Shunsuke Kobayashi [6] [] [7] u t = D 2 u x 2 + fu, v + s L ut, xdx, L x 0.L, t > 0, Neumann 0 v t = D 2 v 2 + gu, v, x 0, L, t > 0. x2 u u v t, 0 = t, L = 0, x x. v t, 0 = t, L = 0.2 x x ut, x R vt, x R D > 0 u, D 2 > 0 v s > 0 [8]..2 X := {u, v H 2 0, L H 2 0, L; u x t, 0 = u x t, L, v x t, 0 = v x t, L = 0}. 24-857 --, E-mail: s.kobayashi.meiji@gmail.com
Assumption. 反応項 f u, v, gu, v は十分滑らかであり f 0, 0 = g0, 0 = 0 かつ fu > 0, fv < 0, gu > 0, gv < 0, fv gu + gv < 0 gv を満たす さらに システム. は自明解においてチューリング不安定性をもつ すなわち fu + gv < 0 かつ fu gv fv gu > 0 が成り立っている したがって 自明解から空間非一様な定常解が分岐しうる ここで fu = f 0, 0, u fv = f 0, 0, v gu = g 0, 0, u gv = g 0, 0. v 現在までに システム. に対してカオス的な挙動をする解が数値的に報告されている [, 9, 0] 例え ば 以下のようにパラメータと反応項をおく D = 0.25, f u, v = u 0v u2 2u3, D2 = 26.8778, s = 2.9890, gu, v = 2u 5v 0.8u2, π = 0.86008. L このときの数値計算によって得られた数値解を図 に示す 図. 3 重臨界点付近におけるシステム. の数値解 上段左図は ut, x の t, x [3000, 4000] [0, L] における鳥瞰図であり 上段右図は高速フーリ エ変換によって得られた フーリエ空間上における u0 t, u t, u2 t の軌道を表している ここで uj
ut, x j ut, x L 2 - u L 2t t log log u L 2t u 2 L 2t u 2 t, x, v 2 t, x u 2 0, x, v 2 0, x = u 0, x, v 0, x + 0 6, 0.. 2. Hopf-Pitchfork Dumortier et al. [2] Hopf-Pitchfork 3 3 ṙ = Reλ + ReJ r 2 + ReJ 2 x 2 r + φ r, x, ẋ = ν 0 + K w 2 + K 2 x 2 x + φ 2 r, x, θ = ω + φ 3 r, x [] 2 3 Hopf-Pitchfork 4 [2] Hopf-Pitchfork 5 2 hidden symmetry ut, x, vt, x X. ũt, x = { ut, x x [0, L], ut, 2L x x [L, 2L]. ṽt, x = { vt, x x [0, L], vt, 2L x x [L, 2L] 2. 0, 2L u t = D u xx + fu, v + s 2L ut, xdx, x 0, 2L, t > 0, 2L 0 v t = D 2 v xx + gu, v, x 0, 2L, t > 0, ut, x = ut, x + 2L, u x t, x = u x t, x + 2L, t > 0, vt, x = vt, x + 2L, v x t, x = v x t, x + 2L, t > 0. 2.2
2.2 X per := {u, v Hper0, 2 2L Hper0, 2 2L; ux, vx = u2l x, v2l x} 2.3 2.2 0, L. 2.2-2.3 ut, x = u m te imkx, vt, x = v m te imkx. m Z m Z 2.2 um v m um Fm = M m +, m Z. 2.4 v m G m M 0 = fu + s f v fu D, M g u g j = j 2 k 2 f v v g u g v D 2 j 2 k 2, j Z \ {0} F m = + G m = + m +m 2 =m m,m 2 Z m +m 2 +m 3 =m m,m 2,m 3 Z m +m 2 =m m,m 2 Z m +m 2 +m 3 =m m,m 2,m 3 Z f uu 2 u m u m2 + f uv u m v m2 + f vv 2 v m v m2 f uuu 6 u m u m2 u m3 + f uuv 2 u m u m2 v m3 + f uvv 2 u m v m2 v m3 + f vvv 6 v m v m2 v m3 +, g uu 2 u m u m2 + g uv u m v m2 + g vv 2 v m v m2 g uuu 6 u m u m2 u