1. Agent-Based Modeling (RePast/Java) of Networks of Logistic Maps with Long-Range Coupling: Synchronization, Pattern Formation and Probability Distributions (Preliminary Results) Anton Burykin * and Balazs Adamcsek ** * University of Southern California (bourykin@usc.edu, http://futura.usc.edu/wgroup/people/anton) ** Eotvos Lorand University, Hungary (balazs@angel.elte.hu) SFI Complex Systems Summer School Santa Fe, NM, June 2005

2. Logistic Map Edge of Chaos (EOC): orderchaos EOC

3. Long Range Interactions Coupled Logistic Maps on 1D and 2D Lattice: C > D – short-range (“extensive”) Coupling: C < D – long-range (“non-extensive”) (D - dimension) => Tsallis Entropy: Boltzmann-Gibbs Entropy:

4. 2D Agent-Based Modeling: Repast/Java Random initial distribution: “agent” – logistic map state – x n (i) black - “-1”; white - “1” (coupled logistic maps on 2D grid)

5. Synchronization: Standard Mean Square Deviation Fully Synchronized (Coherent) States:

6. 1D: Periodic Dynamics x(n) histogram average histogram synchronization (SMSD) average state i

7. 2D: Periodic Dynamics x(n) histogram average histogram synchronization (SMSD) average state i

8. 2D: Chaotic Dynamics x(n) histogram average histogram synchronization (SMSD) average state i

9. 2D: Chaotic Dynamics + Patterns x(n) histogram average histogram synchronization (SMSD) average state i

10. Future Work: Moving Agents Periodic Boundary Conditions Adaptation and Effect of Noise (Self-Adjusting Logistic Map with Noise): More Complex Networks (other then a grid): random, scale-free, etc… Non-Uniform Probability Distribution at the EOC (Tsallis Entropy S q ) (f n – low pass filter of {x n })

11. Acknowledgments: Constantino Tsallis; Tom Carter; Alfred Hubler; Melanie Mitchell; CSSS 2005 & SFI