Bifurcation Structure of a Novel Car-following Model with Relative Velocity Effect 1 Akiyasu Tomoeda 2 Tomoyuki Miyaji 3 2011 AUTO Hopf Kota Ikeda 1 [1, 2] [3, 4, 5]. [4] 1 1 / / JST CREST 2 / 3 / 54

(, ) [6, 7] [8, 9, 10, 11]. Gasser [8] [6] t j x j (t) ẍ j (t) = a {V (h j (t)) ẋ j (t)} (1) h j (t) = x j+1 (t) x j (t) j (j + 1 ) V (h j (t)) 1 ( NEXCO ) 55

No.1 (2016) [6] V (h) = tanh(h 2) + tanh(2) Gasser Hopf Hopf Orosz [9, 10, 11] ẍ j (t) = a {V (h j (t τ)) ẋ j (t)} (2) Gasser Hopf (2) τ T = 1/a 2011 (STNN ) [12] 2 STNN AUTO [13] 3 4 2 STNN 2.1 STNN 2011 (STNN ) [12] STNN [12] STNN v ( ) j v j = a b (h j d) 2 exp cḣj γv j. (3) a, b, c, d, γ a a d b, c, γ 56

(, ) STNN (3) ẍ j = a ẋ j W (h j, ḣj). (4) (3) W W (h j, ḣj) = b ( ) (h j d) 2 exp cḣj + γ (5) 2.2 STNN (3) [12] (3) N (4) (x j, v j ) = (x j, v j ) x j (t) = vt + (j 1) N, v j (t) = v, (j = 1, 2,..., N). (6) v = N > d (7) a W (/N, 0) (8) w 0 = W (/N, 0), w 1 = W h (/N, 0), w 2 = W v (/N, 0) W h W v W ε (x j, v j ) = (x j + εϕ j, v j + εψ j) ε { ϕj = ψ j, ψ j = w 0 ψ j vw 1 (ϕ j+1 ϕ j ) vw 2 (ψ j+1 ψ j ) (9) ϕ 1, ϕ 2,..., ϕ N Fourier N 2 ( ) 0 1 vw 1 (1 ω n ) vw 2 (1 ω n ) w 0 (10) n = 0, 1,..., N 1 ω n = e 2πin/N λ ± n = 1 2 ( p ± p 2 4q), p = w 0 vw 2 (1 ω n ) q = vw 1 (1 ω n ) 57

No.1 (2016) ω N n = ω n λ ± N n = λ± n (3) w 0 > 0, w 1 < 0, w 2 < 0 Reλ + n 0 (w 0 2vw 2 ) 2 v(w 1 + w 2 (w 0 2vw 2 )) 1 + cos 2nπ N (11) (11) n, n. 3 AUTO [12] AUTO [13] STNN AUTO 1 AUTO N x N+1 2 y j [14] 58

(, ) x N+1 = x 1 + h N = x 1 + x N, (12) N h j = (13) j=1 N x = 0 y = (y 1,..., y N ) N y y = (y 1,..., y N ) = (h 1,..., h N 1, x 1 ) (14) ( 2)j 1 1 j 1 x 1 = y N, x j = y N + y k, (j = 2,..., N) (15) x j y (3) N ÿ 1 = (ẏ N + ẏ 1 )W (y 2, ẏ 2 ) + ẏ N W (y 1, ẏ 1 ), j j 1 ÿ j = (ẏ N + ẏ k )W (y j+1, ẏ j+1 ) + (ẏ N + ẏ k )W (y j, ẏ j ), ( N ) ÿ N 1 = ẏ k W ( ÿ N = a ẏ N W (y 1, ẏ 1 ). N 1 N 1 y k, ẏ k ) + (ẏ N + N 2 ẏ k )W (y N 1, ẏ N 1 ), j = 2, 3,..., N 2 (16) y N y N ẏ N (3) (16) (16) AUTO (16) STNN c = 0 c c 2 c 0 [14] Hopf Hopf STNN [12] a = 0.73, b = 3.25, d = 5.25, γ = 0.0517 N = 30 (16) 3.1 (c = 0) c = 0 (11) 30 12 Hopf 59

