Shunsuke Kobayashi 1 [6] [11] [7] u t = D 2 u 1 x 2 + f(u, v) + s L u(t, x)dx, L x (0.L), t > 0, Neumann 0 v t = D 2 v 2 + g(u, v), x (0, L), t > 0. x

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1 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} , s.kobayashi.meiji@gmail.com

2 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 = , s = , gu, v = 2u 5v 0.8u2, π = 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

3 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, 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 >

4 2.2 X per := {u, v Hper0, 2 2L Hper0, 2 2L; ux, vx = u2l x, v2l x} , L 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, 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 s, D 2, k } m Z

5 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 D n= n=2 n= 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 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 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

6 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 a 4 αα 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 e 4 α 0 α 2 + O α 0, α, α 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 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 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.

7 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 r, x r, x 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 b > 0, c < 0, d =, d bc > 0, µ < 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 }.

8 µ < µ, µ = µ, µ > µ. α = 2 c/ bc, β = 2 + b/ bc, γ = + b/ c, bc > 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 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

9 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 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

10 5 {r, x, θ r, x = r, x }, r, x 0 4. Hopf Hopf-Pitchfork 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, , [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, [6] E. N. Lorentz, Deterministic Nonperiodic Flow, J. Atom. Sci., [7] Y. Nishiura and D. Ueyama, Spatio-temporal chaos for the Gray-Scott model, Physica D [8] T. Ogawa, Degenerate Hopf instability in oscillatory reaction-diffusion equations, Discrete and Continuous Dynamical Systems Supplements, Special vol. 2007, [9] T. Ogawa and T. Okuda, Oscillatory dynamics in a reaction-diffusion system in the presence of 0::2 resonance, Networks and Heterogeneous Media, [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 [] W. Tucker, A Rigorous ODE Solver and Smale s 4th Problem, Found. Comput. Math

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