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Bifurcations near homoclinic orbits with symmetry
Volume 103A, number 4
PHYSICS LETTERS
2 July 1984
BIFURCATIONS NEAR HOMOCLINIC ORBITS WITH SYMMETRY Paul GLENDINN1NG Department o f Applied Mathematics and Theoretical Physics, Sih,er Street, Cambridge CB3 9EW, UK
Reccived 19 April 1984 The effcct of symmetry on bifurcations associated with homoclinic orbits to saddle-foci is analysed. With symmetry each homoclhfic bifurcation contributes three periodic orbits to the global bifurcation picture as opposed to a single orbit in the general case. Bifurcations on these orbits are studied : there are sequences of saddle-node and period-doubting bifurcations, chaos and more complicated homoclinic orbits.
1. I n t r o d u c t i o n . 15amilies o f differential equations containing parameter intervals at which chaotic trajectories are observed can frequently be understood in terms of bifurcations associated with the existence of a homoclinic orbit (an orbit which approaches a stationary point o f the flow as t ~ _+oo)at some parameter value. In this letter we prove some results for systems which are invariant under the symmetry
(x, y, z) -. (-x, -y, - z )
(1)
and which, after a trivial change of coordinates and possibly time reversal, can be written in the form ~c = - p x - toy + P ( x , y , z ; u ) , = cox - p y + Q(x, y , z ; u ) , k = kz +R(x,y,
z;u),
(2)
where/9, k > 0, P, Q and R are analytic functions which vanish with their first derivatives at the origin and/a is a real parameter. Note that (1) is the only possible symmetry o f (2) which admitshomoclinic orbits [1 ]. We further assume that there are a pair of homoclinic orbits to the stationary point at the origin when /~=0. In the general case, without the imposed symmetry, there exist a countably infinite number o f periodic orbits and uncountably many aperiodic trajectories at the parameter value (~t = 0 here) at which a homoclinic orbit exists providing 6 - 0/;~ < 1 [2,3]. The bifurcation structure of systems near homoclinicity is investigated in refs. [4,5]. For the interesting case, 6 < 1, 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
there are infinite sequences of saddle-node and perioddoubling bifurcations and more complicated homoclinic orbits. The most important difference between bifurcations of (2) near homoclinic orbits with and without symmetry is that in the symmetric case each homoclinic bifurcation can be thought of as contributing three periodic orbits (a pair of asymmetric orbits in/a > 0 and a symmetric orbit in/a < 0) to the bifurcation diagram far from/.t = 0, whilst in the general case there is only o n e periodic orbit created by the homoclinic orbit that exists outside some neighbourhood of~t = 0 [4]. Despite the intuitively appealing argument that the symmetry should enhance the stability and observability of chaotic motion near homoclinicity the example of section 3 shows that this is not necessarily the case although in other examples it may be possible to observe a stable symmetric orbit, period-doubling, chaos and hysteresis and finally a pair of stable asymmetric orbits as the parameter is increased. 2. B i f u r c a t i o n s a n d h o m o c l i n i c orbits. Using a standard construction [1,4,6] we obtain the following return map on the planes Y.+ = {(x, y, z ) : y = 0,x > 0} and Y~- = {(x, y , z): y = O , x < 0 ) (see fig. 1):
~ + ~ ~+ (z > 0 , x > 0) x ' = r + at~ + axz ~ cos(/j In z + ~1 ) , z ' = la + ~ x z a cos(~ In z + 0 2 ) ,
(3)
163
Volume 103A, number 4
PHYSICS LETTERS
2 July 1984
under the symmetry. The analysis of (3) and (6) follows ref. [4] exactly. We shall summarise the results for (3). The same results hold for (6) by symmetry. To leading order the fixed points of (3) have z-coordinates with
z = la + f3(r + al~)Z ~ cos(~ In z + cI~2) . +
Fig. 1. The two branches of the unstable manifold of the origin are shown with the planes E+- on which the return map is constructed. 1~-+are chosen so that trajectories which intersect Z + (respectively ~ - ) do not intersect Z-(respectively E +) as they spiral closer to the origin.
