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Topology in dynamical systems
Volume
93A, number
PHYSICS
4
10 January
LETTERS
1983
TOPOLOGY IN DYNAMICAL SYSTEMS Tatsuya
UEZU
Department Received
of Physics, Kyoto University, Kyoto, Japarl
20 September
1982
Previously the torsion number and the relative torsion number have been introduced to describe the topological character of periodic solutions of a three-dimensional ordinary differential equation. In this letter, considering the rewinding mechanism of tangent vectors, we investigate the change of the torsion numbers and the relative torsion numbers as perioddoubling bifurcation cascades.
The mechanisms of bifurcation, especially the cascade of period-doubling bifurcation, have been greatly investigated in one-dimensional mappings [ 11. In flow systems, reduction to one-dimensional mappings is usually performed, either by taking a Lorenz plot or a Poincare mapping, and many studies have been done on these reduced mappings. Few studies on the mechanism of bifurcation have considered the flow systems themselves [2]. To clarify the mechanism of bifurcation in flow systems, it is necessary to study the topological structures of the solutions. In another paper [3], we studied in three-dimensional flow systems the topological structures of periodic solutions and the changes in the topological character of periodic solutions when subharmonic or pitchfork bifurcation takes place. In this letter, we investigate the case in which period-doubling bifurcation cascades. We treat only autonomous systems. Let us consider a three-dimensional ordinary differential equation, dx (t)ldt = F(x, cl) ,
(1)
where ~1is a bifurcation parameter and x, F(x,p)E R3. A periodic solution x0 of this system is characterized topologically by its knot type, torsion numbers ni and relative torsion numbers ri. First, we give brief definitions Of ni and ri [3]. A solution w(t) of the linearized equation around x0 is given by
I
ds w(0)=S(t)w(0),
0 031-9163/83/0000-0000/S
(2)
03.00 0 1983 North-Holland
where exp[ ] is an ordered exponential. Let hi be a real eigenvalue of S(T) (T is the period of x0) and ei the eigenvector belonging to $( I Al I > I Xl I, 10 = 1). Wi (t) is defined to be S(t)ei (i = 1,2). The link number between two loops Cl and C2 is given by the Gauss integral; L(C1, C,) = -(114n)[
4 (r2 - ‘I> drl x dr2) ) (3) 1 2 11’1- r2 II3
where II II denotes the euclidean norm, and ( , ) and X an inner- and a vector product, respectively [4]. Using this notation, Iii and ri are defined as follows; 2ni = L(
[x0(t);
0 f c< T] )
ho(f) + ~~~(~)/ll~~(~)II;0 G t< ZTI)
’ C$z(t)Gj3(t)
1 ‘i = TF
s
0
- 4*(f)93(f)
cxf2(t) + cq3(t)
’
(5)
where dots denote the derivative with respect to t, and oij(t) = (Wi(t),fi(t)),fl =i~/lli~II~f2 =iO x g()/lliO X _?, I( andf3 =fl X f2. ni is one-half of the link number betweenxO and xis and describes the torsion of the tangent space around the periodic orbit x0 (see fig. 1). ‘i is the rotation number around the direction of velocity vector 20 is the moving coordinate system { fl , f2 ,f3}. Hi and ri are related by the following equation ni=ri
+L(xo +<‘fi,xu),
[‘<
1)
(6) 161
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93A, number
PHYSICS
4
LETTERS
of 2k period solutionx@) integer ko. From this, we get
Ai<0
hi>0
IO January
An\‘) = r\‘)(i) - ry)(f)
1983
for any k not less than some
E Arc:) ,
k 2 k, .
(9)
We call Arfk) the relative rewinding number. (2) The rewinding number An p’ changes as An:“’ = (-1)k
2lli=-1
2lli=+2
Fig. 1. Explanation of torsion number. Each point at whichxir crosses under x0, count +I for $ and - 1 for {. The sum of these equals 2lri [4].
