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- 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 emergeeven 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 Walk0 < 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 100Case 3 2 < a < 3 n 0.82 < a < 3 1 0.8 0.6 The value of θ converges toIt oscillates at 0.6 x the fixed value.the beginning, x 0.4 0.4 The trail is on a circle wherebut 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 Walk3 < 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. 0.6x 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. a0 1 2 a2 3 3.56... 4
- 13. Not so interesting... How these interestingpatterns can be generated? w ? Ho
- 14. chaos + finitude
- 15. finitudea finite state or quality. - Random House Dictionary,the quality or condition of being finite. - The American Heritage Dictionary of the English LanguageFrom 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 thenumber of possible states in the target system. d d represents that the value of x is rounded off to d decimal placesat 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 forrepresenting chaos in strict sense. •A system consisting of the finite number of possible stateseventually 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 stateseventually 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 =7The patterns generated by chaotic walks with the logistic map forthe 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 eachvalues of a and dThe average length of attractorincreases exponentially as thefinitude parameter d increases.
- 28. Diversity and Robustness of Patternsdiversification of generatedpatterns by varying thefinitude parameter d. The box represents the region that has completely same types of attractors.As the number of possiblestates increases,- the diversity increases- the robustness decreasesThe finitude parametercontrols the degree ofdiversity and robustnessof 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 ina deterministic way only with the combination of deterministicchaos 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 dThe 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 dramaticchange of behaviors in phenomena that we have considered as random walk.
- 34. Diverse complex patterns can emergeeven in the universe governed by deterministic laws.
- 35. Diverse complex patterns can emergeeven in the universe governed by deterministic laws. with the combination of Chaos + Finitude
- 36. Some More Information ...
- 37. Get and Try!ChaoticWalkerA New Vehicle for Exploring Patterns Hidden in Chaoshttp://www.chaoticwalk.org/
- 38. Get and Feel!The Chaos BookNew Explorations for Order Hidden in Chaos
- 39. Come and Talk!Today’s Poster Session Chaos + Finitude[Poster 70]"Hidden Order in Chaos: TheNetwork-Analysis ApproachTo 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

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