(7)
For 6 > 1 there is a single fixed point o f (3) f o r / l > 0 which forms the homoclinic orbit at/a = 0, and for/a < 0 there are no simple periodic orbits near the unstable manifold o f the origin which remain entirely in z > 0 (c.f. ref. [6]). The orbit in/a > 0 is stable. As /a tends to zero from above the period of this orbit increases like - X -1 In/a. For 8 ,< 1 there are many fixed points in both/~ > 0 and/~ < 0, and when/~ = 0 there is a countably infinite number of fixed points. These fixed points are created and destroyed in pairs as/~ varies, and there is a sense in which they are all part o f a single periodic orbit (the principal periodic orbit) which approaches homoclinicity at/a = 0 (see fig. 2). There are sequences of saddle-node bifurcations and period-doubling bifurcations on the branches o f the principal periodic orbit [4]. The period o f the periodic orbits when/a = 0 in-
~ - -+ £+ (z > O , x < 0 )
x'=r+au-~lxlz
~ cos(~lnz +~1),
z' = u - 13Ix Iz ~ cos(~ In z + q ' 2 ) , Y,+ ~ Y,- (z < 0 , x
period I
I\\~ 1JI
(4)
J__
period-doubling
_l-__
symmetry breaking
>0)
x ' = - r - a# + a x Izl8 cos(tj in Izl + ~ 1 ) , z' = - U +/~x Izl6 cos(~ In Iz[ + ~ 2 ) ,
(5)
~ - - * Z - (z < 0 , x < 0 ) x ' = - r - a/a - t~lxllz I~ cos(~ In Izi + ~ 1 ) , z' = -t~ - ~ Ix IIz ~ cos(~ In [z I + ~ 2 ) ,
(6)
where 6 = O/X, ~ = - w / X , or, [3, dpl, ~ 2 , a and r are constants. A full description o f the dynamics when 6 > 1 is given in ref. [6], here we constrate on the simple f'txed points of ( 3 ) - ( 6 ) in the case 5 < 1 for which the analysis of refs. [2,3] is valid in the absence o f symmetry. Using the maps (3) and (6) fLxed points o f the return map can be found in z > 0 and z < 0 respectively. These solutions correspond to pairs o f asymmetric periodic orbits which are mapped onto each other 164
,"
sy o,rLj_. . . . . . .
X
-----_-2. . . . . .
t,,o D Ia
Fig. 2. The period of the asymmetric orbits and half the period of the symmetric orbit are plotted against the parameter, v, near t~ = 0 showing the bifurcations and stability of the simple periodic orbits as they approach the homoclinic orbit at # = 0 with ~ < 1. Dashed lines correspond to nonstable orbits and undashed lines correspond to stable (respectively unstable) orbits for 6 > 1/2 (respectively 6 < 1/2).
Volume 103A, number 4
PHYSICS LETTERS
creases like lira I P i + I - Pil = ~/Io-'1
(8)
and the parameter values at which successive saddlenode bifurcations occur satisfy lim (lai+l/lai) = - e x p ( - n p leo I).
(9)
i-+ o o
There is an infinite sequence of more complicated homoclinic orbits (fig. 3a) which loop twice in z > 0 before tending towards the origin [7]. These doublepulse homoclinic orbits (after ref. [8]) occur at paraineter values which accumulate at/a = 0 from above at the rate lim (lai+l/lai) = e x p ( - r r X / i ~ l ) .
i~
(10)
oo
We now turn our attention to orbits which loop once in z > 0 and once in z < 0. Using (4) we look for initial points ( - x * , z*) on E - which map to (x*, - z * ) on g +. Solutions correspond to the simplest symmetric orbits. To lowest order the z-coordinate of such periodic orbits is given by
z* = -la+{3(r+a#)z .6 cos(~ In z* +~b2).
