We will show that diverse complex patterns can emerge even in the universe governed by deterministic laws. See the details of this study on our paper: Iba, T. & Shimonishi, K. (2011), "The Origin of Diversity: Thinking with Chaotic Walk," in Unifying Themes in Complex Systems Volume VIII: Proceedings of the Eighth International Conference on Complex Systems, New England Complex Systems Institute Series on Complexity (Sayama, H., Minai, A. A., Braha, D. and Bar-Yam, Y. eds., NECSI Knowledge Press, 2011), pp.447-461.

1. The Origin of Diversity
Thinking with Chaotic Walk
Takashi Iba
Ph.D. in Media and Governance
Associate Professor, Faculty of Policy Management
Keio University, Japan
iba@sfc.keio.ac.jp
Kazeto Shimonishi
Interdisciplinary Information Studies
The University of Tokyo, Japan

2. Diverse complex patterns can emerge
even in the universe governed by deterministic laws.

3. xn+1 = a xn ( 1 - xn )

4. Logistic Map
xn+1 = a xn ( 1 - xn )
a simple population growth model (non-overlapping generations)
xn ... population (capacity) 0 < xn < 1 (variable)
a ... a rate of growth 0 < µ < 4 (constant)
x0 = an initial value
n=0 x1 = a x0 ( 1 - x0 )
n=1 x2 = a x1 ( 1 - x1 )
May, R. M. Biological populations with nonoverlapping generations:
stable points, stable cycles, and chaos. Science 186, 645–647 (1974).
n=2 x3 = a x2 ( 1 - x2 ) May, R. M. Simple mathematical models with very complicated
dynamics. Nature 261, 459–467 (1976).

5. Chaotic Walk
A chaotic walker who walk and turns around at the
angle calculated by the logistic map function.

6. Chaotic Walk
Plotting the dots on the two-dimensional space,
as follows.
θn = 2πxn xn+1 = a xn ( 1 - xn )
s
fo otprint
s
of chao 0. Assigning a starting point and an initial direction.
1. Calculating next value of x and then θ.
2. Turning around at θ angle.
3. Moving ahead a distance L.
4. Drawing a dot (small circle).
5. Repeat from step 1.
The trail left by such a walker is investigated.
K. Shimonishi & T. Iba, "Visualizing Footprints of Chaos", 3rd International Nonlinear
Sciences Conference (INSC2008), 2008
K. Shimonishi, J. Hirose & T. Iba, "The Footprints of Chaos: A Novel Method and
Demonstration for Generating Various Patterns from Chaos", SIGGRAPH2008, 2008

7. xn+1 = a xn ( 1 - xn )
The behavior depends on the value of control parameter a.
The system converges to the fixed point.
1 1 1
0.8 0.8 0.8
0.6 0.6 0.6
x x x
0.4 0.4 0.4
0.2 0.2 0.2
0 0 0
0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
n n n
0 < a < 1 1 < a < 2 2 < a < 3
a
0 1 2 3 3.56... 4
3 < a < 1+ 6 1+ 6 < a < 4
1 1
0.8 0.8
0.6 0.6
x x
0.4 0.4
0.2 0.2
0 0
0 20 40 60 80 100 0 20 40 60 80 100
n n
The system oscillates. The system exhibits chaos.

8. xn+1 = a xn ( 1 - xn ) θn = 2πxn
Chaotic Walk
0 < a < 1 1
Case 1
0.8
1
0.8 0 < a < 1 The value of θ converges to
0.6
It converges to x
0.6
θ* = 0.
x
0.4
0.2
zero state. 0.4
The trail represents a line
0
0 20 40 60 80 100
0.2
that goes straight ahead.
n
0
0 20 40 60 80 100
n

9. xn+1 = a xn ( 1 - xn ) θn = 2πxn
Chaotic Walk
1 < a < 2
Case 2 1
1 < a < 2
1 0.8
The value of θ converges to
the fixed value.
0.8
0.6
x
It converges to
0.6
x
a nonzero state.
0.4
0.4
The trail is on a circle where
0.2
0
0 20 40 60 80
0.2
100
the turn-angle is fixed.
n
0
1 0 20 40 60 80 100
Case 3
2 < a < 3 n
0.8
2 < a < 3
1
0.8 0.6
The value of θ converges to
It oscillates at
0.6 x the fixed value.
the beginning,
x
0.4 0.4
The trail is on a circle where
but converges to
0.2
0.2
the turn-angle is fixed.
a nonzero state.
0
0 20 40 60 80 100
n
0
0 20 40 60 80 100
n

10. xn+1 = a xn ( 1 - xn ) θn = 2πxn
Chaotic Walk
3 < a < 1+ 6
1
Case 4
1
0.8 The value of θ oscillates on
0.8
3 < a < 1+ 6 0.6 successive iterations.
0.6
x
It oscillates. x
The trail represents multiple
0.4
0.4
0.2
0 0.2 circles.
0 20 40 60 80 100
n
0
0 20 40 60 80 100
n

