Yo Horikawa and Hiroyuki Kitajima Kagawa University Japan

1. Exponential Transient Oscillations and Their Stabilization in a Bistable Ring of Unidirectionally Coupled Maps Yo Horikawa and Hiroyuki Kitajima Kagawa University Japan 1

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. l 1. Background Examples of exponential transients 1. Bistable reaction-diffusion equation Kink, pulse patterns → Spatially homogeneous states 3

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

5. 1. Background 1. Bistable reaction-diffusion equation 2. Ring neural network Symmetric bistability Common mechanism dl/dt～ -exp(-l) l: width of patterns Purpose of this study Whether exponential transients exist in discrete-time systems: coupled map lattices (CMLs). 5

6. 2 1 N 3 4 5 6 7 8 2. Model Bistable Ring of Unidirectionally Coupled Maps n：index of sites, N：the number of sites (N ≥ 3) t: discrete time xn(t): state of nth site at time t c: coupling strength Bistable steady states： xn = ±c1/2 (1 ≤ n ≤ N) 6

7. 2. Model Random initial states: xn(0) ~ N(0, 0.12) → Traveling pulse wave (xn(t): c1/2⇄ -c1/2 ) simulation Fig. 1. Transient pulse waves and oscillations (c = 0.2, N = 20) 7

8. lh lh 3. Propagation of pulse waves N: even (N = 2M) Initial states： → Symmetric pulse wave Stable in the subspace: xn = -xN/2+n (1≤ n ≤ N/2) 8

9. lh 3. Propagation of pulse waves Propagation time of pulse fronts per site Δt： xn-1(t) = 0 → xn(t +Δt) = 0 9

10. lh 3. Propagation of pulse waves Propagation time of pulse fronts per site:Δt (Δt： xn-1(t) = 0 → xn(t +Δt) = 0) vs Pulse width: lh Fig. 2. Propagation time of pulse fronts per site(c = 0.2) (5) 10

11. Δt l 3. Propagation of pulse waves Supposition: Propagation time of a pulse front per site:Δtdepends on length: l to its backward pulse front. Changes in pulse width: l → Difference between the speeds of two pulse fronts (6) 11

12. 4. Duration of pulse waves 1. Duration of asymmetric pulse waves: T(l0; N) Initial states： (l0≠N/2) → Unstable pulse waves with initial pulse width: l0 and N - l0 l(T) = 0 → (7) (8) 12

13. 4. Duration of pulse waves 1. Duration of asymmetric pulse waves: T(l0; N) Letting N → ∞ T(l0)～exp(l0) ･･･ Increases exponentially with pulse width (9) 13

14. l0 4. Duration of pulse waves 1. Duration of asymmetric pulse waves: T(l0; N) T(l0)～exp(l0) ･･･ Increases exponentially with pulse width Fig. 3. Duration: T of pulse waves with initial pulse width: l0 (c = 0.2, N = 21) 14

15. 4. Duration of pulse waves 2. Random initial states: xn(0) ~ N(0, 0.12) → Pulse waves with initial pulse width obeying the uniform distribution: l0 ~ U(0, N/2) Distribution h(T) of duration of such pulse waves (10) (11) 15

16. Tc 4. Duration of pulse waves 2. Distribution h(T) of duration of pulse waves occurring from random initial states Cut-off： Tc = exp(αN/2)/(αβ) ≈ 3×106 (N = 20) Prob{T > Tc} ≈ 4exp(-2)/(αN) ≈ 0.357/N ≈ 0.018 (N = 20) (12) (13) Fig. 4. Distribution of duration of random pulse waves (c = 0.2, N = 20) 16

17. 4. Duration of pulse waves 2. Distribution h(T) of duration of pulse waves occurring from random initial states (14) Mean: m(T)～exp(N) SD:σ(T) ～exp(N) Coefficient of variation: CV(T) > 1 Fig. 5. Mean, SD and CV of duration of random pulse waves vs N 17

18. 5. Stabilization of pulse waves Coupling strength：c increases → Approaches to steady states are oscillatory → ･Period doubling bifurcations → ･Neimark-Sacker bifurcations Linearization at c1/2 Fig. 7. Eigenvalues of Jacobian Matrix (N = 20) 18

19. 5. Stabilization of pulse waves Steady state → Period doubling bifurcation (c = 0.5 ) → Periodic solution → Chaos Bifurcation diagram Fig. 9. Stabilized pulses (a), Chaotic pulses (b)(c) simulation 19

20. 6. Conclusion Bistable ring of unidirectionally coupled maps → Exponential transient oscillations ･Duration of transient oscillations increases exponentially with system size T ∝ exp(N) ･Stabilization of oscillations due to bifurcations of steady states Future problem: Analytical derivation of pulse kinematics 20