Bo Deng
Outline:
Department of Mathematics
 Small Chaos – Logistic Map
University of Nebraska – Lincoln
 Poincaré Return Map – Spike Renormalization
 All Dynamical Systems Considered
 Big Chaos
Logistic Map
Logistic Map:
xn1  f r ( xn )  rxn (1  xn ) : [0,1]  [0,1]
Orbit with initial point x0
 x  {x0 , x1 , x2 , ... xn , ...}
0
Fixed Point: x0  f ( x0 )
Periodic Point of Period n :
x0  xn  f n ( x0 )  f ( f (... f ( x0 )...))
A periodic orbit  p is globally stable if
xn   p as n  
for all non-periodic initial points x0
Cobweb Diagram
x2
x1
x0 x1 x2 …
Robert May 1976
Period Doubling Bifurcation
Period Doubling Bifurcation
Period-Doubling Cascade, and Universality
cycle of period
2n
1
2
rn
2
4
3.449490
3
8
3.544090
4
16
3.564407
5
32
3.568750
6
64
3.56969
7
128
3.56989
8
256
3.569934
9
512
3.569943
10
1024
3.5699451
11
2048
3.569945557
n
3
∞ Onset of Chaos r*=3.569945672
Feigenbaum’s Universal Number (1978)
rn  rn 1
 4.6692016
4.6692016… :  as n  
rn 1  rn
n
r

r
*
1
/

i.e. n
at a geometric rate
Renormalization
Feigenbaum’s Renormalization, R n ( f )
--- Zoom in to the center square of the graph of f
2n
--- Rotate it 180o if n = odd
--- Translate and scale the square to [0,1]x[0,1]
--- R : U  U where U is the set of unimodal maps
Renormalization
Feigenbaum’s Renormalization at
r  r*
R n ( f r* )  g * as n  
Geometric View of Renormalization
E
u
Rn ( fr )
R( f r )
...
E
fr
f 3.59
s
g*
f r*
U
The Feigenbaum Number α = 4.669…
is the only expanding eigenvalue of
the linearization of R at the fixed
point g*
Chaos at r*
 At r = r* = 3.5699… almost all orbits
converge to a chaotic set A which is
a Cantor set of zero measure.
 At r = 4, f is chaotic in A = [0,1].
Def.: A map f : A → A is chaotic if
 the set of periodic points in A is
dense in A
 it is transitive, i.e. having a dense
orbit in A
 it has the property of sensitive
dependence on initial points, i.e.
there is a δ0 > 0 so that for every εneighborhood of any x there is a y ,
both in A with |y-x| < ε, and n so that
| f n(y) - f n(x) | > δ0
Period three implies chaos,
T.Y. Li & J.A. Yorke, 1975
Poincaré Return Map
Poincaré Return Map (1887) reduces the trajectory
of a differential equation to an orbit of the map.
Poincaré Time-1 Map: φτ(x0) = x(τ, x0), for which x(t, x0)
is the solution with initial condition x(0, x0) = x0
Poincaré Return Map
I pump
Poincaré Return Map
Ipump
Poincaré Return Map
Vc
c0
1
f
0
c0
1
Ipump
INa
Poincaré Return Map
Vc
C -1
c0
1
R( f )
0
C -1/C0
1
Ipump
INa
Poincaré Map Renormalization
1  f c
0
c0
f
R
f2
 Renormalized
1  f 2 c
0
c0
Poincaré maps are Poincaré maps, and
every Poincaré map is between two successive
renormalizations of a Poincaré map.
 R : Y → Y,
where Y is the set of functions from [0,1]
to itself each has at most one discontinuity, is both
increasing and not below the diagonal to the left of
the discontinuity, but below it to the right.
MatLab Simulation 1 …
Spike Return Maps

1
1
 →0
f

0
1
c0
1
e-k/

f0
0
c0
1
0=id
 →0

0
1 
1
0
   x, 0  x  1  
  x   
1   x  1
0,
1
1
Bifurcation of Spikes -- Natural Number Progression
Silent Phase
  Is /C
Discontinuity for
Spike Reset
6th th
5
4th
rd
3
2nd
1st Spike
μ∞=0 ← μn
…
μ10 μ9 μ8
μ7
Scaling Laws : μn ~ 1/n and
(μn - μn-1)/(μn+1 - μn) → 1
μ6
μ
μ5
Poincaré Return Map
 At the limiting bifurcation point μ = 0, an equilibrium
point of the differential equations invades a family
of limit cycles.
Vc
Homoclinic Orbit at μ = 0
c0
1
f0
0
c0
1
Ipump
INa
Bifurcation of Spikes
1 / IS ~ n ↔ IS ~ 1 / n
Dynamics of Spike Map Renormalization -- Universal Number 1
Y
f μn ]
 R[0]=0
 R[]= / 1
 R[1/n1 ]= 1/n
μ1
f μn ]
μ2
f μn
1
W={
1

R

universal
0
1 1
constant 1
0
1
 / 1
1
the set of elements of
Y , each has at least
one fixed point in [0,1].
 /1
0
},
1
Universal Number 1
f μn]] R[ ]  1 (  ) ||  || 
|| R[
 R[
0]=0 /(1  )   ||

