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Bo DengDepartment of Mathematics
University of Nebraska – Lincoln
Outline: Small Chaos – Logistic Map Poincaré Return Map – Spike Renormalization All Dynamical Systems Considered Big Chaos
Logistic Map:
Orbit with initial point
Fixed Point:
Periodic Point of Period n :
A periodic orbit is globally stable if
for all non-periodic initial points x0
Logistic Map
]1,0[]1,0[:)1()(1 nnnrn xrxxfx
...},...,,,{ 2100 nx xxxx
0x
)...))((...()( 000 xfffxfxx nn
)( 00 xfx
p as nx pn
Period Doubling Bifurcation
Robert May 1976
Cobweb Diagram
x0 x1 x2 …
x1
x2
Period Doubling Bifurcation
n cycle of period 2n rn
1 2 32 4 3.4494903 8 3.5440904 16 3.5644075 32 3.5687506 64 3.569697 128 3.569898 256 3.5699349 512 3.569943
10 1024 3.569945111 2048 3.569945557∞ Onset of Chaos r*=3.569945672
Period-Doubling Cascade, and Universality
Feigenbaum’s Universal Number (1978)
i.e. at a geometric rate
Feigenbaum’s Universal Number (1978)
i.e. at a geometric rate
nrr
rr
nn
nn as 4.6692016 :1
1
*rrn n/1
4.6692016…
Renormalization
Feigenbaum’s Renormalization,
--- Zoom in to the center square of the graph of
--- Rotate it 180o if n = odd
--- Translate and scale the square to [0,1]x[0,1]
--- where U is the set of unimodal maps
n
f 2
)( fRn
UUR :
Renormalization
Feigenbaum’s Renormalization at Feigenbaum’s Renormalization at
ngfR rn as *)( *
*rr
Geometric View of Renormalization
The Feigenbaum Number α = 4.669… is the only expanding eigenvalue of the linearization of R at the fixed point g*
*g
)( rfR
59.3f
*rf
rf
. . .
)( rn fRE u
E s
U
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].
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 ε-neighborhood 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 (1887) reduces the trajectory of a differential equation to an orbit of the map.
Poincaré Return 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 Time-1 Map: φτ(x0) = x(τ, x0), for which x(t, x0) is the solution with initial condition x(0, x0) = x0
I pump
Poincaré Return Map
Poincaré Return Map
Ipump
0 c0 1
f1
Ipump
V c
INa
c0
Poincaré Return Map
1
c0
V c
INa
C -1
R( f )
0 1 C -1/C0
Ipump
Poincaré Return Map
Poincaré Map Renormalization
f
00
1 cfc
02
0
1 cfc
R
Renormalized 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.
Renormalized 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 …
f 2
11
10
,0
,
x
xxx
→0
0 c0 1
f1
0 1
1
0 1
=id
1
0 c0 1
f1
e-k/
→0
Spike Return Maps
1st Spike2nd
3rd4th
5th6th
Discontinuity for Spike Reset
Is /C
Sile
nt P
hase
Bifurcation of Spikes -- Natural Number Progression
μ∞=0 ← μn … μ10 μ9 μ8 μ7 μ6 μ5
Scaling Laws : μn ~ 1/n and (μn - μn-1)/(μn+1 - μn) → 1
Scaling Laws : μn ~ 1/n and (μn - μn-1)/(μn+1 - μn) → 1
μ
Ipump
V c
INa
Homoclinic Orbit at μ = 0
At the limiting bifurcation point μ = 0, an equilibrium point of the differential equations invades a family of limit cycles.
0 c0 1
f1
c0
Poincaré Return Map
Bifurcation of Spikes
1 / IS ~ n ↔ IS ~ 1 / n
Y
universalconstant 1
0 1
1
W = { } ,
the set of elements of Y , each has at least one fixed point in [0,1].
Dynamics of Spike Map Renormalization -- Universal Number 1
0 1
R1
0 1
1
R[0]=0
R[]=
R[n]= n
μ1
μ2
μnf μn ]
μnf μn ]
f μn
R[0]=0
R[]=
R[n]= n
1 is an eigenvalue of DR[0]
Universal Number 1
dxxff
L
YYY
Y
norm the withequipped
being with, :
1
0
1
|)(|||||
R
μ1
μ2
μnf μn ]
μnf μn ]
f μn
20
)1/(00
||||2
34
||||||)(1][][||
1-
2
RR
20
)1/(00
||||2
34
||||||)(1][][||
1-
2
RR
0 1
R1
0 1
1
0 1
R1
0 1
1
R[0]=0
R[]=
R[n]= n
1 is an eigenvalue of DR[0]Theorem of One (BD, 2011):
The first natural number 1 is a new universal number .