m3 + g uuv 2 u m u m2 v m3 + g uvv 2 u m v m2 v m3 + g vvv 6 v m v m2 v m3 +. f uu = 2 f u 2 0, 0 2.4 X F { X F := {u m, v m } m Z ; u m, v m = u m, v m, {u m, v m } m Z 2 X F = m Z+m } 2 2 u m, v m 2 <. Pu, v : X per X F ; { Pu, v = 2L 2L 0 ut, x, vt, xe imkx dx X per ut, x R vt, x R 2. u m, v m = u m, v m R R m 0 2.4 s, D 2, k } m Z
Definition. Det M m = 0 D 2, k m Z D 2 k; m = δ := f u g v f v g u g v D m 2 k 2 δ m 2 k 2 D m 2 k 2 f u 50 40 D 2 30 20 0 n= n=2 n=3 0 0 0.5 n=4.5 k 2 2. f u =, f v = 0, g u = 2, g v = 5 D = /4 n =, 2, 3, 4 0 : : 2 0 s, D 2, k = s, D 2, k s, D 2, k 3 2.4 α0 T 0 = β 0 f gv vg u g v g u g u = T 0 u0, T m = v 0, αm β m = T m um v m, m =, 2 gv + D2m 2 k 2 f u D m 2 k 2 α 0 = gv 2 + f v g F 0 + f v G0, u g v β 0 = Tr M 0 β 0 + F gv 2 0 + g v G0 + f v g u g u α = { g u F + D k,2 2 f u dett G }, g u, β = Tr M β + { g u F + D,2 2 dett k,2 2 g v G }, α 2 = { g u F2 + 4D k,2 2 f u dett G } 2, 2 β 2 = Tr M 2 β 2 + { g u F2 + 4D,2 2 dett k,2 2 g v G } 2, 2 u m u m F m = M m + m 3. v m v m G m g u m =, 2 2.5
F m G m α m β m 2.5 [4, 5] 3 Theorem. 3 s, D 2, k = s, D2, k α 0 = µ 0 α 0 + A α0 2 + A 2 α 2 + A 3 α2 2 + a α0 2 + a 2 α 2 + a 3 α2α 2 0 + a 4 αα 2 2 + O α 0, α, α 2 4, α = µ α + B α 0 α + B 2 α α 2 + b α0 2 + b 2 α 2 + b 3 α2α 2 + b 4 α 0 α α 2 + O α 0, α, α 2 4, α 2 = µ 2 α 2 + E α 0 α 2 + E 2 α 2 + e α0 2 + e 2 α 2 + e 3 α2α 2 2 + e 4 α 0 α 2 + O α 0, α, α 2 4. 2.6 A j, B j, E j, a j, b j, e j R. D fu, v gu, v µ j µ j s, D 2, k α 0, α, α 2 α 0, α, α 2 3 α 0 = µ 0 α 0 + A α0 2 + A 2 α 2 + A 3 α2 2 + a α0 2 + a 2 α 2 + a 3 α2α 2 0 + a 4 αα 2 2, α = µ α + B α 0 α + B 2 α α 2 + b α0 2 + b 2 α 2 + b 3 α2α 2 + b 4 α 0 α α 2, 2.7 α 2 = µ 2 α 2 + E α 0 α 2 + E 2 α 2 + e α0 2 + e 2 α 2 + e 3 α2α 2 2 + e 4 α 0 α 2 A j 0, B j 0, E j 0, a j 0, b j 0, e j 0 3 Hopf-Pitchfork α 0, α, α 2 = α0, 0, α2 Lemma. A 3 E < 0 ρ R \ {0} 2ρ 2 e 3 + a + A ρ 2 A 3 = 0 ρ α 0 = ρ2 A 3 A 2a + ρ 2 e 3, α 2 = ρα 0, µ 0 = µ 0 := α 0{A + a α 0 + ρ 2 A 3 + a 3 α 0}, µ = µ := α 0{B + ρb 2 + b + ρb 4 + ρ 2 b 3 α 0}, µ 2 = µ 2 := α 0{E + e + ρ 2 e 3 α 0} α 0, α, α 2 = α 0, 0, α 2 0 α 0, 0, α 2 Hopf [5] Hopf-Pitchfork { ż = λ + J z 2 + J 2 x 2 z + O z, z, x 4, ẋ = ν + K z 2 + K 2 x 2 x + O z, z, x 4. 3.