11. xn+1 = a xn ( 1 - xn ) θn = 2πxn
1+ 6 < a < 4 Chaotic Walk
Case 5 1
0.8
1+ 6 < a < 4
1
0.8
θ takes various values.
0.6
x
It shows chaotic
0.6 x
behaviors.
0.4 0.4
The trail represents complex
0.2
0
0.2 pattern.
0 20 40 60 80 100
n
0
0 20 40 60 80 100
n

12. xn+1 = a xn ( 1 - xn )
The behavior depends on the value of control parameter a.
a
0 1 2
a
2 3 3.56... 4

13. Not so interesting...
How these interesting
patterns can be generated?
w ?
Ho

14. chaos + finitude

15. finitude
a finite state or quality.
- Random House Dictionary,
the quality or condition of being finite.
- The American Heritage Dictionary of the English Language
From finite + -titude, from Latin fīnītus + -dō
(having been limited or bounded) (signifying a noun of state)

16. finitude
We introduce the parameter for finitude, which controls the
number of possible states in the target system. d
d represents that the value of x is rounded off to d decimal places
at every time step.
xn xn+1
0.1 0.36 0.4
d =1 0.2 0.64 0.6
0.3 0.84 0.8
f round-off
•In principle, the infinite number of possible states is required for
representing chaos in strict sense.
•A system consisting of the finite number of possible states
eventually exhibits periodic cycle.
•To tune this parameter means to vary the degree of chaotic behavior.

17. •A system consisting of the finite number of possible states
eventually exhibits periodic cycle.
•To tune this parameter means to vary the degree of chaotic behavior.
d =1 d =8 d =16
regular irregular
all patterns are generated with a =3.76 (in the chaotic regime)

18. d =1
d =2
d =3
d =4
d =5
d =6
d =7
The patterns generated by chaotic walks with the logistic map for
the finitude parameter d varying from 1 to 7 in the chaotic regime.

19. The trails of 10 periodic cycles in the case d = 1.

20. The trails of 10 periodic cycles in the case d = 2.

21. The trails of 10 periodic cycles in the case d = 3.

22. The trails of 10 periodic cycles in the case d = 4.

23. The trails of 10 periodic cycles in the case d = 5.

24. The trails of 10 periodic cycles in the case d = 6.

25. The trails of 10 periodic cycles in the case d = 7.

26. The trails of 10 periodic cycles in the case d = 8.

27. Average lengths of periodic cycle of attractors against each
values of a and d
The average length of attractor
increases exponentially as the
finitude parameter d increases.

28. Diversity and Robustness of Patterns
diversification of generated
patterns by varying the
finitude parameter d. The box represents
the region that has
completely same
types of attractors.
As the number of possible
states increases,
- the diversity increases
- the robustness decreases
The finitude parameter
controls the degree of
diversity and robustness
of order!

29. Implication 1
the origin of diversity

30. (Theoretical) Hypothesis about the origin of diversity
how to generate and climb up the ladder of diversity in a
deterministic way without random mutation and natural selection.
A system starts with small number of possible states, and then
increases the possible states, consequently increases their diversity.
Diversification can occur just by
changing the number of possible states.

31. This is just a hypothesis, however it seems to be plausible.
•In the primitive stage of evolution, it must be quite difficult for
the system to maintain a lot of possible states.
•It is quite difficult to memorize detailed information.
•Therefore, starting with small number of possible states is reasonable.
•Also, it is probable that the system does not have sensitivity
against the parameter.
•It must be difficult to keep the parameter value for calculation with a
high degree of precision.
•In the further stage of evolution, the system would be able to
afford to have larger number of possible states.
•As the number of possible states increases, the system decreases the
robustness to the parameter value.
Thus, the diversification of primitive forms would be explained in
a deterministic way only with the combination of deterministic
chaos and finitude.

32. Implication 2
chaotic walk

33. The parameter for tuning the finitude
would be
another hidden control parameter of complex systems
d =1 d =8 d =16
irregular
regular d
The parameter for finitude controls the number of possible states,
and, as a result, it controls the system’s behavior.
More practically, it may provide a new way of understanding a dramatic
change of behaviors in phenomena that we have considered as random walk.

34. Diverse complex patterns can emerge
even in the universe governed by deterministic laws.

35. Diverse complex patterns can emerge
even in the universe governed by deterministic laws.
with the combination of
Chaos + Finitude

36. Some More Information ...

37. Get and Try!
ChaoticWalker
A New Vehicle for Exploring Patterns Hidden in Chaos
http://www.chaoticwalk.org/

38. Get and Feel!
The Chaos Book
New Explorations for Order Hidden in Chaos

39. Come and Talk!
Today’s Poster Session
Chaos + Finitude
[Poster 70]
"Hidden Order in Chaos: The
Network-Analysis Approach
To Dynamical Systems"
(Takashi Iba)

40. The Origin of Diversity
Thinking with Chaotic Walk
http://www.chaoticwalk.org/
Takashi Iba
Ph.D. in Media and Governance
Associate Professor, Faculty of Policy Management
Keio University, Japan
iba@sfc.keio.ac.jp
Kazeto Shimonishi
Interdisciplinary Information Studies
The University of Tokyo, Japan