[‘fi ,~~)~L([x~(t)t~'f~(t);O~ r < 7'1, [x0(t); 0 < t < T]).While X, is real, ‘21 = n2 and r1
where L(xo t
= rz hold, so hereafter we concentrate our attention on the case i = 1. Furthermore, n I is constant in the region where X, is real. Next, we summarize rules for the change of nl and r1 when pitchfork or subharmonic (period-doubling) bifurcation takes place. See ref. [3] for details. Case 1. Pitchfork bifurcation, h, = 1. Two periodic solutions x’ and x”’ are generated, and IZ; = rz;’ = tz, , r; = r’; = rl, 15(x’, x”) = tzl. The knot type ofx’ and x” are the same as that ofxo. Case 2. Subharmonic
bifurcation,
AI = -I.
One periodic solution x”’ is generated, and ny = 4n I, r;’ = 2r,. The knot type ofxn’ is that ofxlC. Now, we investigate the case in which perioddoubling bifurcation cascades. The eigenvalue ilk) changes as positive + complex + negative in each Zk period region. Let n [‘j(i), Hi@(f), r’:‘(i) and rik)(f) be torsion numbers and relative torsion numbers for the 2k period regibn, where i is for h ‘) > 0 and f for Xik) < 0. The rewinding number An{ 1) and L@) are defined as by)
= iI\‘)(i) - n?‘(f)
L(k) = L(x CM+ @‘,
, x(k)) )
(7) (8)
where x(k) is the 2k period solution with period T(k) and @) = i @) X E(k)/// i (k) X ;‘(k) 11.Then, the following two statements hold. (1) L(k) does not change anywhere in the stable region 162
I/? ,
(10)
where 1 is an integer. The reason for (1) is as follows: when period-doubling bifurcation cascades, the global topological structure of the system does not change, and if k is large enough, the stable region ofx@) is so small that the shape of x ck) does not change greatly, therefore L ck) does not change. Next, we investigate the rewinding mechanism (10). To clarify the situation, we first discuss the case of one-dimensional mappings. In a one-dimensional mappingx,,+l = G(x,), the change of the sign of the derivative dG ‘k(x)/dx Ix+)
= dG2k(xy))
(xJk) is one of 2k period points), from positive to negat/ve,is necessary in each 2k period region for the cascade of period-doubling bifurcation. When the period Zk point x6 k, which is nearest to the critical point X (i.e. dG/dx Ix=.? = 0) passes through X, the derivative dGZk(_yik)) changes from positive to negative, becoming zero at xhk) =X. Period-doubling bifurcation occurs when dG 2k(xik)) becomes ~ 1, and the new period 2k+1 solution is generated with dG2k+1(xik+1)) = +I. The new period 2k+l point xhk+l) nearest tox moves toward X and passes through ,U with dG 2k+‘(xr+‘)) turning negative, so the direction of the movement of x6 k+l) is opposite to that of the movement of xr’. This process is repeated (see fig. 2). Similar behavior is seen in a three-dimensional flow system. In the region where period-doubling bifurcation cascades, the topological structure of the system can be described qualitatively by the paper sheet model [2]. A fold plays the similar role of the critical point in a onedimensional mapping. When a periodic orbit xck) passes through the fold, the eigenvalue than es its sign, and the orientation of the eigenvector e ?1k) changes by the angle 2nZ/2 relative to the sheet. After bifurcation, the new solution~(~+l) moves toward the fold and passes through it, so the orientation of the eigenvector
Volume
93A. number
PHYSICS
4
10 January
LETTERS
Solving eqs. (lo)-(
1983
12), we obtain for (k a ko) .
r(k+ko)(i)= Tk [r\h)(i) 1
_ i(-l)ko[
“k+ko)(f) 1
= T, [r ye)(i)
ny+ko)(i)
(i) - $(-l)koZ = T: [n (ko) 1
+ $(-i)k(-l)kel],
_ 4 (-l)ko/
-
k(-$)k(-l)k”Z]
,
+ $(-$)k(-l)kor]
,
r~y+~o)(f) = Tt [awoke) ~ $(-l)ko/
~ &:)k(-l)kel] L(k+kO) t
mapping. of xAk’ m a one-dimensional The ordinate isx and the abscissa is a bifurcation parameter. The solid curves are xgk). The yk\hed-dotted curve denotes and the dashed line the the trace of the movement of x0 critical point 2.