(11)
Comparing this to (7) we see that the same results as obtained above apply (to lowest order) to (11) with the sign of/a changed. Thus for 6 < 1 there are sequences of saddle-node bifurcations as described in (9) and the
2 July 1984
period of the symmetric orbits at/a = 0 increases at twice the rate given in (8). The stability results are interpreted as a symmetry-breaking bifurcation followed by a (conjectured) sequence o f period-doubling bifurcations of the asymmetric orbits produced. This sequence then reverses, finally restabilising the simple symmetric orbit just before the saddle-node bifurcation as shown in fig. 2. Not surprisingly there are also double-pulse homoclinic orbits (fig. 3) in/a < 0. For/a < 0 the unstable + manifold of the origin first strikes Y-+ at ~u = (r + a/~, /a), so from (5) the condition for a double-pulse homoclinic orbit like the one in fig. 3b is
O=-ta+3(r+ala)ltli 8 cos(~ lnl#l + q'2).
(12)
Thus there is an infinite sequence of double-pulse homoclinic orbits when 6 < 1. This sequence converges to/a > 0. Numerical experiments suggest that the double-pulse homoclinic orbits in gt > 0 are formed by the orbits which are produced in the period-doubling bifurcations on the branches of the pair of asymmetric principal periodic orbits and those in # < 0 are created by orbits produced in symmetry-breaking bifurcations on the branches of the simple symmetric periodic orbit. The analysis above is valid only asymptotically for Iz I and I/al small and ~ definitely less than one. In considering examples we must expect variations at large Izl and I#l and for ~ near to one. For a fuller discussion of such variations see ref. [4].
3. An example. To illustrate the above results consider the system [ 1,9] ic=y,
/-
,,, ,.--i:...
\ ',,L'
i
~
li
,' \,
/
f
\\ (a) la>O
/
J
i
/
(b) la
Fig. 3. Schematic illustration o f double-pulse homoclinic orbits: (a) # > 0, (b) # < 0. The dashed lines are the images o f the undashed lines u n d e r the s y m m e t r y .
/
fp=z,
~ =-bz-y+cx(1-x2).
(13)
Periodic orbits have been followed numerically with b = 0.4 and c ('>0) increasing. The characteristic equation of the linearised flow at the origin has a pair of complex conjugate eigenvalues and one real eigenvalue in the region we are interested in below, The real root is always positive (c > 0) and 8 = 1 when c = c 1 = 0.528 for the choice b = 0.4, 6 < 1 for c > c I and 8 > 1 f o r c < c 1. There are stationary points of (13) at B_+ = (+1, 0, 0) as well as at the origin, O. For C < 0.2 these are stable, and when c = 0.2 there is a supercritical Hopf bifurcation producing a stable pair of periodic orbits which are mapped onto each other under the symmetry. As 165
Volume 103A, number 4
PHYSICS LETTERS periud ~5 I
Y 0.6
I
i
l
t
2 July 1984
I:L]
I
!
0.0
J" {t
J - 1 tC
o'.0
'
i ~c
o.:e
o'.c
i
]*.¢
(,)
i
X
1.9
(<
Fig. 4. (a) One of the asymmetric periodic orbits with b = 0.4, c = 0.926127 and period 11.704 starting from x (0) = -0.2510313, y(0) = 0,17159594 and z(0) = 0.27136. (b) The results of following the simple period orbits shown in (a) and (c) numerically (c.f. fig. 2). The dashed line corresponds to the symmetric orbit and the undashed line corresponds to the asymmetric orbits. (c) The simple symmetric orbit with b = 0.4, c = 0.98022 and half-period of 11.735 starting from x (0) = 0.9574206, y (0) = 0.4072612 and z(0) = 0.01384886.
,]
l
I
I
]
i
t
a period-doubling bifurcation o f the periodic orbit o f fig. 4a when c = 0.376. As c increases most trajectories are attracted to infinity b e f o r e the h o m o c l i n i c orbit at c = 0.995 in a similar way to the n o n - s y m m e t r i c analogue o f (13) discussed in refs. [10] and [4]. Many other h o m o c l i n i c orbits have been found. In particular it is interesting that the s y m m e t r i c orbit o f fig. 4c approaches h o m o clinicity with increasing c at c = 1.175.