No.1 (2016) STNN 3 3 3 Hopf = 1333.43 Hopf ( dh 1 dt = 0) ( = 1333.43) ( = 1333.41 = 2619.25) 1333.43 < < 2619.25 STNN [8] 4 0.006 3 0.004 dh 1 /dt 2 dh 1 /dt 0.002 1 0 0 0 500 1000 1500 2000 2500 SE UE SC UC 1333 1333.5 1334 SE UE SC UC 3 ( ) STNN [14] ( ) ( = 1333.43) [14] Stable Equilibrium SE Unstable Equilibrium UE Stable Cycle SC Unstable Cycle UC. 3.2 (c 0) c 0. c Hopf (, c) 2, c c = 0 Hopf ((, c) = (205.612, 0), (1333.41, 0)), c 4 2 Hopf (, c) = (395.55, 1.955) c > 1.995 Hopf 60

(, ) c Hopf c > 1.955 (, c) = (1019.23, 0.6664) c > 0.6664 3 0 < c < 0.6664, 0.6664 < c < 1.955, 1.955 < c 2 1.5 HB SN 0.75 0.5 HB SN c 1 c 0.5 0.25 0 0 500 1000 1500 2000 2500 0 1000 1100 1200 1300 1400 4 ( ) Hopf HBSN (, c) [14] ( ) [14] 3.3 Hopf Hopf Hopf 205.612 < < 1333.41 STNN [15, 16] c = 0 Hopf 5 5 = 1200 = 1200 6 (, c) 2 Hopf 7 c Hopf 61

No.1 (2016) 4 3 dh 1 /dt 2 1 0 0 500 1000 1500 2000 2500 SE UE SC UC 5 Hopf 4 AUTO STNN c = 0 Hopf Hopf c 0 (, c) Hopf c > 1.955 Hopf (, c) = (1019.23, 0.6664) c > 0.6664 Hopf Hopf Hopf MIMS JSPS (B)(No. 25790099, No. 15K17594) 62

(, ) (a) 2 (b) 3 6 (c) 4 (d) 5 Hopf (c = 0) [1] (1989) [2] (2008). [3] D. Chowdhury,. Santen, and A. Schadschneider, Phys. Rep. 329, 199 (2000). [4] D. Helbing, Rev. Mod. Phys. 73, 1067 (2001). [5] A. Schadschneider, D. Chowdhury, and K. Nishinari, Stochastic Transport in Complex Systems from Molecules to Vehicles (Elsevier, Amsterdam, 2010). [6] M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama, Phys. Rev. E 51, 1035 (1995). [7] Y. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, S. Tadaki and S. Yukawa, New J. Phys. 10, 033001 (2008). [8] I. Gasser, G. Sirito, and B. Werner, Physica D 197, 222 (2004). 63

No.1 (2016) 2 HB 1.5 c 1 0.5 0 0 500 1000 1500 7 2 Hopf [9] G. Orosz, R. E. Wilson, and B. Krauskopf, Phys. Rev. E 70, 026207 (2004). [10] G. Orosz, B. Krauskopf, and R. E. Wilson, Physica D 211, 277 (2005). [11] G. Orosz and G. Stepan, Proc. R. Soc. A 462, 2643 (2006). [12] D. Shamoto, A. Tomoeda, R. Nishi, and K. Nishinari, Phys. Rev. E 83, 046105 (2011). [13] E. J. Doedel and B. E. Oldeman, Concordia University, Montreal, Canada (2012). [14] A. Tomoeda, T. Miyaji and K. Ikeda, Proceedings of Traffic and Granular Flow 15, Springer (to appear). [15] E. Tomer,. Safonov and S. Havlin, Phys. Rev. ett., 84, 382 (2000). [16] K. Nishinari and D. Takahashi, J. Phys. A: Math. Gen. 32, 93 (1999). (: 2016 1 6 ; : 2016 1 25 ) 64