,
= Tk[Tk(nikO)(i)
;(-l)$]
+ &l)k’kO
- g(_l)kO/
_ r:kO)(i)
)
(13)
Fig. 2. The movement
where Tk = Zk. The mean relative torsion number (Y) per mean one period (Tck)/2k) (i.e. per cycle) is (r) = iF_
G’rlk)
= (rtko’(i) - f(-l)ker)/r,o
(14)
e(k+‘)
changes by -2rr1/2 (fig. 3). The repetition of tiis process induces the alternating change of the sign of the rewinding number. Thus we get eq. (10). The value of E reflects the global topological structure of the system, i.e. the net number of foldings. From the result of case (2) we get n:)(i)=
4ny-
‘j(f) ,
r\“(i)
= 2’$-
l)(f)
(11)
and from eq. (6) and (1) n$)(i) = rik)(i) + Lck) ,
(k2
.y)(fJ
(k > k,J .
= rik’(f-) + Lck) ,
kO) >
Fig. 3. Paper sheet model. The dashed-dotted folds and solid curves periodic solutions.
(1.3
curves denote
and the mean linking number (n) is (,z)=
lim $&9 1 -- (nl(ko) (1) . - ;(-l)koZ)/T;o
k+-
(15)
Though we introduce (r) as the limit of ‘ik)/Tk, (I) can be defined for any trajectory [3]. (I-) contains information on the rotation of tangent vectors relative to that trajectory. As for (n), it is also generalized for any trajectory. (n) is different from the “mean linking coefficient” [S]. The latter is concerned with the linking number between two trajectories. On the other hand, (n) is the mean linking number per cycle between a trajectory and another trajectory near to it. 2, (r) and (n>, together with the “mean linking coefficient” and the “mean linking index” [ 51, may be used to classify strange attractors topologically. Knowledge of the topology of periodic solutions may be used to judge whether one solution is a bifurcating solution of another. If the topological character of two stable periodic solutionsx, (say at parameter pt) andx2(p2) cannot be related by any of the rules obtained above, one can conclude that the two solutions are not related by bifurcation, and if the system has a single basin, one may expect chaos somewhere between p’I and p2. Numerical calculation. As three-dimensional flow systems, we consider (a) two-dimensional non-autonomous systems and (b) three-dimensional autonomous systems. (Three-dimensional non-autonomous
163
Volume 93A, number 4
PHYSICS LETTERS
3 I
m I
r-4 .
164
z I
-7
10 January 1983
L(k)
Arik)
rp’
rl’lk’
1
41
-3
38
-312 41
-512
7712
159
-5
154
l/2 159
-1112
30712
r = 333.4 (i) r = 333.3 (f)
r = 333.5 (f)
trivia1
l/2
r = 334(i)
trivial
knot type
1
0
-l/2
112
k = 4 (a)
0
L(k)
0
1
1
1
r = 400(f)
k = 3 (a)
1
rik)
A@
1
np)
r = 420(i)
__-
r = 480
r = 500
r = 600
-
k = 0 (a)
(a)
(9
1
-l/2
l/2
r = 360(f)
-1
3
2
trivial
-l/2
r = 356(i)
k = 1 (a)
3
-l/2
512
r= 340(f)
-1 11
IO
7-2s
l/2
r = 388(i)
k = 2 (a)
11
-312
1912
r = 334.5(f)
The Lorenz system; dx/df = -U(X - ,v), d)r/dt = rx - JJ - xz, dz/dt = XJJ - bz, 0 = 16, b = 4. T2,5 denotes a torus knot 141. (s) or (a) denotes a symmetric or an bifurcation takes place, and then period-doubling bifurcation asymetric limit cycle. Italic numbers are the values evaluated by using eq. (13). At r = 493, pitchfork cascades.
Table 2
CA
3
E r
Z WY
.
