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