Reprint/Preprint Download at: http://www.math.unl.edu/~bdeng1

Bo DengDepartment of Mathematics

University of Nebraska – Lincoln

Outline of Talk Bursting Spike Phenomenon Bifurcation of Bursting Spikes Definition of Renormalization Dynamics of Renormalization

Phenomenon of Bursting Spikes

Rinzel & Wang (1997)Neurosciences

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:

Bifurcation of Spikes

),( IVgdtdI

IIdtdVC

VRIVdt

dIL

L

LEL

1-d Return Map at = 0

g (V, I) = 0

1-d map

2 time scale system: with ideal situation at = 0.

IIL

V

Bifurcation of Spikes

),( IVgdtdI

IIdtdVC

VRIVdt

dIL

L

LEL

c0

IIL

V

Bifurcation of Spikes

),( IVgdtdI

IIdtdVC

VRIVdt

dIL

L

LEL

c0

f1

10

c0

IIL

V

Homoclinic Orbit at = 0

Food Chains

Phenomenon of Bursting Spikes

Bifurcation of Spikes

),( IVgdtdI

IIdtdVC

VRIVdt

dIL

L

LEL

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].

0 c0 1

f1

Def of Isospike c0

IIL

V

Bifurcation of Spikes

),( IVgdtdI

IIdtdVC

VRIVdt

dIL

L

LEL

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

IIL

V

Bifurcation of Spikes

),( IVgdtdI

IIdtdVC

VRIVdt

dIL

L

LEL

c0

Isospike of 3 spikes c0

IIL

V

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].

Bifurcation of Spikes

Bifurcation of Spikes

1st

2nd3rd

4th

5th

6th

Spike Reset

C/L

Numeric

Silent Phase

Renormalization

Feigenbaum’s Renormalization Theory (1978)• Period-doubling bifurcation for f(x)=x(1-x)• Let n = the 2n-period-doubling bifurcation

parameters, n 0

_

• A renormalization can be defined at each n , referred to as Feigenbaum’s renormalization.• It has a hyperbolic fixed point with eigenvalue

((n+1) - n )/((n+2) - (n+1)) 4.6692016…

which is a universal constant, called the

Feigenbaum number.

Feigenbaum

Renormalization

f

Def of R

Renormalization

f

f 2

Renormalization

f

f 2

00

1 cfc

02

0

1 cfc

Renormalization

f

f 2

00

1 cfc

02

0

1 cfc

R

Renormalization

f

f 2

00

1 cfc

02

0

1 cfc

R

dxxfffYY Y |)(| |)0(||||| with ,:1

0RR

C-1/C0

R( f )1

10

C-1

IIL

Vc0

2 familiesRenormalization

0 c0 1

f

1

0 1

1

0 1

=id

1

11

10

,0

,

x

xxx

0 c0 1

f

1

e-K/

Renormalization

R[0]=0

Y

universalconstant 1

0 1

1

W = { }

Renormalization

R[0]=0

R[]=

0 1

R1

0 1

1

Renormalization

R[0]=0

R[]=

R[n]= n

0 1

R1

0 1

1

Renormalization

R[0]=0

R[]=

R[n]= n

1 is an eigenvalue of DR[0]

20

2

)1/(00

|||| 2

34

-1

|||| ||)(1][][||

RR

20

2

)1/(00

|||| 2

34

-1

|||| ||)(1][][||

RR

0 1

R1

0 1

1

0 1

R1

0 1

1

Renormalization

R[0]=0

R[]=

R[n]= n

1 is an eigenvalue of DR[0]Lemma

1lim1

12

nn

nn

n

q

p

nqn

qnpqn

n

lim

Renormalization

R[0]=0

R[]=

R[n]= n

1 is an eigenvalue of DR[0] - Lemma &

Theorem 1:

q

p

nqn

qnpqn

n

lim

= id

W

U=

Invariant

Invariant

Fixed Point

Eigenvalue:

Renormalizationsuperchaos

YY :R

W

id

Renormalization

YY :R

Theorem 2: R has fixed points whose stable spectrum contains 0 < < 1 in W For any >1 there exists a fixed point repelling at rate and normal to W

0 1

Fixed Points= { }

1

Let W = X0 U X1 with

W

X0

X1

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.

Renormalization

0 1

X0 = { }

1

YY :R

Theorem 2: R has fixed points whose stable spectrum contains 0 < < 1 in W For any >1 there exists a fixed point repelling at rate and normal to W

0 1

X1 = { }

1

id

0 1

X0 = { }

1

Theorem 2: R has fixed points whose stable spectrum contains 0 < < 1 in W For any >1 there exists a fixed point repelling at rate and normal to W

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

Renormalization

…

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

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.

W

X0

X1

id

Renormalization

The conjugacy preserves f ’s Lyapunov number L if L <

YY :R

Theorem 2: R has fixed points whose stable spectrum contains 0 < < 1 in W For any >1 there exists a fixed point repelling at rate and normal to W

Every n-dimensional dynamical system can be conjugate embedded into X0 in infinitely many ways.

1 ,: nDRDf n

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.

W

X0

X1

id

Renormalization

f

The conjugacy preserves f ’s Lyapunov number L if L <

Rmk: Neuronal families f through

100 XXf YY :R

Theorem 2: R has fixed points whose stable spectrum contains 0 < < 1 in W For any >1 there exists a fixed point repelling at rate and normal to W

Every n-dimensional dynamical system can be conjugate embedded into X0 in infinitely many ways.

1 ,: nDRDf n

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.

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.

Summary