1. 1 Exponential Transient Oscillations and Standing Pulses in Rings of Coupled Symmetric Bistable Maps Yo Horikawa Kagawa University Japan

2. 2 1. Background Exponential transients Initial states → Transient states → Asymptotic states ↑ Duration (Life time) of transients increases exponentially with system size. T ∝ exp(N) T: duration of transients N: system size

3. 3 1. Background Systems never reach their asymptotic states in a practical time. → Transient states play important roles. Two kinds of exponential transients I. Metastable dynamics in reaction-diffusion systems (Kawasaki and Ohta 1982) in ring neural networks (Horikawa and Kitajima 2008) II. Transient chaos in coupled map lattices (Crutchfield and Kaneko 1988) in neural networks (Bastolla and Parisi 1998)

4. 4 1. Background Examples of exponential transients 1. Bistable reaction-diffusion equation Transient kink, pulse patterns → Spatially homogeneous states: u = ±1 l

5. 5 1. Background Examples of exponential transients 2. Ring neural network Transient traveling waves and oscillations → Spatially homogeneous states 4 5 N 12 6 7 8 3 l

6. 6 1. Background Examples of exponential transients 3. Bistable ring of directly coupled maps Traveling waves → Spatially homogeneous states 4 5 N 12 6 7 8 3 l

7. 7 1. Background 1. Bistable reaction-diffusion equation 2. Ring neural network 3. Ring of directly coupled maps Symmetric bistability Common kinematics dl/dt ～ –exp(–l) l: width of patterns Purpose of this study Whether exponential transients exist in lattices of coupled circle maps.

8. 8 2. Unidirectionally coupled maps Ring of unidirectionally coupled bistable symmetric circle maps n ： index of sites, N ： the number of sites t: discrete time x n (t): state of nth site at time t ε: coupling strength Bistable steady states ： x n = ± 1/2 (1 ≤ n ≤ N) 4 5 N 12 6 7 8 3 (1a), (2)

9. 9 2. Unidirectionally coupled maps Random initial states: x n (0) ~ N(0, 0.1 2 ) → Traveling pulse waves (x n (t): 1/2 ⇄ –1/2) Fig. 1(a). Transient pulse waves ( ε = 0.2, K = 0.5, N = 20) simulation

10. 10 2. Unidirectionally coupled maps Random initial states: x n (0) ~ N(0, 0.1 2 ) → Traveling pulse waves (x n (t): 1/2 ⇄ –1/2) Fig. 1(b). Transient pulse waves ( ε = 0.8, K = 0.5, N = 20) simulation

11. 11 2. Unidirectionally coupled maps N: even (N = 2M) Initial states ： → Unstable symmetric pulse wave → Saddle manifold in the state space Stable in the subspace: x n = -x N/2+n (1≤ n ≤ N/2) lhlh lhlh (3)

12. 12 3. Bidirectionally coupled maps Ring of bidirectionally coupled bistable symmetric circle maps n ： index of sites, N ： the number of sites t: discrete time x n (t): state of nth site at time t ε: coupling strength Bistable steady states ： x n = ± 1/2 (1 ≤ n ≤ N) 4 5 N 12 6 7 8 3 (1b), (2)

13. 13 3. Bidirectionally coupled maps Random initial states: x n (0) ~ N(0, 0.1 2 ) → Standing pulses (x n (t): 1/2 ⇄ –1/2) Fig. 1(c). Standing pulse ( ε = 0.5, K = 0.1, N = 40) simulation

14. 14 3. Bidirectionally coupled maps N: even (N = 2M) Initial states ： → Unstable symmetric standing pulses → Saddles in the state space Stable in the subspace: x n = -x N/2+n (1≤ n ≤ N/2) (3) lhlh lhlh

15. 4. Changes in pulse width Locations of pulse fronts: n 1, n 2 Speeds of pulse fronts: v 1 = Δ n 1 / Δ t, v 2 = Δ n 2 / Δ t Changes in pulse width l → Difference between the speeds of two pulse fronts 15 l n1n1 n2n2 N – l x1x1 xNxN v1v1 v2v2 dl/dt = Δ( n 1 – n 2 )/ Δ t = v 2 – v 1

16. 4. Changes in pulse width Changes in pulse width: l ~ exponentially small with pulse width: l and N – l 16 l n1n1 n2n2 N – l x1x1 xNxN v1v1 v2v2 (5) α = 2.375, β = 1.304 in unidirectionally coupled maps α = 0.651, β = 0.487 in bidirectionally coupled maps

17. 4. Changes in pulse width Changes in pulse width: l Initial pulse width: l(0) = l 0 < N/2 → l(T) = 0 → T(l 0 ; N): Duration of pulses with initial pulse width l 0 17 (5) (6) (7)

18. 4. Changes in pulse width Simple forms by letting N → ∞ T(l 0 ) ～ exp(l 0 ) ･･･ Duration increases exponentially with initial pulse width 18 (8)

19. 19 5. Duration of transient pulses 1. Duration of asymmetric pulses: T(l 0 ) T(l 0 ) ～ exp(l 0 ) Fig. 4. Duration T vs initial pulse width l 0 in unidirectionally coupled maps (ε = 0.2, K = 0.5, N = 21) l0l0 Fig. 7. Duration T vs initial pulse width l 0 in bidirectionally coupled maps

20. 20 5. Duration of transient pulses 2. Randomly generated pulses Random initial states: x n (0) ~ N(0, 0.1 2 ) → Pulses with initial pulse width obeying the uniform distribution: l 0 ~ U(0, N/2) Distribution h(T) of duration T of these pulses (9) (10)

21. 21 5. Duration of transient pulses 2. Distribution h(T) of duration T of randomly generated pulses Cut-off ： T c = exp(αN/2)/(αβ) ≈ 3×10 6 (N = 20) Prob{T > T c } ≈ 4exp(-2)/(αN) ≈ 0.357/N ≈ 0.018 (N = 20) Fig. 5. Distribution of duration of random traveling pulses in unidirectionally coupled maps (ε = 0.2, K = 0.5, N = 20) (11) (12)

22. 22 5. Duration of tansient pulses 2. Distribution h(T) of duration T of randomly generated pulses Cut-off ： T c = exp(αN/2)/(αβ) ≈ 1.1×10 6 (N = 40) Prob{T > T c } ≈ 4exp(-2)/(αN) ≈ 0.832/N ≈ 0.021 (N = 40) Fig. 8. Distribution of duration of random standing pulses in bidirectionally coupled maps (ε = 0.5, K = 0.1, N = 40) (11) (12)

23. 23 6. Conclusion ･ Rings of unidirectionally and bidirectionally coupled maps → Transient traveling pulses and standing pulses ･ Duration T of transient pulses increases exponentially with initial pulse width l 0. T ∝ exp(l 0 ) ･ Duration T of transient pulses generated under random initial conditions is distributed in a power law form. h(T) ~ 1/T

24. 24 2-2. Duration of pulse waves 2. Duration of pulse waves occurring from random initial states Fig. 6. Mean, SD and CV of duration of random pulse waves vs N Mean: m(T) ～ exp(N) SD:σ(T) ～ exp(N) Coefficient of variation: CV(T) > 1 (14)