z C x R λ R ν R J j, K j C z = rte θt ṙ = ν + r 2 + bx 2 r + φ r, x, ẋ = ν 2 + cr 2 + dx 2 x + φ 2 r, x, 3.2 θ = ω + φ 3 r, x ν, ν 2 R d = ± ω R φ j r, x r, x 4 3.2 r, x r, x 2.6 4 3.2 3.2 θ [3] S -symmetry Dumortier et al. [2] S -symmetric φ n C Dumortier et al. [2] 3.2 blow-up ε > 0 µ = ε 2 µ, µ 2 = ε 2, r = ε r, x = ε x dτ = ε 2 dt r = rµ + r 2 + bx 2 + φ εr, εx/ε 3, x = x + cr 2 + dx 2 + φ 2 εr, εx/ε 3, 4. θ = ω/ε 2 + φ 3 εr, εx/ε 2 τ Oε { r = rµ + r 2 + bx 2, x = x + cr 2 + dx 2. 4.2 4.2 b > 0, c < 0, d =, d bc > 0, µ < 0. 4.3 O := 0, 0, p := 0,, p 2 := cµ b + µ bc,, p 3 := µ, 0 bc 4. p µ,ε p2 µ,ε p3 µ,ε p3 µ,ε C µ,ε 4.2 µ = µ := b + /c Hr, x Hr, x = r α x β { µ + r 2 + γx 2 }.
4. 4.2 µ < µ, µ = µ, µ > µ. α = 2 c/ bc, β = 2 + b/ bc, γ = + b/ c, bc > 0.4. 3 φ j r, x S -symmetric 3 [3] generic 2.7 Hopf-Pitchfork 3. 4 θ O p p 3 µ,ε C µ,ε Σ: r, x, θ 2 Σ := {r, x, θ x = b + µ/ bc} C u : Σ W u p µ,ε C s : Σ W s C µ,ε q: W u O Σ Definition 2. Exceptional set E E := {µ, ε 0, 0 0, 0, 0 } Definition 3. 4. forward trapping C u C s.4. C u C s Σ 2 W s C µ,ε W u p µ,ε forward trapping region 4. backward trapping C s C u.4. backward trapping region 4.3 b + c < 0 µ := c /b + < 0 µ, ε exceptional set E O p µ,ε forward trapping backward trapping C u C s.4.2 Σ q q C s Σ O p 3 µ,ε µ, ε µ, ε = µ, ε O O.4.3
3.2 [] p µ,ε p µ,ε p 2 µ,ε C s C u q C u p 2 µ,ε q C s O C µ,ε p 3 µ,ε O C µ,ε p 3 µ,ε 4.2 4. µ < µ backward trapping µ > µ forward trapping ε µ, ε ε, µ µ +, ε + γ µ O µ 4.3 µ ε µ, ε backward trapping µ +, ε + forward trapping µ < µ < µ + < 0 exceptional set E γ µ, ε µ +, ε + 4. µ, ε γ
5 {r, x, θ r, x = r, x }, r, x 0 4. Hopf Hopf-Pitchfork. 2.. 2. References [] S. Kobayashi and T. O. Sakamoto, Hopf Bifurcation and Hopf-Pitchfork Bifurcation in an Integro-Differential Reaction-Diffusion System, submitted. [2] F. Dumortier F, S. Ibáñez, H. Kokubu and C. Simó, About the unfolding of a Hopf-zero singularity, Discrete and Continuous Dynamical Systems series A, 33 203, 4435 447. [3] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 983. [4] M. Haragus and G. Iooss, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite- Dimentional Dynamics Systems, Springer, 200. [5] Yu A. Kuznetsov, Elements of Applied Bifurcation Theory, 3 rd edition, Springer-Verlag, New York, 2004. [6] E. N. Lorentz, Deterministic Nonperiodic Flow, J. Atom. Sci., 20 963 30 4. [7] Y. Nishiura and D. Ueyama, Spatio-temporal chaos for the Gray-Scott model, Physica D 50 200 37 62. [8] T. Ogawa, Degenerate Hopf instability in oscillatory reaction-diffusion equations, Discrete and Continuous Dynamical Systems Supplements, Special vol. 2007, 784-793. [9] T. Ogawa and T. Okuda, Oscillatory dynamics in a reaction-diffusion system in the presence of 0::2 resonance, Networks and Heterogeneous Media, 7 4 202 893 926. [0] T. Ogawa and T. O. Sakamoto, Chaotic dynamics in an integro-differential reaction-diffusion system in the presence of 0::2 resonance, Mathematical Fluid Dynamics, Present and Future 206, Nov. 53 562. [] W. Tucker, A Rigorous ODE Solver and Smale s 4th Problem, Found. Comput. Math. 2 2002 53 7.