0

0
 4R[

3
]=/1
 
  
  2 
 ||    0 || 2
 R[21/n1 ]= 1/n
 1- 
μ1 
f μn ]
μ2
 1 is an eigenvalue
of DR[0]
f μn
R : Y  Y , with Y being
equipped with the L1 norm
1
1
R
 / 1



0
1 1
|| f ||Y   | f ( x) | dx
1
0
 /1
0
1
Universal Number 1
f μn ]
 R[0]=0
 R[]=/1
 R[1/n1 ]= 1/n
 1 is an eigenvalue
of DR[0]
μ1
f μn ]
μ2
Theorem of One (BD, 2011):
f μn
1
R
0
1
 / 1



1 1
The first natural
number 1 is a new
universal number .
 /1
0
1
n 2  n1
lim
1
n 
n 1   n
Renormalization Summary
1
X0 = {
: the right
0
1
most fixed point is 0. }
Eigenvalue: l  1
1
U={}
X1 = {
Invariant
} = W \ X0
0
0 = id
Fixed Point
X1
X0
W = X0 U X1 Invariant
R :Y Y
1
All Dynamical Systems Considered
Cartesian Coordinate (1637),
Lorenz Equations (1964)







dX
  (Y  X )
dt
dY
 X (  Z )  Y
dt
dZ
 XY  Z
dt
and Smale’s Horseshoe Map (1965)
MatLab Simulation 2 …
Time-1 Map Orbit
All Dynamical Systems Considered
Theorem of Big: Every dynamical system of
f
any finite dimension can be embedded into R n  R n
the spike renormalization R : X0 → X0


infinitely many times. That is, for any
X 0  X 0
n and every map f : R n → R n there are
R
n
infinitely many injective maps θ : R → X0
so that the diagram commutes.

W
id
1
X1
X0
X0 = {
}
0
R: Y  Y
1
Big Chaos
Theorem of Chaos: The spike renormalization R : X0 → X0
is chaotic, i.e. (1) has a dense set of periodic orbits; (2) has
the property of sensitive dependence on initial conditions;
and (3) has a dense orbit.
l1

W
id
1
X1
X0
X0 = {
}
0
R: Y  Y
1
Big Chaos
Theorem of Chaos: The spike renormalization R : X0 → X0
is chaotic, i.e. (1) has a dense set of periodic orbits; (2) has
the property of sensitive dependence on initial conditions;
and (3) has a dense orbit.
l1
n
R (f)
f

g
l>
1
W
…
|| f  gX
|| 1 
id
Y
X0
R: Y  Y
1
0
| f ( x)  g ( x) | dx  
R
n
R (g)
n
l>
1
|| R n ( f )  R n ( g ) ||Y >  0  1 / 4
Big Chaos
Theorem of Chaos: The spike renormalization R : X0 → X0
is chaotic, i.e. (1) has a dense set of periodic orbits; (2) has
the property of sensitive dependence on initial conditions;
and (3) has a dense orbit.
l1

W
id
X1
X0
R: Y  Y
 Use concatenation on a countable dense set
(as L1 is separable) to construct a dense orbit
Universal Number
Theorem of Almost Universality: Every number is
an eigenvalue of the spike renormalization.
Slope = λ > 1
l>1
gμ
l1
f

gμ
μ
g0
|| R ( g  )  R ( g 0 )  l ( g   g 0 ) ||Y
W
id
l1
 || g   g 0 ||Y2
X1
1
g0
R: Y  Y
X0
1
R
f0
0
c0
1
0
0=id
1
Summary
 Zero is the origin of everything.
 One is a universal constant.
 Everything has infinitely many parallel copies.
 All are connected by a transitive orbit.
Summary
 Zero is the origin of everything.
 One is a universal constant.
 Everything has infinitely many parallel copies.
 All are connected by a transitive orbit.
 Small chaos is hard to prove, big chaos is easy.
 Hard infinity is small, easy infinity is big.
Phenomenon of Bursting Spikes
Rinzel & Wang (1997)
Excitable Membranes
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

Big Chaos
All Dynamical Systems Considered
slope = l
y0

 (x0)
Let W = X0 U X1 with
1
1
X0 = {
y1
l>1
y2
…
}, X1 = {
0
1
}
0
l1

Every n-dimensional dynamical
system
f : D  R n  D, 1  n  
id
can be conjugate embedded into
For each orbit { xX
0 1, x1= f (x0), x2= f (x1), …} in [0,1],
X0 yin =
infinitely
let y = S(x ), y = R-1S(x ),
R-2S(xmany
), … ways.
0
0
X0
1
1
W
R :Y Y
2
2
 f : D  D,   : D  Y , s.t
  f ( x)  R   ( x)
1
Bifurcation of Spikes
Vc
c0
c0
Ipump
I
INa
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
Vc
Isospike of 3 spikes
c0
c0
Ipump
I
INa
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].
Universal Number 1
f μn ]
μ1
f μn ]
μ2
f μn
1
R
0
1 1
Theorem of One (BD, 2011):
The first natural
number 1 is a new
universal number .
1
 / 1



 R[0]=0
 R[]=/1
 R[1/n1 ]= 1/n
 1 is an eigenvalue
of DR[0]
 /1
0
n nq2pnn1 q p
lim
lim
1
n
 
n


q
n

q
n

1
n
1