Volume 93A, number 4
PHYSICS
LETTERS
10 January
1983
des. Results are shown in tables 1 and 2. In case (a), kn = 1, I = 1, r:‘“)(i) = -1, nl’e)(i) = -2, and so (r) =-i,(n)=-:. Incase(b),ko=O,f= l,ri’O)(i)=O, n (k 1 0) (1) -- 1, and (I) = -f , (n) = z. Thus in both cases, the number of foldings is one. The mean rotation angle per cycle relative to the reference trajectory is (a) -2n. i,(b) -2n 5 and the mean linking number per cycle is(a) -g,(b):. The extension of the above discussion to systems of dimension higher than 3 has still to be studied.
Fig. 4. A braid. It is regarded trefoil knot.
as a knot.
In this case, it is the
systems are essentially four-dimensional autonomous systems, and so we do not treat them.) For a twodimensional non-autonomous system, if the system is time-periodic (period Tf>, periodic solutions are regarded as “braids”. If points on a braid which have the same (_q v) coordinates on the two surfaces t = 0 and t = Tf are connected without making any links, the braid is regarded as a knot and nj, ‘i, and L can also be defined, i.e. we can consider the system as a threedimensional autonomous system on Sl X R2 (see fig. 4). We investigated numerically (a) the forced Brusselator [6] and (b) the Lorenz model in the parameter regions where period-doubling bifurcation casca-
166
The author would like to thank Professor K. Tomita and the members of his research group and Professor Y. Kuramoto for valuable discussions. He is also grateful to Mr. P.C. Davis for correcting his English. References [l] [2] [ 31 [4] [5] [6]
M.J. Fcigenbaum, J. Stat. Phys. 19 (1978) 25; P. Manneville and Y. Pomeau, Physica 1D (1980) 219. O.E. Rosslcr, Bull. Math. Biol. 39 (1977) 275. T. Uezu and Y. Aizawa, Prog. Theor. Phys. 68 (1982), to be published. D. Rolfsen, Knots and links (Publish or Perish, Berkeley, 1976). Y. Aizawa and T. Uezu, Prog. Theor. Phys. Lett. 67 (1982) 982. K. Tomita andT. Kai, Suppl. Prog. Theor. Phys. 64 (1978) 280.
93A, number
PHYSICS
4
10 January
LETTERS
1983
TOPOLOGY IN DYNAMICAL SYSTEMS Tatsuya
UEZU
Department Received
of Physics, Kyoto University, Kyoto, Japarl
20 September
1982
Previously the torsion number and the relative torsion number have been introduced to describe the topological character of periodic solutions of a three-dimensional ordinary differential equation. In this letter, considering the rewinding mechanism of tangent vectors, we investigate the change of the torsion numbers and the relative torsion numbers as perioddoubling bifurcation cascades.
The mechanisms of bifurcation, especially the cascade of period-doubling bifurcation, have been greatly investigated in one-dimensional mappings [ 11. In flow systems, reduction to one-dimensional mappings is usually performed, either by taking a Lorenz plot or a Poincare mapping, and many studies have been done on these reduced mappings. Few studies on the mechanism of bifurcation have considered the flow systems themselves [2]. To clarify the mechanism of bifurcation in flow systems, it is necessary to study the topological structures of the solutions. In another paper [3], we studied in three-dimensional flow systems the topological structures of periodic solutions and the changes in the topological character of periodic solutions when subharmonic or pitchfork bifurcation takes place. In this letter, we investigate the case in which period-doubling bifurcation cascades. We treat only autonomous systems. Let us consider a three-dimensional ordinary differential equation, dx (t)ldt = F(x, cl) ,
(1)
where ~1is a bifurcation parameter and x, F(x,p)E R3. A periodic solution x0 of this system is characterized topologically by its knot type, torsion numbers ni and relative torsion numbers ri. First, we give brief definitions Of ni and ri [3]. A solution w(t) of the linearized equation around x0 is given by
