1. From Small To One Big Chaos
Bo Deng
Department of Mathematics
University of Nebraska – Lincoln
Outline:
Small Chaos – Logistic Map
Poincaré Return Map – Spike Renormalization
All Dynamical Systems Considered
Big Chaos
Reprint/Preprint Download at:

2. Orbit with initial point
Logistic Map
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

3. Cobweb Diagram x2 x1 x0 x1 x2 … Period Doubling Bifurcation
Robert May 1976

4. Period Doubling Bifurcation

5. rn 4.6692016… Feigenbaum’s Universal Number (1978)
Period-Doubling Cascade, and Universality n
cycle of period 2n rn 1 2 3 4 8 16 5 32 6 64 7
128
256 9
512 10
1024 11
2048 ∞
Onset of Chaos
r*=
Feigenbaum’s Universal Number (1978)
i.e at a geometric rate …

6. 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

7. Feigenbaum’s Renormalization at

8. U E u E s The Feigenbaum Number α = 4.669…
Geometric View of Renormalization
E u
E s U
The Feigenbaum Number α = 4.669…
is the only expanding eigenvalue of
the linearization of R at the fixed
point g*

9. At r = 4, f is chaotic in A = [0,1].
Chaos at r*
At r = r* = … 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

10. Period three implies chaos,
T.Y. Li & J.A. Yorke, 1975

11. 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

12. Poincaré Return Map
I pump

13. Poincaré Return Map
Ipump

14. Poincaré Return Map
V c c0 1 f
INa
Ipump c

15. Poincaré Return Map
V c
C -1 c0 1
R( f )
INa
Ipump
C -1/C0

16. Renormalized Poincaré maps are Poincaré maps, and
Poincaré Map Renormalization f R
f 2
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 …

17. e-k/m fm f0 m →0 ym y0=id m 1-m Spike Return Maps 1 0 c0 1 0 c0 1 0 1 ym
y0=id m
1-m c fm
e-k/m

18. μ m = Is /C μ∞=0 ← μn … μ10 μ9 μ8 μ7 μ6 μ5 Scaling Laws : μn ~ 1/n and
Bifurcation of Spikes -- Natural Number Progression
m = Is /C
Silent Phase
Discontinuity for Spike Reset
6th
5th
4th
3rd
2nd
1st Spike
μ∞=0 ← μn … μ10 μ9 μ μ μ μ5 μ
Scaling Laws : μn ~ 1/n and
(μn - μn-1)/(μn+1 - μn) → 1

19. At the limiting bifurcation point μ = 0, an equilibrium
Poincaré Return Map
At the limiting bifurcation point μ = 0, an equilibrium
point of the differential equations invades a family
of limit cycles.
V c
Homoclinic Orbit at μ = 0 c0 1 f0 c
INa
Ipump

20. Bifurcation of Spikes
1 / IS ~ n ↔ IS ~ 1 / n

21. R[y0]=y0 R[ym]=ym / (1-m) R[y1/(n+1) ]= y1/n R m / (1-m)
Dynamics of Spike Map Renormalization -- Universal Number 1 Y μn
f μn ]
R[y0]=y0
R[ym]=ym / (1-m)
R[y1/(n+1) ]= y1/n μn
f μn ] μ1 μ2
f μn 1
W = { } ,
the set of elements of
Y , each has at least
one fixed point in [0,1]. ym R 1 m
1-m
ym /(1-m)
m / (1-m)
universal constant 1

22. 1 is an eigenvalue of DR[y0] R m / (1-m) R[y0]=y0 R[ym]=ym/(1-m)
Universal Number 1 μn
f μn ]
R[y0]=y0
R[ym]=ym/(1-m)
R[y1/(n+1) ]= y1/n
1 is an eigenvalue
of DR[y0] μn
f μn ] μ1 μ2
f μn ym R 1 m
1-m
m / (1-m)
ym /(1-m) m

23. R m / (1-m) R[y0]=y0 R[ym]=ym/(1-m) R[y1/(n+1) ]= y1/n
Universal Number 1 μn
f μn ]
R[y0]=y0
R[ym]=ym/(1-m)
R[y1/(n+1) ]= y1/n
1 is an eigenvalue
of DR[y0]
Theorem of One (BD, 2011):
The first natural
number 1 is a new
universal number . μn
f μn ] μ1 μ2
f μn ym R 1 m
1-m
m / (1-m)
ym /(1-m) m

24. most fixed point is 0. } Eigenvalue: l = 1 U={ym} X1 X0 Invariant
Renormalization Summary
X0 = { : the right
most fixed point is 0. } 1
Eigenvalue:
l = 1
X1 = { } = W \ X0 1
U={ym}
Invariant
y0 = id
Fixed Point
chaos X1 X0
W = X0 U X1 Invariant

25. Cartesian Coordinate (1637), Lorenz Equations (1964)
All Dynamical Systems Considered
Cartesian Coordinate (1637),
Lorenz Equations (1964)
and Smale’s Horseshoe Map (1965)
Time-1 Map Orbit
MatLab Simulation 2 …
because of Cartwright-Littlewood-Levinson (1940s)

26. All Systems ym W id Theorem of Big: Every dynamical system of
All Dynamical Systems Considered
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. ym W id
X0 = { } 1 X1
All Systems X0

27. Chaos ym W id Theorem of Chaos: The spike renormalization R : X0 → X0
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 ym W
chaos id
X0 = { } 1 X1
Chaos X0

28. Chaos ym W id Theorem of Chaos: The spike renormalization R : X0 → X0
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 …
Rn( f )
Rn(g) f g
l = 1 ym W
chaos id X1
Chaos X0

29. Chaos ym W id Theorem of Chaos: The spike renormalization R : X0 → X0
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 ym W
chaos id X1
Chaos X0
Use concatenation on a countable dense set
(as L1 is separable) to construct a dense orbit

30. Pan-Chaos fm gμ ym g0 gμ W id g0
Universal Number
Theorem of Almost Universality: Every number is
an eigenvalue of the spike renormalization.
Slope = λ > 1 gμ
l = 1
l > 1 fm ym g0 gμ μ W
Pan-Chaos
chaos id
l < 1 X1
l = 0 1 c0
y0=id f0 g0 X0

31. 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.

32. 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.

33. Phenomenon of Bursting Spikes
Rinzel & Wang (1997)
Excitable Membranes

34. Phenomenon of Bursting Spikes
Food Chains
Dimensionless Model:

35. Big Chaos

36. … X1 For each orbit { x0 , x1= f (x0), x2= f (x1), …} in [0,1],
All Dynamical Systems Considered
slope = l …
Let W = X0 U X1 with y0
q (x0)
X1 = { } 1
X0 = { }, 1 y1
l > 1 y2
l = 1 ym
Every n-dimensional dynamical
system
can be conjugate embedded into
X0 in infinitely many ways.
chaos id X1
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), … X0 W

37. Bifurcation of Spikes c0
V c I
INa
Ipump c0
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].

38. Bifurcation of Spikes c0
V c
Isospike of 3 spikes I
INa
Ipump c0
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].

39. R m / (1-m) R[y0]=y0 R[ym]=ym/(1-m) R[y1/(n+1) ]= y1/n
Universal Number 1 μn
f μn ]
R[y0]=y0
R[ym]=ym/(1-m)
R[y1/(n+1) ]= y1/n
1 is an eigenvalue
of DR[y0]
Theorem of One (BD, 2011):
The first natural
number 1 is a new
universal number . μn
f μn ] μ1 μ2
f μn ym R 1 m
1-m
m / (1-m)
ym /(1-m) m