DengRenormalization.ppt

Bo Deng
Department of Mathematics
University of Nebraska – Lincoln
Reprint/Preprint Download at: http://www.math.unl.edu/~bdeng1
Outline of Talk
 Bursting Spike Phenomenon
 Bifurcation of Bursting Spikes
 Definition of Renormalization
 Dynamics of Renormalization
Phenomenon of Bursting Spikes
Rinzel & Wang (1997)
Neurosciences
Phenomenon of Bursting Spikes
Food Chains
Dimensionless Model:
y
 x  x(1  x 
) : xf ( x, y )
1  x
 x
z 
y  y 
 (1   1 y ) 
 : yg ( x, y, z )


x


y
 1
2

 y

z   z 
 ( 2   2 z )  : zh( y, z )
 2  y

Bifurcation of Spikes
dI
L L  VE  RI L  V
dt
C dV  I L  I
dt
 dI  g (V , I )
dt
2 time scale system:
0 <  << 1,
with ideal situation at
 = 0.
1-d Return Map at  = 0
V
g (V, I) = 0
IL
1-d map
I
Bifurcation of Spikes
dI
L L  VE  RI L  V
dt
C dV  I L  I
dt
 dI  g (V , I )
dt
c0
V
IL
I
Bifurcation of Spikes
dI
L L  VE  RI L  V
dt
C dV  I L  I
dt
 dI  g (V , I )
dt
Homoclinic Orbit at  = 0
c0
V
1
f
0
c0
1
IL
I
Phenomenon of Bursting Spikes
Food Chains
Bifurcation of Spikes
dI
L L  VE  RI L  V
dt
C dV  I L  I
dt
 dI  g (V , I )
dt
Def of Isospike
c0
V
1
f
0
c0
1
IL
I
Def: System is isospiking of n spikes if for every c0 < x0 <=1, there
are exactly n points x1, x2, … xn in [0, c0) and xn+1 returns to [c0,1].
Bifurcation of Spikes
dI
L L  VE  RI L  V
dt
C dV  I L  I
dt
 dI  g (V , I )
dt
c0
V
c0
IL
I
Def: System is isospiking of n spikes if for every c0 < x0 <=1, there
are exactly n points x1, x2, … xn in [0, c0) and xn+1 returns to [c0,1].
Bifurcation of Spikes
dI
L L  VE  RI L  V
dt
C dV  I L  I
dt
 dI  g (V , I )
dt
Isospike of 3 spikes
c0
c0
V
IL
I
Def: System is isospiking of n spikes if for every c0 < x0 <=1, there
are exactly n points x1, x2, … xn in [0, c0) and xn+1 returns to [c0,1].
# of Spikes
Bifurcation of Spikes
n
Isospike Distribution
1/x
3
2
1
0 … 1/n … 1/3 1/2
1
Bifurcation of Spikes
Silent Phase
Spike Reset
6th
5th
4th
2nd
1st
m  C/L
Numeric
3rd
Renormalization
Feigenbaum
Feigenbaum’s Renormalization Theory (1978)
• Period-doubling bifurcation for
fl(x)=lx(1-x)
• Let ln = the 2n-period-doubling
bifurcation
_
parameters, ln  l0
• A renormalization can be defined at each ln ,
referred to as Feigenbaum’s renormalization.
• It has a hyperbolic fixed point with eigenvalue
(l(n+1) - ln )/(l(n+2) - l(n+1))  4.6692016…
which is a universal constant, called the
Feigenbaum number.
Renormalization
f
Def of R
Renormalization
f
f2
Renormalization
1  f c
0
c0
f
f
2
1  f 2 c
0
c0
Renormalization
1  f c
0
c0
f
f
2
R
1  f 2 c
0
c0
Renormalization
1  f c
0
c0
f
f
2
R
1  f 2 c
0
c0
1
R : Y  Y , with || f ||Y | f (0) |   | f ( x) | dx
R
0
C-1
V
c0
1
R( f )
IL
0
C-1/C0
1
I
2 families m
Renormalization
1
1
fm
m
m
0
1
c0
1
f0
0
e-K/m
0
1
ym
m
0
c0
1
y0=id
m
0
1m
1
0
 m  x, 0  x < 1  m
y m x   
1 m  x  1
0,
1
Renormalization
Y
 R[y0]=y0
1
W={
}
0
universal
constant 1
1
Renormalization
 R[y0]=y0
 R[ym]=ym / 1m
1
R
m / 1m
ym
m
0
1
1m 1
ym /1m
0
1
Renormalization
 R[y0]=y0
 R[ym]=ym/1m
 R[y1/n1 ]= y1/n
1
R
m / 1m
ym
m
0
1
1m 1
ym/1m
0
1
Renormalization
 R[y0]=y0
 R[ym]=ym/1m
 R[y1/n1 ]= y1/n
 1 is an eigenvalue
of DR[y0]
|| R[y m ]  R[y 0 ]  1 (y m y 0 ) ||  || y m /(1 m ) y m ||
 m
 4  3m
 