I
ds w(0)=S(t)w(0),
0 031-9163/83/0000-0000/S
(2)
03.00 0 1983 North-Holland
where exp[ ] is an ordered exponential. Let hi be a real eigenvalue of S(T) (T is the period of x0) and ei the eigenvector belonging to $( I Al I > I Xl I, 10 = 1). Wi (t) is defined to be S(t)ei (i = 1,2). The link number between two loops Cl and C2 is given by the Gauss integral; L(C1, C,) = -(114n)[
4 (r2 - ‘I> drl x dr2) ) (3) 1 2 11’1- r2 II3
where II II denotes the euclidean norm, and ( , ) and X an inner- and a vector product, respectively [4]. Using this notation, Iii and ri are defined as follows; 2ni = L(
[x0(t);
0 f c< T] )
ho(f) + ~~~(~)/ll~~(~)II;0 G t< ZTI)
’ C$z(t)Gj3(t)
1 ‘i = TF
s
0
- 4*(f)93(f)
cxf2(t) + cq3(t)
’
(5)
where dots denote the derivative with respect to t, and oij(t) = (Wi(t),fi(t)),fl =i~/lli~II~f2 =iO x g()/lliO X _?, I( andf3 =fl X f2. ni is one-half of the link number betweenxO and xis and describes the torsion of the tangent space around the periodic orbit x0 (see fig. 1). ‘i is the rotation number around the direction of velocity vector 20 is the moving coordinate system { fl , f2 ,f3}. Hi and ri are related by the following equation ni=ri
+L(xo +<‘fi,xu),
[‘<
1)
(6) 161
Volume
93A, number
PHYSICS
4
LETTERS
of 2k period solutionx@) integer ko. From this, we get
Ai<0
hi>0
IO January
An\‘) = r\‘)(i) - ry)(f)
1983
for any k not less than some
E Arc:) ,
k 2 k, .
(9)
We call Arfk) the relative rewinding number. (2) The rewinding number An p’ changes as An:“’ = (-1)k
2lli=-1
2lli=+2
Fig. 1. Explanation of torsion number. Each point at whichxir crosses under x0, count +I for $ and - 1 for {. The sum of these equals 2lri [4].
[‘fi ,~~)~L([x~(t)t~'f~(t);O~ r < 7'1, [x0(t); 0 < t < T]).While X, is real, ‘21 = n2 and r1
where L(xo t
= rz hold, so hereafter we concentrate our attention on the case i = 1. Furthermore, n I is constant in the region where X, is real. Next, we summarize rules for the change of nl and r1 when pitchfork or subharmonic (period-doubling) bifurcation takes place. See ref. [3] for details. Case 1. Pitchfork bifurcation, h, = 1. Two periodic solutions x’ and x”’ are generated, and IZ; = rz;’ = tz, , r; = r’; = rl, 15(x’, x”) = tzl. The knot type ofx’ and x” are the same as that ofxo. Case 2. Subharmonic
bifurcation,
AI = -I.
One periodic solution x”’ is generated, and ny = 4n I, r;’ = 2r,. The knot type ofxn’ is that ofxlC. Now, we investigate the case in which perioddoubling bifurcation cascades. The eigenvalue ilk) changes as positive + complex + negative in each Zk period region. Let n [‘j(i), Hi@(f), r’:‘(i) and rik)(f) be torsion numbers and relative torsion numbers for the 2k period regibn, where i is for h ‘) > 0 and f for Xik) < 0. The rewinding number An{ 1) and L@) are defined as by)
= iI\‘)(i) - n?‘(f)
L(k) = L(x CM+ @‘,
, x(k)) )
(7) (8)
where x(k) is the 2k period solution with period T(k) and @) = i @) X E(k)/// i (k) X ;‘(k) 11.Then, the following two statements hold. (1) L(k) does not change anywhere in the stable region 162
I/? ,
(10)
where 1 is an integer. The reason for (1) is as follows: when period-doubling bifurcation cascades, the global topological structure of the system does not change, and if k is large enough, the stable region ofx@) is so small that the shape of x ck) does not change greatly, therefore L ck) does not change. Next, we investigate the rewinding mechanism (10). To clarify the situation, we first discuss the case of one-dimensional mappings. In a one-dimensional mappingx,,+l = G(x,), the change of the sign of the derivative dG ‘k(x)/dx Ix+)
= dG2k(xy))
