Geometric Singular Perturbation Method 3D (Ch 5, page 86)

Clear["Global`*"]
Off[General::spell1]

Regular Perturbation Expansion

Z[x,y,ϵ]:=(Subscript[Z, 0])[x,y]+ϵ (Subscript[Z, 1])[x,y]+(ϵ^2) (Subscript[Z, 2])[x,y]+(ϵ^3) (Subscript[Z, 3])[x,y]

q:=ϵ (Subscript[Z, 1])[x,y]+(ϵ^2) (Subscript[Z, 2])[x,y]+(ϵ^3) (Subscript[Z, 3])[x,y]

D[Z[x,y,ϵ],x]

D[Z[x,y,ϵ],y]

Taylor series expansion of the vector field

F[x_,y_,z_]:=f[x,y,z] + q(∂_z) f[x,y,z]+((q^2)/2)(∂_(z,z))f[x,y,z]+((q^3)/6)(∂_(z,z,z))f[x,y,z]

G[x_,y_,z_]:=g[x,y,z] + q(∂_z) g[x,y,z]+((q^2)/2)(∂_(z,z))g[x,y,z]+((q^3)/6)(∂_(z,z,z))g[x,y,z]

H[x_,y_,z_]:=h[x,y,z] + q(∂_z) h[x,y,z]+((q^2)/2)(∂_(z,z))h[x,y,z]+((q^3)/6)(∂_(z,z,z))h[x,y,z]

Invariance (Darboux - Fenichel)

IDF=D[Z[x,y,ϵ],x] F[x,y,z]+ϵ D[Z[x,y,ϵ],y] G[x,y,z]- ϵ H[x,y,z]

Collect[IDF,{ϵ,(ϵ^2),(ϵ^3)}]

Order (ϵ^0)

sp0[x_,y_,z_]:=Solve[f[x,y,z]==0,z]

sp0[x,y,z]

(Subscript[Z, 0])[x_,y_]:=

(Subscript[Z, 0])[x,y]

Order (ϵ^1)

Coefficient[IDF,ϵ]

sp1[x_,y_,z_]:=Solve[-h[x,y,z]+g[x,y,z] Derivative[0,1][(Subscript[Z, 0])][x,y]+f[x,y,z] Derivative[1,0][(Subscript[Z, 1])][x,y]+(Subscript[Z, 1])[x,y] Derivative[1,0][(Subscript[Z, 0])][x,y] Derivative[0,0,1][f][x,y,z]==0,(Subscript[Z, 1])[x,y]]

sp1[x,y,z]

sp1[x,y,(Subscript[Z, 0])[x,y]]/.{f[x,y,(Subscript[Z, 0])[x,y]]->0}

(Subscript[Z, 1])[x_,y_]:=((h[x,y,(Subscript[Z, 0])[x,y]]-g[x,y,(Subscript[Z, 0])[x,y]] Derivative[0,1][(Subscript[Z, 0])][x,y])/(Derivative[1,0][(Subscript[Z, 0])][x,y] Derivative[0,0,1][f][x,y,(Subscript[Z, 0])[x,y]]))

(Subscript[Z, 1])[x,y]

Order (ϵ^2)

Coefficient[IDF,(ϵ^2)]

sp2[x_,y_,z_]:=Solve[g[x,y,z] Derivative[0,1][(Subscript[Z, 1])][x,y]+f[x,y,z] Derivative[1,0][(Subscript[Z, 2])][x,y]+(Subscript[Z, 2])[x,y] Derivative[1,0][(Subscript[Z, 0])][x,y] Derivative[0,0,1][f][x,y,z]+(Subscript[Z, 1])[x,y] Derivative[1,0][(Subscript[Z, 1])][x,y] Derivative[0,0,1][f][x,y,z]+(Subscript[Z, 1])[x,y] Derivative[0,1][(Subscript[Z, 0])][x,y] Derivative[0,0,1][g][x,y,z]-(Subscript[Z, 1])[x,y] Derivative[0,0,1][h][x,y,z]+(1/2) (((Subscript[Z, 1])[x,y])^2) Derivative[1,0][(Subscript[Z, 0])][x,y] Derivative[0,0,2][f][x,y,z]==0,(Subscript[Z, 2])[x,y]]

