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The Origin of DiversityThinking with Chaotic Walk
Ph.D. in Media and GovernanceAssociate Professor, Faculty of Policy Management
Keio University, [email protected]
Takashi Iba
Interdisciplinary Information StudiesThe University of Tokyo, Japan
Kazeto Shimonishi
Diverse complex patterns can emergeeven in the universe governed by deterministic laws.
xn+1 = a xn ( 1 - xn )
a simple population growth model (non-overlapping generations)
May, R. M. Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos. Science 186, 645–647 (1974).May, R. M. Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976).
xn+1 = a xn ( 1 - xn )
0 < xn < 10 < µ < 4 (constant)
(variable)xn population (capacity)...
a a rate of growth...
Logistic Map
x1 = a x0 ( 1 - x0 ) n = 0
n = 1 x2 = a x1 ( 1 - x1 )
n = 2 x3 = a x2 ( 1 - x2 )
x0 = an initial value
A chaotic walker who walk and turns around at the angle calculated by the logistic map function.
Chaotic Walk
θn = 2πxn
K. Shimonishi & T. Iba, "Visualizing Footprints of Chaos", 3rd International Nonlinear Sciences Conference (INSC2008), 2008K. Shimonishi, J. Hirose & T. Iba, "The Footprints of Chaos: A Novel Method and Demonstration for Generating Various Patterns from Chaos", SIGGRAPH2008, 2008
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.
Chaotic WalkPlotting the dots on the two-dimensional space, as follows.
xn+1 = a xn ( 1 - xn ) footprints
of chaos
2
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100n
x
1 < a < 2
1
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100n
x
0 < a < 1
3.56...
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100
x
n
3 < a < 1+ 6
a0 4
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100
x
n
1+ 6 < a < 4
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100
x
n
2 < a < 3
3
xn+1 = a xn ( 1 - xn )
The behavior depends on the value of control parameter a.
The system converges to the fixed point.
The system oscillates. The system exhibits chaos.
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100n
x
0 < a < 1Case 1
It converges tozero state.
The value of θ converges to θ* = 0.
0 < a < 1
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100n
x
xn+1 = a xn ( 1 - xn ) θn = 2πxn
The trail represents a line that goes straight ahead.
Chaotic Walk
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100n
x
1 < a < 2Case 2
It converges toa nonzero state.
The value of θ converges to the fixed value.
1 < a < 2
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100n
x
xn+1 = a xn ( 1 - xn ) θn = 2πxn
The trail is on a circle where the turn-angle is fixed.
The value of θ converges to the fixed value.
The trail is on a circle where the turn-angle is fixed.
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100
x
n
2 < a < 3Case 3
It oscillates at the beginning, but converges toa nonzero state.0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100
x
n
2 < a < 3
Chaotic Walk
The value of θ oscillates on successive iterations.
xn+1 = a xn ( 1 - xn ) θn = 2πxn
The trail represents multiple circles.
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100
x
n
3 < a < 1+ 6Case 4
It oscillates.
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100
x
n
3 < a < 1+ 6 Chaotic Walk
θ takes various values.
xn+1 = a xn ( 1 - xn ) θn = 2πxn
The trail represents complex pattern.
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100
x
n
1+ 6 < a < 4Case 5
It shows chaoticbehaviors.
1+ 6 < a < 4
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100
x
n
Chaotic Walk
0 1 2
2 3.56... 43
a
a
xn+1 = a xn ( 1 - xn )
The behavior depends on the value of control parameter a.
How these interestingpatterns can be generated?
How?
Not so interesting...
finitudechaos +
the quality or condition of being finite.
a finite state or quality.- Random House Dictionary,
- The American Heritage Dictionary of the English Language
From finite + -titude, from Latin fīnītus + -dō
(signifying a noun of state)(having been limited or bounded)
finitude
dWe introduce the parameter for finitude, which controls the number of possible states in the target system.
d represents that the value of x is rounded off to d decimal places at every time step.
0.1 0.36 0.40.2 0.64 0.60.3 0.84 0.8
xn+1xn
f round-off
•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
•In principle, the infinite number of possible states is required for representing chaos in strict sense.
finitude
d =1 d =8 d =16
•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.
regular irregular
all patterns are generated with a =3.76 (in the chaotic regime)
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.
The trails of 10 periodic cycles in the case d = 1.
The trails of 10 periodic cycles in the case d = 2.
The trails of 10 periodic cycles in the case d = 3.
The trails of 10 periodic cycles in the case d = 4.
The trails of 10 periodic cycles in the case d = 5.
The trails of 10 periodic cycles in the case d = 6.
The trails of 10 periodic cycles in the case d = 7.
The trails of 10 periodic cycles in the case d = 8.
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.
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 robustnessof order!
the origin of diversity
Implication 1
(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.
•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.
This is just a hypothesis, however it seems to be plausible.
chaotic walk
Implication 2
The parameter for tuning the finitudewould be
another hidden control parameter of complex systems
d =1 d =8 d =16
regular irregulard
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.
Diverse complex patterns can emergeeven in the universe governed by deterministic laws.
Diverse complex patterns can emergeeven in the universe governed by deterministic laws.
Chaos + Finitudewith the combination of
Some More Information ...
Get and Try!ChaoticWalker
http://www.chaoticwalk.org/
A New Vehicle for Exploring Patterns Hidden in Chaos
Get and Feel!The Chaos BookNew Explorations for Order Hidden in Chaos
Come and Talk!Today’s Poster Session
[Poster 70]"Hidden Order in Chaos: The Network-Analysis Approach To Dynamical Systems"(Takashi Iba)
Chaos + Finitude
The Origin of DiversityThinking with Chaotic Walk
Ph.D. in Media and GovernanceAssociate Professor, Faculty of Policy Management
Keio University, [email protected]
Takashi Iba
Interdisciplinary Information StudiesThe University of Tokyo, Japan
Kazeto Shimonishi
http://www.chaoticwalk.org/
