DengRenormalization.ppt
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Transcript 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
1m
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 / 1m
1
R
m / 1m
ym
m
0
1
1m 1
ym /1m
0
1
Renormalization
R[y0]=y0
R[ym]=ym/1m
R[y1/n1 ]= y1/n
1
R
m / 1m
ym
m
0
1
1m 1
ym/1m
0
1
Renormalization
R[y0]=y0
R[ym]=ym/1m
R[y1/n1 ]= 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 / 1m
ym
m
0
1
1m 1
ym/1m
0
1
Renormalization
R[y0]=y0
R[ym]=ym/1m
R[y1/n1 ]= y1/n
1 is an eigenvalue
of DR[y0]
l Lemma
1
R
0
1
m / 1m
ym
m
1m 1
n nq2pnn1 q p
lim
lim
1
n
n
q
nnq1 n
ym/1m
0
1
Theorem 1:
R[y0]=y0
R[ym]=ym/1m
R[y1/n1 ]= y1/n
1 is an eigenvalue
of DR[y0]
nq p nq p
lim
l- Lemma & n
nq n
q
Renormalization
Renormalization
superchaos
Eigenvalue:
l1
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
l1
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
l1
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
…
l1
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.
l1
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
l1
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.