sp2[x,y,(Subscript[Z, 0])[x,y]]/.{f[x,y,(Subscript[Z, 0])[x,y]]->0}

(Subscript[Z, 2])[x,y]:=(1/(2 Derivative[1,0][(Subscript[Z, 0])][x,y] Derivative[0,0,1][f][x,y,(Subscript[Z, 0])[x,y]]))(-2 g[x,y,(Subscript[Z, 0])[x,y]] Derivative[0,1][(Subscript[Z, 1])][x,y]-2 (Subscript[Z, 1])[x,y] Derivative[1,0][(Subscript[Z, 1])][x,y] Derivative[0,0,1][f][x,y,(Subscript[Z, 0])[x,y]]-2 (Subscript[Z, 1])[x,y] Derivative[0,1][(Subscript[Z, 0])][x,y] Derivative[0,0,1][g][x,y,(Subscript[Z, 0])[x,y]]+2 (Subscript[Z, 1])[x,y] Derivative[0,0,1][h][x,y,(Subscript[Z, 0])[x,y]]-(((Subscript[Z, 1])[x,y])^2) Derivative[1,0][(Subscript[Z, 0])][x,y] Derivative[0,0,2][f][x,y,(Subscript[Z, 0])[x,y]])

(Subscript[Z, 2])[x,y]

Order (ϵ^3)

Coefficient[IDF,(ϵ^3)]

sp3[x_,y_,z_]:=Solve[g[x,y,z] Derivative[0,1][(Subscript[Z, 2])][x,y]+f[x,y,z] Derivative[1,0][(Subscript[Z, 3])][x,y]+(Subscript[Z, 3])[x,y] Derivative[1,0][(Subscript[Z, 0])][x,y] Derivative[0,0,1][f][x,y,z]+(Subscript[Z, 2])[x,y] Derivative[1,0][(Subscript[Z, 1])][x,y] Derivative[0,0,1][f][x,y,z]+(Subscript[Z, 1])[x,y] Derivative[1,0][(Subscript[Z, 2])][x,y] Derivative[0,0,1][f][x,y,z]+(Subscript[Z, 2])[x,y] Derivative[0,1][(Subscript[Z, 0])][x,y] Derivative[0,0,1][g][x,y,z]+(Subscript[Z, 1])[x,y] Derivative[0,1][(Subscript[Z, 1])][x,y] Derivative[0,0,1][g][x,y,z]-(Subscript[Z, 2])[x,y] Derivative[0,0,1][h][x,y,z]+(Subscript[Z, 1])[x,y] (Subscript[Z, 2])[x,y] Derivative[1,0][(Subscript[Z, 0])][x,y] Derivative[0,0,2][f][x,y,z]+(1/2) (((Subscript[Z, 1])[x,y])^2) Derivative[1,0][(Subscript[Z, 1])][x,y] Derivative[0,0,2][f][x,y,z]+(1/2) (((Subscript[Z, 1])[x,y])^2) Derivative[0,1][(Subscript[Z, 0])][x,y] Derivative[0,0,2][g][x,y,z]-(1/2) (((Subscript[Z, 1])[x,y])^2) Derivative[0,0,2][h][x,y,z]+(1/6) (((Subscript[Z, 1])[x,y])^3) Derivative[1,0][(Subscript[Z, 0])][x,y] Derivative[0,0,3][f][x,y,z]==0,(Subscript[Z, 3])[x,y]]