Universal Number 1
11
12
nn
nn
n
lim
μ1
μ2
μnf μn ]
μnf μn ]
f μn
W = X0 U X1 Invariant
U=Invariant
Eigenvalue:
YY :R
0 1
X0 = { : the right
most fixed point is 0. }
1
0 1
X1 = { } = W \ X0
1
Renormalization Summary
X1
= id Fixed Point
X0
All Dynamical Systems Considered
Cartesian Coordinate (1637), Lorenz Equations (1964)
and Smale’s Horseshoe Map (1965)
ZXYdt
dZ
YZXdt
dY
XYdt
dX
)(
)(
MatLab Simulation 2 …
Time-1 Map Orbit
All Dynamical Systems Considered
W
X0
X1
YY :R
Theorem of Big: Every dynamical system of any finite dimension can be embedded into the spike renormalization R : X0 → X0 infinitely many times. That is, for any n and every map f : R n → R n there are infinitely many injective maps θ : R n → X0 so that the diagram commutes.
Theorem of Big: Every dynamical system of any finite dimension can be embedded into the spike renormalization R : X0 → X0 infinitely many times. That is, for any n and every map f : R n → R n there are infinitely many injective maps θ : R n → X0 so that the diagram commutes.
id
00 XX
Rf
R nn
R
0 1
X0 = { }
1
W
X0
YY :R
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.
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.
Big Chaos
id X1
0 1
X0 = { }
1
W
X0
YY :R
Big Chaos
id X1
…
n R
4/1||)()(|| 0 Ynn gf RR dxxgxfgf Y
1
0|)()(|||||
Rn( f ) Rn(g)f g
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.
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.
W
X0
YY :R
Big Chaos
id X1
Use concatenation on a countable dense set (as L1 is separable) to construct a dense orbit
Use concatenation on a countable dense set (as L1 is separable) to construct a dense orbit
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.
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.
W
X0
YY :R
f
Theorem of Almost Universality: Every number is an eigenvalue of the spike renormalization.
Theorem of Almost Universality: Every number is an eigenvalue of the spike renormalization.
Universal Number
Slope = λ > 1
20
00
||||
||)()()(||
Y
Y
gg
gggg
RR
gμ
g0μ
g0
gμ
X1id
0 1
1
c0 0 1
=id
1
Rf0
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.
Summary
Phenomenon of Bursting Spikes
Rinzel & Wang (1997)Excitable Membranes
Food Chains Phenomenon of Bursting Spikes
1
1 11 2
2 22
(1 ) : ( , )
( ) : ( , , )
( ) : ( , )
yx x x xf x y
x
x zy y y yg x y z
x y
yz z z zh y z
y
Dimensionless Model:
Big Chaos
W
X0
X1
id
Every n-dimensional dynamical system can be conjugate embedded into X0 in infinitely many ways.
1 ,: nDRDf n
)()(
s.t ,: ,:
xxf
YDDDf
R
…
slope =
For each orbit { x0 , x1= f (x0), x2= f (x1), …} in [0,1], let y0 = S(x0), y1 = R-1S(x1), y2 = R-2S(x2), …
y0
y1
y2
(x0)
YY :R
Let W = X0 U X1 with
0 1
X0 = { },
1
0 1
X1 = { }
1
All Dynamical Systems Considered
Bifurcation of Spikes
c0
Def: System is isospiking of n spikes if for every c0 < x0 <=1, thereare exactly n points x1, x2, … xn in [0, c0) and xn+1 returns to [c0,1].
c0
IIpump
V c
INa
Bifurcation of Spikes
c0
Isospike of 3 spikes
Def: System is isospiking of n spikes if for every c0 < x0 <=1, thereare exactly n points x1, x2, … xn in [0, c0) and xn+1 returns to [c0,1].
c0
I
V c
INaIpump
0 1
R1
0 1
1
R[0]=0
R[]=
R[n]= n
1 is an eigenvalue of DR[0]Theorem of One (BD, 2011):
The first natural number 1 is a new universal number .
Universal Number 1
11
12
nn
nn
n
lim
μ1
μ2
μnf μn ]
μnf μn ]
f μn
q
p
nqn
qnpqn
n
lim