(xJk) is one of 2k period points), from positive to negat/ve,is necessary in each 2k period region for the cascade of period-doubling bifurcation. When the period Zk point x6 k, which is nearest to the critical point X (i.e. dG/dx Ix=.? = 0) passes through X, the derivative dGZk(_yik)) changes from positive to negative, becoming zero at xhk) =X. Period-doubling bifurcation occurs when dG 2k(xik)) becomes ~ 1, and the new period 2k+1 solution is generated with dG2k+1(xik+1)) = +I. The new period 2k+l point xhk+l) nearest tox moves toward X and passes through ,U with dG 2k+‘(xr+‘)) turning negative, so the direction of the movement of x6 k+l) is opposite to that of the movement of xr’. This process is repeated (see fig. 2). Similar behavior is seen in a three-dimensional flow system. In the region where period-doubling bifurcation cascades, the topological structure of the system can be described qualitatively by the paper sheet model [2]. A fold plays the similar role of the critical point in a onedimensional mapping. When a periodic orbit xck) passes through the fold, the eigenvalue than es its sign, and the orientation of the eigenvector e ?1k) changes by the angle 2nZ/2 relative to the sheet. After bifurcation, the new solution~(~+l) moves toward the fold and passes through it, so the orientation of the eigenvector
Volume
93A. number
PHYSICS
4
10 January
LETTERS
Solving eqs. (lo)-(
1983
12), we obtain for (k a ko) .
r(k+ko)(i)= Tk [r\h)(i) 1
_ i(-l)ko[
“k+ko)(f) 1
= T, [r ye)(i)
ny+ko)(i)
(i) - $(-l)koZ = T: [n (ko) 1
+ $(-i)k(-l)kel],
_ 4 (-l)ko/
-
k(-$)k(-l)k”Z]
,
+ $(-$)k(-l)kor]
,
r~y+~o)(f) = Tt [awoke) ~ $(-l)ko/
~ &:)k(-l)kel] L(k+kO) t
mapping. of xAk’ m a one-dimensional The ordinate isx and the abscissa is a bifurcation parameter. The solid curves are xgk). The yk\hed-dotted curve denotes and the dashed line the the trace of the movement of x0 critical point 2.
,
= Tk[Tk(nikO)(i)
;(-l)$]
+ &l)k’kO
- g(_l)kO/
_ r:kO)(i)
)
(13)
Fig. 2. The movement
where Tk = Zk. The mean relative torsion number (Y) per mean one period (Tck)/2k) (i.e. per cycle) is (r) = iF_
G’rlk)
= (rtko’(i) - f(-l)ker)/r,o
(14)
e(k+‘)
changes by -2rr1/2 (fig. 3). The repetition of tiis process induces the alternating change of the sign of the rewinding number. Thus we get eq. (10). The value of E reflects the global topological structure of the system, i.e. the net number of foldings. From the result of case (2) we get n:)(i)=
4ny-
‘j(f) ,
r\“(i)
= 2’$-
l)(f)
(11)
and from eq. (6) and (1) n$)(i) = rik)(i) + Lck) ,
(k2
.y)(fJ
(k > k,J .
= rik’(f-) + Lck) ,
kO) >
Fig. 3. Paper sheet model. The dashed-dotted folds and solid curves periodic solutions.
(1.3
curves denote
and the mean linking number (n) is (,z)=
lim $&9 1 -- (nl(ko) (1) . - ;(-l)koZ)/T;o
k+-
(15)
Though we introduce (r) as the limit of ‘ik)/Tk, (I) can be defined for any trajectory [3]. (I-) contains information on the rotation of tangent vectors relative to that trajectory. As for (n), it is also generalized for any trajectory. (n) is different from the “mean linking coefficient” [S]. The latter is concerned with the linking number between two trajectories. On the other hand, (n) is the mean linking number per cycle between a trajectory and another trajectory near to it. 2, (r) and (n>, together with the “mean linking coefficient” and the “mean linking index” [ 51, may be used to classify strange attractors topologically. Knowledge of the topology of periodic solutions may be used to judge whether one solution is a bifurcating solution of another. If the topological character of two stable periodic solutionsx, (say at parameter pt) andx2(p2) cannot be related by any of the rules obtained above, one can conclude that the two solutions are not related by bifurcation, and if the system has a single basin, one may expect chaos somewhere between p’I and p2. Numerical calculation. As three-dimensional flow systems, we consider (a) two-dimensional non-autonomous systems and (b) three-dimensional autonomous systems. (Three-dimensional non-autonomous
163
Volume 93A, number 4
PHYSICS LETTERS
3 I
m I
r-4 .
