burrus, c. s. & h. resnikoff 2009 - dynamics of social systems

8/13/2019 Burrus, C. S. & H. Resnikoff 2009 - Dynamics of Social Systems

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Dynamics of Social Systems

By:C. Sidney Burrus

Howard Resnikoff


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Dynamics of Social Systems

By:C. Sidney Burrus

Howard Resnikoff

Online:< http://cnx.org/content/col10587/1.7/ >

C O N N E X I O N S

Rice University, Houston, Texas


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F = M A


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x y s

x s yx y s

s y x

x + a x + bx = 0 x (o) x (o) x (t)

x v


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v + x

12 v 2x

x1 y1 x2 y2 x1 + x2

y1 + y2 x1 a a

F = M A

x1 y1 x2 y2

(x1 + x2) (x1 + y2)

ax 1 ay1

t

x (t ) y (t) x (t + T ) y (t + T )


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r i r s

n

M (n + 1) = M (n ) + r i M (n ) r s M (n ) r

r = r i r s R

R = ( r + 1)

M (n + 1) = (1 + r i r s ) M (n )

= (1 + r ) M (n )

M (n + 1) = R M (n ) .

M (n ) n = 0 , 1, 2, ...

M (1) = R M (0)

M (2) = R M (1) = R2 M (0)


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M (3) = R M (2) = R3 M (0)

M (n ) = M (0) Rn

M (0) n R (r 0) n R (r 0)

p (t) r b rd p (t) t

dp

dt = ( r b

rd ) p

r = rb rd

dpdt

= rp

po

p (t ) = po ert

po = p (0)

r rb rd r r b rd

p (0) p (t )

p (t ) T pn p (t )

pn = p (nT ) n = 0 , 1, 2, ...

pn = p (nT ) = prnT o e = po erT n

R = erT


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R

T

R = erT 1 + rT

x (n + 1) x (n )T

= r x (n ) ,

x (n + 1) = x (n ) + rT x (n )

= (1 + rT ) x (n )

x (n ) = x (0) (1 + rT )n

x = k ert ,

dxdt

= k r e rt

r = 1

xdxdt

.

r x

T d

T d = 1

r loge 2 = 0 .6931472

1r


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r T d

p (t ) = 3 e0.021 t

p (t ) t


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x (t )

x (t ) = k ert

e

log x = log k + rt

x t x t r k


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r = p (t 1) p (t 2)

t1 t2

=

log 107

log 105

1823 1670

= 16.118 11.513

153 = 0 .03

r e = 2 .72


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r


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r

r

r = ro (1 p) r o p

p r r p


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r o p p

1

dpdt

= r p

= ro(l p

) p

dpdt

= ro p ro p2

p (t ) = ke r o t

1 + ke r o t=

1 + k 1e r o t

k = p (o)1 p (o)

p (o)


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p = ro (1 pk ) p

k p () k = 1

p (t )

x = p = t + log k ,

x ( ) = 11 + e

p (t)

l p

= + k 1 e ro t

log l p = log k t


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r r p

dpdt

= f ( p)

f ( p) p p (t)

f ( p)

r = ro ( 1 + Bp Cp 2 )

dpdt

= ro p + ro Bp 2 r3o Cp


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k


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dpdt

= rp

dpdt =

p (nT + T )

p (nT )T

T

t = nT n = 0 , 1, 2, ...

p (nT + T ) = p (nT ) + T rp (nT ) p (nT + T ) = (1 + rT ) p (nT )

T

p (nT ) = x (n )

x (n + 1) = (1 + rT ) x (n )

x (n )

dpdt

= 1 pk

ro p


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x (n + 1) = x (n ) + T r o x (n ) T r o

k x(n )2

x (n ) r o = 0 .1

po = 100 = 0 .0001 k = 10 , 000

T = 2

k

k k k =

T T = 0 .2

T T = 20 T

r o = 0 .1 M = .2 k =10, 000 ro

r o k M = .5


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dp1dt = f ( p1 , p2)

dp2dt

= g ( p1 , p2)

p1 p1 p2 p2 p1 p2

p1 p2 p1 p2

f g

dp1dt

= a p1 b p1 p2

dp2dt

= c p2 d p1 p2


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a c b d

dc

p1 = a dc p1 ba p2 dc p1 ba

p2 = c ba p2 dc p1 ba p2 x = dc p1 y =

ba p2

x = a ( x xy )

y = c ( y

xy )

x y p1 p2a c

t

dp1dp2

= f ( p1 , p2 )

g ( p1 , p2 )

p1 , p2

dxdy

= ac

x xyy xy


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A x y t B

A


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B


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C y x

dxdt dydt x y

x = 0 y = 0 x = 1 y = 1

f ( p1 , p2 ) = 0


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p1

dp 1dt = 0

dp 1dt

f = 0

p2 p1 f = 0 p2 p1

p2 f = 0 psub 1 p2

p2

g ( p1 , p2 ) = 0

p1 p2 g = 0


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p1 p2 f = 0 g = 0


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x = a ( x xy ) = 0

y = 1

y = c ( y xy ) = 0

x = 1


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f = 0 g = 0

m

f g

= m

a = c = 1

x xyy xy

= m

x y

x = my

1 + ( m 1) y

m = 0 y = 1 m = x = 1 m = 1

x = y


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m = 1

x = y

(1 2y) .