sp3[x,y,(Subscript[Z, 0])[x,y]]/.{f[x,y,(Subscript[Z, 0])[x,y]]->0}

(Subscript[Z, 3])[x,y]:=(1/(6 Derivative[1,0][(Subscript[Z, 0])][x,y] Derivative[0,0,1][f][x,y,(Subscript[Z, 0])[x,y]]))(-6 g[x,y,(Subscript[Z, 0])[x,y]] Derivative[0,1][(Subscript[Z, 2])][x,y]-6 (Subscript[Z, 2])[x,y] Derivative[1,0][(Subscript[Z, 1])][x,y] Derivative[0,0,1][f][x,y,(Subscript[Z, 0])[x,y]]-6 (Subscript[Z, 1])[x,y] Derivative[1,0][(Subscript[Z, 2])][x,y] Derivative[0,0,1][f][x,y,(Subscript[Z, 0])[x,y]]-6 (Subscript[Z, 2])[x,y] Derivative[0,1][(Subscript[Z, 0])][x,y] Derivative[0,0,1][g][x,y,(Subscript[Z, 0])[x,y]]-6 (Subscript[Z, 1])[x,y] Derivative[0,1][(Subscript[Z, 1])][x,y] Derivative[0,0,1][g][x,y,(Subscript[Z, 0])[x,y]]+6 (Subscript[Z, 2])[x,y] Derivative[0,0,1][h][x,y,(Subscript[Z, 0])[x,y]]-6 (Subscript[Z, 1])[x,y] (Subscript[Z, 2])[x,y] Derivative[1,0][(Subscript[Z, 0])][x,y] Derivative[0,0,2][f][x,y,(Subscript[Z, 0])[x,y]]-3 (((Subscript[Z, 1])[x,y])^2) Derivative[1,0][(Subscript[Z, 1])][x,y] Derivative[0,0,2][f][x,y,(Subscript[Z, 0])[x,y]]-3 (((Subscript[Z, 1])[x,y])^2) Derivative[0,1][(Subscript[Z, 0])][x,y] Derivative[0,0,2][g][x,y,(Subscript[Z, 0])[x,y]]+3 (((Subscript[Z, 1])[x,y])^2) Derivative[0,0,2][h][x,y,(Subscript[Z, 0])[x,y]]-(((Subscript[Z, 1])[x,y])^3) Derivative[1,0][(Subscript[Z, 0])][x,y] Derivative[0,0,3][f][x,y,(Subscript[Z, 0])[x,y]])

(Subscript[Z, 3])[x,y]

Example 5.4: Chua's System (Ch 5, page 88)

Vector field

f[x_,y_,z_]:=z-α (x^3)-β (x^2)-μ x

g[x_,y_,z_]:=-z

h[x_,y_,z_]:=-a x +y+b z

Taylor series expansion of the vector field

F[x_,y_,z_]:=f[x,y,z] + q(∂_z) f[x,y,z]+((q^2)/2)(∂_(z,z))f[x,y,z]+((q^3)/6)(∂_(z,z,z))f[x,y,z]

G[x_,y_,z_]:=g[x,y,z] + q(∂_z) g[x,y,z]+((q^2)/2)(∂_(z,z))g[x,y,z]+((q^3)/6)(∂_(z,z,z))g[x,y,z]

H[x_,y_,z_]:=h[x,y,z] + q(∂_z) h[x,y,z]+((q^2)/2)(∂_(z,z))h[x,y,z]+((q^3)/6)(∂_(z,z,z))h[x,y,z]

Invariance (Darboux - Fenichel)

Ordre (ϵ^0)

sp0[x_,y_,z_]:=Solve[f[x,y,z]==0,z]

sp0[x,y,z]

(Subscript[Z, 0])[x_,y_]:=(x^3) α+(x^2) β+x μ

(Subscript[Z, 0])[x,y]

Order (ϵ^1)