(b)
Fig. 5. (a) An asymmetric periodic orbit with b = 0.4, c = 0.580985, x(0) = 0.5839862, y(0) = -0.92304785, z(0) = 0.0. (b) The symmetric periodic orbit created in a homoclinic bifurcation with the orbit of (a), b = 0.4,c = 0.580962, x(0) = -0.23511183. y(0) = 0.29117478, z(0) = 0.07092.
I w o u l d like to thank A. Bernoff, P. Coullet, C. Sparrow, C. Tresser and N. Weiss for useful and interesting conversations. This w o r k was supported by an S E R C studentship.
References c is increased we observe a period-doubling sequence followed by a p p a r e n t l y chaotic orbits and then, for larger c, m o s t trajectories eventally leave the region near the origin and go o f f to infinity. The periodic orbit which bifurcates f r o m B÷ w h e n c = 0.2 (fig. 4a) has been followed w i t h increasing c. It is observed to a p p r o a c h h o m o c l i n i c i t y at c = 0.995 in the way illustrated in fig. 4b. A s y m m e t r i c orbit (fig. 4c) has also been f o u n d , which approaches h o m o clinicity at the same parameter value. Fig. 5 shows the orbits involved in a h o m o c l i n i c bifuraction at c = 0.5816. The a s y m m e t r i c orbit (fig. 5 a ) i s created in
166
[lJ [2] [3J [4] [5] [6] [71 [8] [9] [ 10]
C. Tresser, These d'Etat, Universit4 de Nice (1981). L.P. Shilnikov, Soy. Math. Dokl. 6 (1965) 163. L.P. Shilnikov, Math. USSR Sb. 10 (1970) 91. P.A. Glendinning and C. Sparrow, J. Stat. Phys. (1984), to be published. P. Gaspard, R. Kapral and G. Nicolis, J. Star. Phys. (1984), to be published. P. Holmes, J. Diff. Eq. 37 (1980) 382. P. Gaspard, Phys. Lett. 97A (1983) 1. S.P. Hastings, SIAM J. Appl. Math. 42 (1982) 247. P. Coullet, C. Tresser and A. Arn6odo, Phys. Lett. 72A (1979) 268. A. Arn~odo, P. Coullet, E. Spiegel and C. Tresser, Asymptotic chaos, submitted to Physica D (1982).
PHYSICS LETTERS
2 July 1984
BIFURCATIONS NEAR HOMOCLINIC ORBITS WITH SYMMETRY Paul GLENDINN1NG Department o f Applied Mathematics and Theoretical Physics, Sih,er Street, Cambridge CB3 9EW, UK
Reccived 19 April 1984 The effcct of symmetry on bifurcations associated with homoclinic orbits to saddle-foci is analysed. With symmetry each homoclhfic bifurcation contributes three periodic orbits to the global bifurcation picture as opposed to a single orbit in the general case. Bifurcations on these orbits are studied : there are sequences of saddle-node and period-doubting bifurcations, chaos and more complicated homoclinic orbits.