p1 a = 0 .1 p2 p2 p1

>

p1 = ap 1 bp1 p2 = ( a bp2) p1


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a = 0 .1 b = 0 .001 p2 p1 p2 p2 p1 c = 0 .1 d = 0 .001

p1 = 0 .1 p1

0.001 p1 p2

p2 = 0 .1 p2 0.001 p1 psb2 .

p1 (n + 1) = p1 (n ) + T a p1 (n ) T b p1 (n ) p2 (n )

p2 (n + 1) = p2 (n ) + T c p2 (n ) T d p1 (n ) p2 (n )

p1 = 25

p2 = 50

p1 = 30

p2 = 35

p1 = 0 .2 p1 0.002 p1 p2

p2 = 0 .05 p2 0.0005 p1 p2

p1 = 10 p2 = 1 p2 p1 p1 = 16 p2 = 1 p2 p1 = 17

p2 = 1 p1 p2 = 500 p1 = 200 p2 p2 = 500 p1 = 250 p1

p1 = ap 1 bp1 p2 e p21

p2 = cp2 dp1 p2 f p22

x = a x a x y k x2

y = c y c x y L y2


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x = dc

p1 k = ec

d

y = ba

p2 L = f a

b

f (x, y ) = ax axy Kx2 = 0

x = (a ay )

K

g (x, y ) = cy cxy Ly2 = 0

x = (c Ly )c


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a,b,K L

L < c K < a

K L

y = 0 x = a

K

y = cL x = 0

x y


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x y a c K L

L > c K > a


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a = c = 1 K = L = 2

L < c K > a


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x = 0 y = cL y x

a = c = 1 K = 2 L = 12

x x = aK y = 0

dp1dt

= a p1 b p1 p2

dp2dt

= c p2 + d p1 p2


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p1 a b p1 p1 p2

p2 c d

p1 p2 x dc p1 y =

ba p2

x = a ( x xy )

y = c ( y xy )

dxdy

= ac

( x xy )( y xy )

f (x, y ) = a ( x xy ) = 0

y = 1

g (x, y ) = c ( y xy ) = 0

x = 1 .


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p1

p1 = 100 p2 = 200

p1 = 0 .05 p1 0.0005 p1 p2

p2 = 0.1 p2 + 0 .0005 p1 p2


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T

p1 = 200 p2 = 50 p1 = 50 p2 = 200

T

p

1 = a p

1 b p

1 p

2 e p2

1

p2 = c p2 + d p1 p2 f p22

e f

x = ax axy Kx2

y = cx cxy Ly2

aK 1

x = aK y = 0 a = c = L = 1 K = 0 .5

K = L = 0

x = 1 y = 2 x = 0 .3 y = 2 x = 2 .5 y = 0 .3

y L = 0 y

L = 4

ak 1

x = aK y = 0 ac = 1 K = 2

x = 1 .5 y = 1


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f ( p1 , p2) g ( p1 , p2)

y

x y xy

x = axy

y = cy + cxy


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a = c = 1

xy

x = ay

x = ax + by + f

y = cy + dx + g


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x y a cb d

f g

ac>bd

bd>ac


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x p1 p2 y p2 p1

x = f (x, y )

y = g (x, y ) .

f g f (x, y ) = 0

x y f (x, y ) = 0 x p1 y p2 p1

y x y x y x y

x g (x, y ) = 0 p2 p1


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p2 p1 p1 A p1 p2 B

C


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dpdt

= rp

r

p1 p2 p3 b1 p1

b2 p2 d1 p1

p1 (n + 1) = b1 p1 (n ) + b2 p2 (n ) +... b10 p10 (n ) p(n + 1) = (1 d1 ) p1 (n ) p2 (n + 1) = (1 d2 ) p2 (n ) p10 (n +

n


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[ p1 (n + 1) p2 (n + 1) p3 (n + 1) p4 (n + 1 ) 3 dotp 10 (n + 1) ] =[ b1 (1 d1) 003dot 0b20 (1 d2) 0 0...... ...b 90 above (1 d9) b100 3dot 0 ] [ p

P [U+0332]

(n + 1) = A P [U+0332]

(n )

> n Punder (0)

P [U+0332]

(n ) = An P [U+0332]

(0)

n A

T T

S i pi p2 T d2 p2

T S 2 ( 1 d2 ) p2 1

T S 2 ( 1 d2) p2


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P 1 = ( n + 1) = b1 p1 (n ) + b2 p2 (n ) + ...

P 2 (n + 1) = T S 1

(1 d1) P 1 (n ) + (1 d2) P 2 (n )

P M (n + 1) = T

S M 1(1 dM 1) P M 1 (n ) +

1 T S M

(1 dM ) P M (n )

bi pi T T

N

C

A

P

R

N = f 1 ( N, C, A, P, R )C = f 2 ( N, C, A, P, R )A = f 3 ( N, C, A, P, R )P = f 4 ( N, C, A, P, R )R = f 5 ( N, C, A, P, R )


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f 1 f 2

f i


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burrus, c. s. & h. resnikoff 2009 - dynamics of social systems

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