Coefficient[IDF,ϵ]

sp1[x_,y_,z_]:=Solve[-h[x,y,z]+g[x,y,z] Derivative[0,1][(Subscript[Z, 0])][x,y]+f[x,y,z] Derivative[1,0][(Subscript[Z, 1])][x,y]+(Subscript[Z, 1])[x,y] Derivative[1,0][(Subscript[Z, 0])][x,y] Derivative[0,0,1][f][x,y,z]==0,(Subscript[Z, 1])[x,y]]

sp1[x,y,z]

sp1[x,y,(Subscript[Z, 0])[x,y]]/.{f[x,y,(Subscript[Z, 0])[x,y]]->0}

(Subscript[Z, 1])[x_,y_]:=((-a x+y+b (x^3) α+b (x^2) β+b x μ)/(3 (x^2) α+2 x β+μ))

(Subscript[Z, 1])[x,y]

Order (ϵ^2)

Coefficient[IDF,(ϵ^2)]

Factor[sp2[x,y,(Subscript[Z, 0])[x,y]]/.{f[x,y,(Subscript[Z, 0])[x,y]]->0}]

(Subscript[Z, 2])[x_,y_]:=(1/(((3 (x^2) α+2 x β+μ))^4))(3 (a^2) (x^3) α-9 a (x^2) y α+6 x (y^2) α-9 a b (x^5) (α^2)+12 b (x^4) y (α^2)+9 (x^7) (α^3)+6 (b^2) (x^7) (α^3)-2 a x y β+2 (y^2) β-11 a b (x^4) α β+16 b (x^3) y α β+21 (x^6) (α^2) β+14 (b^2) (x^6) (α^2) β-2 a b (x^3) (β^2)+4 b (x^2) y (β^2)+16 (x^5) α (β^2)+10 (b^2) (x^5) α (β^2)+4 (x^4) (β^3)+2 (b^2) (x^4) (β^3)-(a^2) x μ+a y μ-8 a b (x^3) α μ+12 b (x^2) y α μ+15 (x^5) (α^2) μ+12 (b^2) (x^5) (α^2) μ-a b (x^2) β μ+4 b x y β μ+22 (x^4) α β μ+16 (b^2) (x^4) α β μ+8 (x^3) (β^2) μ+4 (b^2) (x^3) (β^2) μ+a b x (μ^2)+7 (x^3) α (μ^2)+6 (b^2) (x^3) α (μ^2)+5 (x^2) β (μ^2)+2 (b^2) (x^2) β (μ^2)+x (μ^3))

(Subscript[Z, 2])[x,y]

Order (ϵ^3)

Coefficient[IDF,(ϵ^3)]

sp3[x_,y_,z_]:=Solve[g[x,y,z] Derivative[0,1][(Subscript[Z, 2])][x,y]+f[x,y,z] Derivative[1,0][(Subscript[Z, 3])][x,y]+(Subscript[Z, 3])[x,y] Derivative[1,0][(Subscript[Z, 0])][x,y] Derivative[0,0,1][f][x,y,z]+(Subscript[Z, 2])[x,y] Derivative[1,0][(Subscript[Z, 1])][x,y] Derivative[0,0,1][f][x,y,z]+(Subscript[Z, 1])[x,y] Derivative[1,0][(Subscript[Z, 2])][x,y] Derivative[0,0,1][f][x,y,z]+(Subscript[Z, 2])[x,y] Derivative[0,1][(Subscript[Z, 0])][x,y] Derivative[0,0,1][g][x,y,z]+(Subscript[Z, 1])[x,y] Derivative[0,1][(Subscript[Z, 1])][x,y] Derivative[0,0,1][g][x,y,z]-(Subscript[Z, 2])[x,y] Derivative[0,0,1][h][x,y,z]+(Subscript[Z, 1])[x,y] (Subscript[Z, 2])[x,y] Derivative[1,0][(Subscript[Z, 0])][x,y] Derivative[0,0,2][f][x,y,z]+(1/2) (((Subscript[Z, 1])[x,y])^2) Derivative[1,0][(Subscript[Z, 1])][x,y] Derivative[0,0,2][f][x,y,z]+(1/2) (((Subscript[Z, 1])[x,y])^2) Derivative[0,1][(Subscript[Z, 0])][x,y] Derivative[0,0,2][g][x,y,z]-(1/2) (((Subscript[Z, 1])[x,y])^2) Derivative[0,0,2][h][x,y,z]+(1/6) (((Subscript[Z, 1])[x,y])^3) Derivative[1,0][(Subscript[Z, 0])][x,y] Derivative[0,0,3][f][x,y,z]==0,(Subscript[Z, 3])[x,y]]