1. I n t r o d u c t i o n . 15amilies o f differential equations containing parameter intervals at which chaotic trajectories are observed can frequently be understood in terms of bifurcations associated with the existence of a homoclinic orbit (an orbit which approaches a stationary point o f the flow as t ~ _+oo)at some parameter value. In this letter we prove some results for systems which are invariant under the symmetry
(x, y, z) -. (-x, -y, - z )
(1)
and which, after a trivial change of coordinates and possibly time reversal, can be written in the form ~c = - p x - toy + P ( x , y , z ; u ) , = cox - p y + Q(x, y , z ; u ) , k = kz +R(x,y,
z;u),
(2)
where/9, k > 0, P, Q and R are analytic functions which vanish with their first derivatives at the origin and/a is a real parameter. Note that (1) is the only possible symmetry o f (2) which admitshomoclinic orbits [1 ]. We further assume that there are a pair of homoclinic orbits to the stationary point at the origin when /~=0. In the general case, without the imposed symmetry, there exist a countably infinite number o f periodic orbits and uncountably many aperiodic trajectories at the parameter value (~t = 0 here) at which a homoclinic orbit exists providing 6 - 0/;~ < 1 [2,3]. The bifurcation structure of systems near homoclinicity is investigated in refs. [4,5]. For the interesting case, 6 < 1, 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
there are infinite sequences of saddle-node and perioddoubling bifurcations and more complicated homoclinic orbits. The most important difference between bifurcations of (2) near homoclinic orbits with and without symmetry is that in the symmetric case each homoclinic bifurcation can be thought of as contributing three periodic orbits (a pair of asymmetric orbits in/a > 0 and a symmetric orbit in/a < 0) to the bifurcation diagram far from/.t = 0, whilst in the general case there is only o n e periodic orbit created by the homoclinic orbit that exists outside some neighbourhood of~t = 0 [4]. Despite the intuitively appealing argument that the symmetry should enhance the stability and observability of chaotic motion near homoclinicity the example of section 3 shows that this is not necessarily the case although in other examples it may be possible to observe a stable symmetric orbit, period-doubling, chaos and hysteresis and finally a pair of stable asymmetric orbits as the parameter is increased. 2. B i f u r c a t i o n s a n d h o m o c l i n i c orbits. Using a standard construction [1,4,6] we obtain the following return map on the planes Y.+ = {(x, y, z ) : y = 0,x > 0} and Y~- = {(x, y , z): y = O , x < 0 ) (see fig. 1):
~ + ~ ~+ (z > 0 , x > 0) x ' = r + at~ + axz ~ cos(/j In z + ~1 ) , z ' = la + ~ x z a cos(~ In z + 0 2 ) ,
(3)
163
Volume 103A, number 4
PHYSICS LETTERS
2 July 1984
under the symmetry. The analysis of (3) and (6) follows ref. [4] exactly. We shall summarise the results for (3). The same results hold for (6) by symmetry. To leading order the fixed points of (3) have z-coordinates with
z = la + f3(r + al~)Z ~ cos(~ In z + cI~2) . +
Fig. 1. The two branches of the unstable manifold of the origin are shown with the planes E+- on which the return map is constructed. 1~-+are chosen so that trajectories which intersect Z + (respectively ~ - ) do not intersect Z-(respectively E +) as they spiral closer to the origin.
(7)
For 6 > 1 there is a single fixed point o f (3) f o r / l > 0 which forms the homoclinic orbit at/a = 0, and for/a < 0 there are no simple periodic orbits near the unstable manifold o f the origin which remain entirely in z > 0 (c.f. ref. [6]). The orbit in/a > 0 is stable. As /a tends to zero from above the period of this orbit increases like - X -1 In/a. For 8 ,< 1 there are many fixed points in both/~ > 0 and/~ < 0, and when/~ = 0 there is a countably infinite number of fixed points. These fixed points are created and destroyed in pairs as/~ varies, and there is a sense in which they are all part o f a single periodic orbit (the principal periodic orbit) which approaches homoclinicity at/a = 0 (see fig. 2). There are sequences of saddle-node bifurcations and period-doubling bifurcations on the branches o f the principal periodic orbit [4]. The period o f the periodic orbits when/a = 0 in-
~ - -+ £+ (z > O , x < 0 )
x'=r+au-~lxlz
~ cos(~lnz +~1),
z' = u - 13Ix Iz ~ cos(~ In z + q ' 2 ) , Y,+ ~ Y,- (z < 0 , x
period I
I\\~ 1JI
(4)
J__
period-doubling
_l-__
symmetry breaking
>0)
x ' = - r - a# + a x Izl8 cos(tj in Izl + ~ 1 ) , z' = - U +/~x Izl6 cos(~ In Iz[ + ~ 2 ) ,
(5)
~ - - * Z - (z < 0 , x < 0 ) x ' = - r - a/a - t~lxllz I~ cos(~ In Izi + ~ 1 ) , z' = -t~ - ~ Ix IIz ~ cos(~ In [z I + ~ 2 ) ,
(6)
where 6 = O/X, ~ = - w / X , or, [3, dpl, ~ 2 , a and r are constants. A full description o f the dynamics when 6 > 1 is given in ref. [6], here we constrate on the simple f'txed points of ( 3 ) - ( 6 ) in the case 5 < 1 for which the analysis of refs. [2,3] is valid in the absence o f symmetry. Using the maps (3) and (6) fLxed points o f the return map can be found in z > 0 and z < 0 respectively. These solutions correspond to pairs o f asymmetric periodic orbits which are mapped onto each other 164
,"
sy o,rLj_. . . . . . .
