Example 6.14: Kapteyn-Bautin system (Ch 6, page 125)

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

System

Parameters

f[x_,y_]:=-y-b (x^2)-C x y+b (y^2)

g[x_,y_]:=x+A x y

Vector field

V={f[x,y],g[x,y]}

Fixed Points & Nullclines

fp=Solve[{f[(Subscript[x, 1]),(Subscript[y, 1])]==0,g[(Subscript[x, 1]),(Subscript[y, 1])]==0},{(Subscript[x, 1]),(Subscript[y, 1])}]

fp1=Sort[fp[[1]]];

{(Subscript[x, 1]),(Subscript[y, 1])}={Extract[fp1,{1}][[2]],Extract[fp1,{2}][[2]]}

fp2=Sort[fp[[2]]];

{(Subscript[x, 2]),(Subscript[y, 2])}={Extract[fp2,{1}][[2]],Extract[fp2,{2}][[2]]}

fp3=Sort[fp[[3]]];

{(Subscript[x, 3]),(Subscript[y, 3])}={Extract[fp3,{1}][[2]],Extract[fp3,{2}][[2]]}

fp4=Sort[fp[[4]]];

{(Subscript[x, 4]),(Subscript[y, 4])}={Extract[fp4,{1}][[2]],Extract[fp4,{2}][[2]]}

Jacobian

a11=Simplify[D[f[x,y],x]]

a12=Simplify[D[f[x,y],y]]

a21=Simplify[D[g[x,y],x]]

a22=Simplify[D[g[x,y],y]]

J={{a11,a12},{a21,a22}}

MatrixForm[J]

Acceleration

γ=J . V

Fixed Points Stability - Eigenvalues

Eigenvalues[J]/.{x-> (Subscript[x, 1]),y->(Subscript[y, 1])}

Eigenvalues[J]/.{x-> (Subscript[x, 2]),y->(Subscript[y, 2])}

Flow Curvature Manifold

ϕ=FullSimplify[Det[{V,γ}]]

Solve[(-1+C x (-1+b y)+b ((A+b) (x^2)+2 y-b (y^2)))==0,y]

FullSimplify[-y+-((-2 b-b C x-b (Sqrt[4 A b+4 (b^2)+(C^2)]) x)/(2 (b^2)))]

FullSimplify[-y+-((-2 b-b C x+b (Sqrt[4 A b+4 (b^2)+(C^2)]) x)/(2 (b^2)))]

ϕ=(((1+A y))^p) (((((2+(C+(Sqrt[4 b (A+b)+(C^2)])) x-2 b y)/(2 b))))^q)(((((2+C x-(Sqrt[4 b (A+b)+(C^2)]) x-2 b y)/(2 b))))^r)

Flow Curvature Invariance

Normal Vector

η={D[ϕ,x],D[ϕ,y]}

Invariance

Ψ=FullSimplify[V . η]

Collect[(2 A p x-2 b (q+r) x-((Sqrt[4 b (A+b)+(C^2)]) (q-r)+C (q+r)) y),{x,y}]

Solve[{(2 A p-2 b (q+r)) ==0,(-(Sqrt[4 b (A+b)+(C^2)]) (q-r)-C (q+r)) ==0},{q,r}]

Let' s choose arbitrarily p = (1/A)

Factor[{q->((A p)/(2 b))-((A C p)/(2 b (Sqrt[4 A b+4 (b^2)+(C^2)]))),r->-((-A C p-A (Sqrt[4 A b+4 (b^2)+(C^2)]) p)/(2 b (Sqrt[4 A b+4 (b^2)+(C^2)])))}/.{p-> (1/A)}]

ϕ=Factor[(((1+A y))^p) (((((2+(C+(Sqrt[4 b (A+b)+(C^2)])) x-2 b y)/(2 b))))^q)(((((2+C x-(Sqrt[4 b (A+b)+(C^2)]) x-2 b y)/(2 b))))^r)/.{q->((-C+(Sqrt[4 A b+4 (b^2)+(C^2)]))/(2 b (Sqrt[4 A b+4 (b^2)+(C^2)]))),r->((C+(Sqrt[4 A b+4 (b^2)+(C^2)]))/(2 b (Sqrt[4 A b+4 (b^2)+(C^2)]))),p-> (1/A)}]

η={D[ϕ,x],D[ϕ,y]}

Invariance

Ψ=FullSimplify[V . η]