Report Arran Tamsett
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d
dtx= f(x) , x= (x1,...,xM)
f(x)
x0(t) = x0(t+T0) x= x+x t= t0 x
x (t) =
x0(t) +x (t) +x (t) t t0x0(t) t < t0
t x (t) x0(t+t)
t
= + 2
d
dt=
f(x)
d
dt
x= f(x) +p (x, t) , p (x, t) = p (x, t+T)
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p
f(x)
+
d
dt1 = 1+I1,2(1, 2)
d
dt2 = 2+I2,1(2, 1)
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=1 2 d
dt = 1 2+I1,2() I2,1()
= f()
f() = 0 (t) = 0
1 2
N
d
dti= i+
Nj=1
Ki jsin (j i)
i g() g() g( +) = g( )
d
dt1 = 1+K1sin (2 1) = 1 K1sin (1 2)
d
dt2 = 2+K2sin (1 2)
=1 2 d
dt = (1 2) (K1+K2)sin()
f()
f()
2
f()
0 f()
0 0
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2
2
1 2
K1 K2
f
2
2
1 2
K1 K2
f
2
2
1 2
K1 K2
f
f()
ddt
= 0 = sin () =(12)(K1+K2)
K1+K2 |1 2|
Ki j = K
N
d
dti = i+
K
N
Nj=1
sin(j i)
ii t i i g() = g()
r (t)
r (t) ei(t) = 1
N
Nj=1
eij(t)
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(t) r (t) r
[0, 1] eik
r (t) ei((t)k) = 1
N
Nj=1
ei(jk)
r (t)sin( k) = 1NN
j=1sin (j k)
d
dti = i K r (t)sin(i )
r (t)
|i| K r (t)
r (t) r (t)
r (t)
1/
N,
r (t ) = r (K) > 0
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r (t) = (t) = = 0
d
dti = i K r sin(i)
|i| K r
ddt
i = 0
i= K r sin(i)
(, ) (, ) d= [, + d] d= 1
t+
[( K r sin()) ] = 0
= K r sin()
(, ) = const
| K r sin()|
(, ) =
2 K2 r2
2
|
K r sin()
|
r=
ei (, ) g() dd
= 0
ei
r=
ei
population=
ei
coherent+
ei
drifting
g()
sin()coherent = 0
ei
coherent = cos()coherent = K r
K r cos( ()) g() d
()
ei
coherent= K r
2
2
cos2 g(K r sin()) d
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r
ei
drift=
||>K r
ei (, ) g() dd
(+, ) = (, ) g() = g() ( >K r, < K r) ( [, 0], [0, ])
I1 =
0
ei (, ) g() d
I2 =
0
ei (, ) g() d
=
I2 =
0
ei ei
+,
g() d
= 0
ei
,
g() d
I1+I2 = 0
ei drift=0
r= K r
2
2
cos2 () g(K r sin()) d
r (t) = 0 = 12
1 = K
2
2
cos2 () g(K r sin()) d
r (K) K
g()
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K r = 1 K = 0 r = 0 K K= Kc
K = Kc g(K r sin()) r
g(K r sin()) =
n=0
1
n!g(n) (0) Kn sinn () rn
r 0 K Kc
1 = Kc
2
2
cos2 () g(0) d
= Kcg(0)
2
Kc = 2
g(0)
r (K) K Kc
r (K) =1 (Kc/K)K > Kc
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i j(t) = cos(i j )I.C.
Dt(T)i j =
1 i j(t)> T
0 i j(t)< T
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T
i j(t)
DT(t)
DT(t)
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Kc= 2g(0)
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