X
-----_-2. . . . . .
t,,o D Ia
Fig. 2. The period of the asymmetric orbits and half the period of the symmetric orbit are plotted against the parameter, v, near t~ = 0 showing the bifurcations and stability of the simple periodic orbits as they approach the homoclinic orbit at # = 0 with ~ < 1. Dashed lines correspond to nonstable orbits and undashed lines correspond to stable (respectively unstable) orbits for 6 > 1/2 (respectively 6 < 1/2).
Volume 103A, number 4
PHYSICS LETTERS
creases like lira I P i + I - Pil = ~/Io-'1
(8)
and the parameter values at which successive saddlenode bifurcations occur satisfy lim (lai+l/lai) = - e x p ( - n p leo I).
(9)
i-+ o o
There is an infinite sequence of more complicated homoclinic orbits (fig. 3a) which loop twice in z > 0 before tending towards the origin [7]. These doublepulse homoclinic orbits (after ref. [8]) occur at paraineter values which accumulate at/a = 0 from above at the rate lim (lai+l/lai) = e x p ( - r r X / i ~ l ) .
i~
(10)
oo
We now turn our attention to orbits which loop once in z > 0 and once in z < 0. Using (4) we look for initial points ( - x * , z*) on E - which map to (x*, - z * ) on g +. Solutions correspond to the simplest symmetric orbits. To lowest order the z-coordinate of such periodic orbits is given by
z* = -la+{3(r+a#)z .6 cos(~ In z* +~b2).
(11)
Comparing this to (7) we see that the same results as obtained above apply (to lowest order) to (11) with the sign of/a changed. Thus for 6 < 1 there are sequences of saddle-node bifurcations as described in (9) and the
2 July 1984
period of the symmetric orbits at/a = 0 increases at twice the rate given in (8). The stability results are interpreted as a symmetry-breaking bifurcation followed by a (conjectured) sequence o f period-doubling bifurcations of the asymmetric orbits produced. This sequence then reverses, finally restabilising the simple symmetric orbit just before the saddle-node bifurcation as shown in fig. 2. Not surprisingly there are also double-pulse homoclinic orbits (fig. 3) in/a < 0. For/a < 0 the unstable + manifold of the origin first strikes Y-+ at ~u = (r + a/~, /a), so from (5) the condition for a double-pulse homoclinic orbit like the one in fig. 3b is
O=-ta+3(r+ala)ltli 8 cos(~ lnl#l + q'2).
(12)
Thus there is an infinite sequence of double-pulse homoclinic orbits when 6 < 1. This sequence converges to/a > 0. Numerical experiments suggest that the double-pulse homoclinic orbits in gt > 0 are formed by the orbits which are produced in the period-doubling bifurcations on the branches of the pair of asymmetric principal periodic orbits and those in # < 0 are created by orbits produced in symmetry-breaking bifurcations on the branches of the simple symmetric periodic orbit. The analysis above is valid only asymptotically for Iz I and I/al small and ~ definitely less than one. In considering examples we must expect variations at large Izl and I#l and for ~ near to one. For a fuller discussion of such variations see ref. [4].
3. An example. To illustrate the above results consider the system [ 1,9] ic=y,
/-
,,, ,.--i:...
