977) 91.

D.J. Mento, C.P. Ervin and L.D. McG-innis, BSSA 76 (1986) 1001. T. Aragnos and A.S. Kiremidjian, Bull. Seismol. Soc. Am. 74 (1984) 2593. L.R. Sykes and R.C. Quittmeyer, in: Earthquake Predict- ion, Maurice Ewing Series 4 (1981) 217.

[4] J.P. Gollub, T.O. Brunner and B.G. Danley, Science 200 (1980) 48. J. Testa, J. Perez and C. leffries, Phys. Rev. Lett. 48 (1984) 714.

[5] M.J. Feigenbaum, J. Stat. Phys. 19 (1978) 25. M.J. Feigenbaum, J. Stat. Phys. 21 (1979) 669. S.J. Schenker, Physica 5D (1982) 405. M.J. Feigenbaum, Physica 7D (1983) 16. P. Cvitanovic, M.H. Jensen, L.P. Kadanoff and I. Procac- cia, Phys. Rev. Lett. 55 (1985) 343.

[6] M.R. Guevara, L. Glass and A. Shrier, Science 214 (1981) 1350. R. Perez and L. Glass, Phys. Lett. A90 (1982) 441. L. Glass and R. Perez, Phys. Rev. Lett. 48 (1982) 1772.

[7] J. Lonmitz-Adler and R. P~rez Pascual, Rev. Mex. Fis. 32-$1 (1986) $221.

[8] T. Allen, Physica 6D (1983) 305. [9] J.P. Keener, Trans. Am. Math. Soc. 261 (1980) 589.

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Coupled relaxation oscillators and circle maps