Factor[sp3[x,y,(Subscript[Z, 0])[x,y]]/.{f[x,y,(Subscript[Z, 0])[x,y]]->0}]

(Subscript[Z, 3])[x_,y_]:=(1/(((3 (x^2) α+2 x β+μ))^7))(-54 (a^3) (x^5) (α^2)+252 (a^2) (x^4) y (α^2)-360 a (x^3) (y^2) (α^2)+162 (x^2) (y^3) (α^2)+171 (a^2) b (x^7) (α^3)-531 a b (x^6) y (α^3)+378 b (x^5) (y^2) (α^3)-216 a (x^9) (α^4)-171 a (b^2) (x^9) (α^4)+270 (x^8) y (α^4)+270 (b^2) (x^8) y (α^4)+270 b (x^11) (α^5)+54 (b^3) (x^11) (α^5)-6 (a^3) (x^4) α β+96 (a^2) (x^3) y α β-198 a (x^2) (y^2) α β+108 x (y^3) α β+222 (a^2) b (x^6) (α^2) β-810 a b (x^5) y (α^2) β+630 b (x^4) (y^2) (α^2) β-558 a (x^8) (α^3) β-423 a (b^2) (x^8) (α^3) β+720 (x^7) y (α^3) β+720 (b^2) (x^7) y (α^3) β+990 b (x^10) (α^4) β+198 (b^3) (x^10) (α^4) β+12 (a^2) (x^2) y (β^2)-32 a x (y^2) (β^2)+20 (y^3) (β^2)+48 (a^2) b (x^5) α (β^2)-304 a b (x^4) y α (β^2)+288 b (x^3) (y^2) α (β^2)-510 a (x^7) (α^2) (β^2)-326 a (b^2) (x^7) (α^2) (β^2)+690 (x^6) y (α^2) (β^2)+642 (b^2) (x^6) y (α^2) (β^2)+1410 b (x^9) (α^3) (β^2)+266 (b^3) (x^9) (α^3) (β^2)+4 (a^2) b (x^4) (β^3)-40 a b (x^3) y (β^3)+44 b (x^2) (y^2) (β^3)-192 a (x^6) α (β^3)-82 a (b^2) (x^6) α (β^3)+280 (x^5) y α (β^3)+220 (b^2) (x^5) y α (β^3)+970 b (x^8) (α^2) (β^3)+158 (b^3) (x^8) (α^2) (β^3)-24 a (x^5) (β^4)-8 a (b^2) (x^5) (β^4)+40 (x^4) y (β^4)+28 (b^2) (x^4) y (β^4)+320 b (x^7) α (β^4)+40 (b^3) (x^7) α (β^4)+40 b (x^6) (β^5)+4 (b^3) (x^6) (β^5)+36 (a^3) (x^3) α μ-90 (a^2) (x^2) y α μ+60 a x (y^2) α μ-6 (y^3) α μ+117 (a^2) b (x^5) (α^2) μ-483 a b (x^4) y (α^2) μ+396 b (x^3) (y^2) (α^2) μ-342 a (x^7) (α^3) μ-354 a (b^2) (x^7) (α^3) μ+450 (x^6) y (α^3) μ+594 (b^2) (x^6) y (α^3) μ+720 b (x^9) (α^4) μ+192 (b^3) (x^9) (α^4) μ+6 (a^3) (x^2) β μ-20 (a^2) x y β μ+14 a (y^2) β μ-50 (a^2) b (x^4) α β μ-140 a b (x^3) y α β μ+234 b (x^2) (y^2) α β μ-576 a (x^6) (α^2) β μ-451 a (b^2) (x^6) (α^2) β μ+810 (x^5) y (α^2) β μ+936 (b^2) (x^5) y (α^2) β μ+1980 b (x^8) (α^3) β μ+486 (b^3) (x^8) (α^3) β μ-12 (a^2) b (x^3) (β^2) μ-16 a b (x^2) y (β^2) μ+44 b x (y^2) (β^2) μ-292 a (x^5) α (β^2) μ-88 a (b^2) (x^5) α (β^2) μ+460 (x^4) y α (β^2) μ+382 (b^2) (x^4) y α (β^2) μ+1960 b (x^7) (α^2) (β^2) μ+392 (b^3) (x^7) (α^2) (β^2) μ-40 a (x^4) (β^3) μ-6 a (b^2) (x^4) (β^3) μ+80 (x^3) y (β^3) μ+56 (b^2) (x^3) y (β^3) μ+820 b (x^6) α (β^3) μ+110 (b^3) (x^6) α (β^3) μ+120 b (x^5) (β^4) μ+12 (b^3) (x^5) (β^4) μ-2 (a^3) x (μ^2)+2 (a^2) y (μ^2)-91 (a^2) b (x^3) α (μ^2)+135 a b (x^2) y α (μ^2)-30 b x (y^2) α (μ^2)-138 a (x^5) (α^2) (μ^2)-108 a (b^2) (x^5) (α^2) (μ^2)+210 (x^4) y (α^2) (μ^2)+282 (b^2) (x^4) y (α^2) (μ^2)+660 b (x^7) (α^3) (μ^2)+204 (b^3) (x^7) (α^3) (μ^2)-16 (a^2) b (x^2) β (μ^2)+30 a b x y β (μ^2)-4 b (y^2) β (μ^2)-118 a (x^4) α β (μ^2)+75 a (b^2) (x^4) α β (μ^2)+220 (x^3) y α β (μ^2)+112 (b^2) (x^3) y α β (μ^2)+1240 b (x^6) (α^2) β (μ^2)+278 (b^3) (x^6) (α^2) β (μ^2)-14 a (x^3) (β^2) (μ^2)+18 a (b^2) (x^3) (β^2) (μ^2)+50 (x^2) y (β^2) (μ^2)+20 (b^2) (x^2) y (β^2) (μ^2)+730 b (x^5) α (β^2) (μ^2)+74 (b^3) (x^5) α (β^2) (μ^2)+130 b (x^4) (β^3) (μ^2)+8 (b^3) (x^4) (β^3) (μ^2)+3 (a^2) b x (μ^3)-a b y (μ^3)-10 a (x^3) α (μ^3)+74 a (b^2) (x^3) α (μ^3)+30 (x^2) y α (μ^3)-42 (b^2) (x^2) y α (μ^3)+240 b (x^5) (α^2) (μ^3)+48 (b^3) (x^5) (α^2) (μ^3)+4 a (x^2) β (μ^3)+15 a (b^2) (x^2) β (μ^3)+10 x y β (μ^3)-8 (b^2) x y β (μ^3)+260 b (x^4) α β (μ^3)-14 (b^3) (x^4) α β (μ^3)+60 b (x^3) (β^2) (μ^3)-4 (b^3) (x^3) (β^2) (μ^3)+2 a x (μ^4)-a (b^2) x (μ^4)+30 b (x^3) α (μ^4)-18 (b^3) (x^3) α (μ^4)+10 b (x^2) β (μ^4)-4 (b^3) (x^2) β (μ^4))

(Subscript[Z, 3])[x,y]

to be continued ...