 m 
 m 2  || y m y 0 || 2
2
 1- m

1
R
m / 1m
ym
m
0
1
1m 1
ym/1m
0
1
Renormalization
 R[y0]=y0
 R[ym]=ym/1m
 R[y1/n1 ]= y1/n
 1 is an eigenvalue
of DR[y0]
 l Lemma
1
R
0
1
m / 1m
ym
m
1m 1
n nq2pnn1 q p
lim
lim
1
n
 
n
q
nnq1   n
ym/1m
0
1
Theorem 1:
 R[y0]=y0
 R[ym]=ym/1m
 R[y1/n1 ]= y1/n
 1 is an eigenvalue
of DR[y0]
 nq p   nq p
lim

 l- Lemma & n
 nq   n
q
Renormalization
Renormalization
superchaos
Eigenvalue:
l1
U={ym}
Invariant
y0 = id
Fixed Point
W
Invariant
R :Y Y
Renormalization
Theorem 2:
 R has fixed points whose stable
spectrum contains 0 < r < 1 in W
 For any l >1 there exists a fixed point
repelling at rate l and normal to W
l>1
l>1
l1
ym
1
Fixed Points= {
id
0
r<1
W
R :Y Y
}
1
Theorem 2:
 R has fixed points whose stable
spectrum contains 0 < r < 1 in W
 For any l >1 there exists a fixed point
repelling at rate l and normal to W
Renormalization
 Let W = X0 U X1 with
Every point in X1 goes to a fixed
point
X0 is a chaotic set: (1) dense set of
periodic orbits; (2) every point is
connected to any other point; (3)
sensitive dependence on initial
conditions; (4) dense orbits.
l>1
l>1
l1
ym
1
id
r<1
X1
X0 = {
}
0
X0
1
1
W
R :Y Y
X1 = {
}
0
1
1
X0 = {
}
0
1
Theorem 2:
slope =stable
l
 R has fixed points whose
spectrum contains 0 < r < 1 in W
y0 any l >1 there exists a fixed point
 For
 (x0)
repelling at rate l and normal to W
Renormalization
 Let W = X0 U X1 with
Every point in X1 goes to a fixed
point
X0 is a chaotic set: (1) dense set of
periodic orbits; (2) every point is
connected to any other point; (3)
sensitive dependence on initial
conditions; (4) dense orbits.
y1
l>1
y2
…
l1
ym
Every n-dimensional dynamical
system
f : D  R n  D, 1  n  
For
x2conjugate
= f (x1), …}
in [0,1],into
id each
r < 1orbit { x0 X,1x1= f (x
0),be
can
embedded
let y0 = S(x0), y1 = R-1S(x1),Xy02in= infinitely
R-2S(x2),many
… ways.
X0
W
R :Y Y
 f : D  D,   : D  Y , s.t
  f ( x)  R   ( x)
Theorem 2:
 R has fixed points whose stable
spectrum contains 0 < r < 1 in W
 For any l >1 there exists a fixed point
repelling at rate l and normal to W
l>1
Renormalization
 Let W = X0 U X1 with
Every point in X1 goes to a fixed
point
X0 is a chaotic set: (1) dense set of
periodic orbits; (2) every point is
connected to any other point; (3)
sensitive dependence on initial
conditions; (4) dense orbits.
l1
ym
id
X1
r<1
X0
W
R :Y Y
Every n-dimensional dynamical
system
f : D  R n  D, 1  n  
can be conjugate embedded into
X0 in infinitely many ways.
The conjugacy preserves f ’s
Lyapunov number L if L < l
Theorem 2:
 R has fixed points whose stable
spectrum contains 0 < r < 1 in W
 For any l >1 there exists a fixed point
repelling at rate l and normal to W
l>1
 Let W = X0 U X1 with
Every point in X1 goes to a fixed
point
fm
l1
ym
id
Renormalization
X1
r<1
X0 is a chaotic set: (1) dense set of
periodic orbits; (2) every point is
connected to any other point; (3)
sensitive dependence on initial
conditions; (4) dense orbits.
Every n-dimensional dynamical
system
f : D  R n  D, 1  n  
can be conjugate embedded into
X0 in infinitely many ways.
The conjugacy preserves f ’s
Lyapunov number L if L < l
X0
W
Rmk: Neuronal families fm through
R :Y Y
f0  X 0  X1
Summary
 Zero is the origin of everything.
 One is a universal constant.
 Infinity is the number of copies every dynamical
system can be found inside a chaotic square.
 It can be taught to undergraduate students who
have learned separable spaces.

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