164
z I
-7
10 January 1983
L(k)
Arik)
rp’
rl’lk’
1
41
-3
38
-312 41
-512
7712
159
-5
154
l/2 159
-1112
30712
r = 333.4 (i) r = 333.3 (f)
r = 333.5 (f)
trivia1
l/2
r = 334(i)
trivial
knot type
1
0
-l/2
112
k = 4 (a)
0
L(k)
0
1
1
1
r = 400(f)
k = 3 (a)
1
rik)
A@
1
np)
r = 420(i)
__-
r = 480
r = 500
r = 600
-
k = 0 (a)
(a)
(9
1
-l/2
l/2
r = 360(f)
-1
3
2
trivial
-l/2
r = 356(i)
k = 1 (a)
3
-l/2
512
r= 340(f)
-1 11
IO
7-2s
l/2
r = 388(i)
k = 2 (a)
11
-312
1912
r = 334.5(f)
The Lorenz system; dx/df = -U(X - ,v), d)r/dt = rx - JJ - xz, dz/dt = XJJ - bz, 0 = 16, b = 4. T2,5 denotes a torus knot 141. (s) or (a) denotes a symmetric or an bifurcation takes place, and then period-doubling bifurcation asymetric limit cycle. Italic numbers are the values evaluated by using eq. (13). At r = 493, pitchfork cascades.
Table 2
CA
3
E r
Z WY
.
Volume 93A, number 4
PHYSICS
LETTERS
10 January
1983
des. Results are shown in tables 1 and 2. In case (a), kn = 1, I = 1, r:‘“)(i) = -1, nl’e)(i) = -2, and so (r) =-i,(n)=-:. Incase(b),ko=O,f= l,ri’O)(i)=O, n (k 1 0) (1) -- 1, and (I) = -f , (n) = z. Thus in both cases, the number of foldings is one. The mean rotation angle per cycle relative to the reference trajectory is (a) -2n. i,(b) -2n 5 and the mean linking number per cycle is(a) -g,(b):. The extension of the above discussion to systems of dimension higher than 3 has still to be studied.
Fig. 4. A braid. It is regarded trefoil knot.
as a knot.
In this case, it is the
systems are essentially four-dimensional autonomous systems, and so we do not treat them.) For a twodimensional non-autonomous system, if the system is time-periodic (period Tf>, periodic solutions are regarded as “braids”. If points on a braid which have the same (_q v) coordinates on the two surfaces t = 0 and t = Tf are connected without making any links, the braid is regarded as a knot and nj, ‘i, and L can also be defined, i.e. we can consider the system as a threedimensional autonomous system on Sl X R2 (see fig. 4). We investigated numerically (a) the forced Brusselator [6] and (b) the Lorenz model in the parameter regions where period-doubling bifurcation casca-
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The author would like to thank Professor K. Tomita and the members of his research group and Professor Y. Kuramoto for valuable discussions. He is also grateful to Mr. P.C. Davis for correcting his English. References [l] [2] [ 31 [4] [5] [6]
M.J. Fcigenbaum, J. Stat. Phys. 19 (1978) 25; P. Manneville and Y. Pomeau, Physica 1D (1980) 219. O.E. Rosslcr, Bull. Math. Biol. 39 (1977) 275. T. Uezu and Y. Aizawa, Prog. Theor. Phys. 68 (1982), to be published. D. Rolfsen, Knots and links (Publish or Perish, Berkeley, 1976). Y. Aizawa and T. Uezu, Prog. Theor. Phys. Lett. 67 (1982) 982. K. Tomita andT. Kai, Suppl. Prog. Theor. Phys. 64 (1978) 280.