\ ',,L'
i
~
li
,' \,
/
f
\\ (a) la>O
/
J
i
/
(b) la
Fig. 3. Schematic illustration o f double-pulse homoclinic orbits: (a) # > 0, (b) # < 0. The dashed lines are the images o f the undashed lines u n d e r the s y m m e t r y .
/
fp=z,
~ =-bz-y+cx(1-x2).
(13)
Periodic orbits have been followed numerically with b = 0.4 and c ('>0) increasing. The characteristic equation of the linearised flow at the origin has a pair of complex conjugate eigenvalues and one real eigenvalue in the region we are interested in below, The real root is always positive (c > 0) and 8 = 1 when c = c 1 = 0.528 for the choice b = 0.4, 6 < 1 for c > c I and 8 > 1 f o r c < c 1. There are stationary points of (13) at B_+ = (+1, 0, 0) as well as at the origin, O. For C < 0.2 these are stable, and when c = 0.2 there is a supercritical Hopf bifurcation producing a stable pair of periodic orbits which are mapped onto each other under the symmetry. As 165
Volume 103A, number 4
PHYSICS LETTERS periud ~5 I
Y 0.6
I
i
l
t
2 July 1984
I:L]
I
!
0.0
J" {t
J - 1 tC
o'.0
'
i ~c
o.:e
o'.c
i
]*.¢
(,)
i
X
1.9
(<
Fig. 4. (a) One of the asymmetric periodic orbits with b = 0.4, c = 0.926127 and period 11.704 starting from x (0) = -0.2510313, y(0) = 0,17159594 and z(0) = 0.27136. (b) The results of following the simple period orbits shown in (a) and (c) numerically (c.f. fig. 2). The dashed line corresponds to the symmetric orbit and the undashed line corresponds to the asymmetric orbits. (c) The simple symmetric orbit with b = 0.4, c = 0.98022 and half-period of 11.735 starting from x (0) = 0.9574206, y (0) = 0.4072612 and z(0) = 0.01384886.
,]
l
I
I
]
i
t
a period-doubling bifurcation o f the periodic orbit o f fig. 4a when c = 0.376. As c increases most trajectories are attracted to infinity b e f o r e the h o m o c l i n i c orbit at c = 0.995 in a similar way to the n o n - s y m m e t r i c analogue o f (13) discussed in refs. [10] and [4]. Many other h o m o c l i n i c orbits have been found. In particular it is interesting that the s y m m e t r i c orbit o f fig. 4c approaches h o m o clinicity with increasing c at c = 1.175.
(b)
Fig. 5. (a) An asymmetric periodic orbit with b = 0.4, c = 0.580985, x(0) = 0.5839862, y(0) = -0.92304785, z(0) = 0.0. (b) The symmetric periodic orbit created in a homoclinic bifurcation with the orbit of (a), b = 0.4,c = 0.580962, x(0) = -0.23511183. y(0) = 0.29117478, z(0) = 0.07092.
I w o u l d like to thank A. Bernoff, P. Coullet, C. Sparrow, C. Tresser and N. Weiss for useful and interesting conversations. This w o r k was supported by an S E R C studentship.
References c is increased we observe a period-doubling sequence followed by a p p a r e n t l y chaotic orbits and then, for larger c, m o s t trajectories eventally leave the region near the origin and go o f f to infinity. The periodic orbit which bifurcates f r o m B÷ w h e n c = 0.2 (fig. 4a) has been followed w i t h increasing c. It is observed to a p p r o a c h h o m o c l i n i c i t y at c = 0.995 in the way illustrated in fig. 4b. A s y m m e t r i c orbit (fig. 4c) has also been f o u n d , which approaches h o m o clinicity at the same parameter value. Fig. 5 shows the orbits involved in a h o m o c l i n i c bifuraction at c = 0.5816. The a s y m m e t r i c orbit (fig. 5 a ) i s created in
166
[lJ [2] [3J [4] [5] [6] [71 [8] [9] [ 10]
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