Výpočet Jakobiho matice pro dvojperiodický bod (±(Subscript[X, 1]),∓(Subscript[X, 1]))

f[x_,y_]=x Exp[r-r x-s y]
g[x_,y_]=y Exp[r-r y-s x]

Simplify[f[f[x,y],g[x,y]]]

Simplify[g[f[x,y],g[x,y]]]

u1=2 r-r x(1+Exp[r-r x-s y])-s y (1+Exp[r-s x-r y])
u2=2 r-r y(1+Exp[r-s x-r y])-s x (1+Exp[r-r x-s y])

d11=D[u1,x]

d11a=D[u1,x]/.{r-r x-s y->2(Subscript[X, 1]),r-s x-r y->-2(Subscript[X, 1]),x->r/(r+s)(1-Tanh[(Subscript[X, 1])]),y->r/(r+s)(1+Tanh[(Subscript[X, 1])])}

D[u1,y]

D[u2,x]

D[u2,y]

A12a=r/(r+s)(1-Tanh[(Subscript[X, 1])])(-(1+(E^(-2(Subscript[X, 1])))) s+(E^(2(Subscript[X, 1]))) r s r/(r+s)(1-Tanh[(Subscript[X, 1])])+(E^(-2(Subscript[X, 1]))) r s r/(r+s)(1+Tanh[(Subscript[X, 1])]))

A21a=r/(r+s)(1+Tanh[(Subscript[X, 1])])(-(1+(E^(2(Subscript[X, 1])))) s+(E^(2(Subscript[X, 1]))) r s r/(r+s)(1-Tanh[(Subscript[X, 1])])+(E^(-2(Subscript[X, 1]))) r s r/(r+s)(1+Tanh[(Subscript[X, 1])]))

A12b=r/(r+s)(1+Tanh[(Subscript[X, 1])])(-(1+(E^(+2(Subscript[X, 1])))) s+(E^(-2(Subscript[X, 1]))) r s r/(r+s)(1+Tanh[(Subscript[X, 1])])+(E^(2(Subscript[X, 1]))) r s r/(r+s)(1-Tanh[(Subscript[X, 1])]))

A21b=r/(r+s)(1-Tanh[(Subscript[X, 1])])(-(1+(E^(-2(Subscript[X, 1])))) s+(E^(-2(Subscript[X, 1]))) r s r/(r+s)(1+Tanh[(Subscript[X, 1])])+(E^(2(Subscript[X, 1]))) r s r/(r+s)(1-Tanh[(Subscript[X, 1])]))

a12a=FullSimplify[A12a]

a21a=FullSimplify[A21a]

a12b=FullSimplify[A12b]

a21b=FullSimplify[A21b]

A11a=1+r/(r+s)(1-Tanh[(Subscript[X, 1])])(d11a)

A11a se dá upravit pomocí vzorců 
(e^(2 (Subscript[X, 1])))  (1-Tanh[(Subscript[X, 1])])=(1+Tanh[(Subscript[X, 1])])
(e^(-2 (Subscript[X, 1])))  (1+Tanh[(Subscript[X, 1])])=(1-Tanh[(Subscript[X, 1])])
((Sech[(Subscript[X, 1])])^2) = 1-((Tanh[(Subscript[X, 1])])^2)
(1-Tanh[(Subscript[X, 1])]) (1+(e^(2 (Subscript[X, 1]))))= 2

A11a=1-((2 (r^2))/(r+s))+(((r^2) (r-s))/(r+s))((Sech[(Subscript[X, 1])])^2)+((2 (r^2)(s^2))/(((r+s))^2))-2(((r^2)(s^2)Tanh[(Subscript[X, 1])])/(((r+s))^2))

A22a=1+r/(r+s)(1+Tanh[(Subscript[X, 1])])(-(1+(E^(-2(Subscript[X, 1])))) r+(E^(2(Subscript[X, 1]))) (s^2) r/(r+s)(1-Tanh[(Subscript[X, 1])])+(E^(-2(Subscript[X, 1]))) (r^2) r/(r+s)(1+Tanh[(Subscript[X, 1])]))

A11b=1+r/(r+s)(1+Tanh[(Subscript[X, 1])])(-(1+(E^(-2(Subscript[X, 1])))) r+(E^(-2(Subscript[X, 1]))) (r^2) r/(r+s)(1+Tanh[(Subscript[X, 1])])+(E^(2(Subscript[X, 1]))) (s^2) r/(r+s)(1-Tanh[(Subscript[X, 1])]))

A22b=1+r/(r+s)(1-Tanh[(Subscript[X, 1])])(-(1+(E^(2(Subscript[X, 1])))) r+(E^(-2(Subscript[X, 1]))) (s^2) r/(r+s)(1+Tanh[(Subscript[X, 1])])+(E^(2(Subscript[X, 1]))) (r^2) r/(r+s)(1-Tanh[(Subscript[X, 1])]))

A22b se dá upravit pomocí vzorců 
(e^(2 (Subscript[X, 1])))  (1-Tanh[(Subscript[X, 1])])=(1+Tanh[(Subscript[X, 1])])
(e^(-2 (Subscript[X, 1])))  (1+Tanh[(Subscript[X, 1])])=(1-Tanh[(Subscript[X, 1])])
((Sech[(Subscript[X, 1])])^2) = 1-((Tanh[(Subscript[X, 1])])^2)
(1-Tanh[(Subscript[X, 1])]) (1+(e^(2 (Subscript[X, 1]))))= 2

A22b=1-((2 (r^2))/(r+s))+(((r^2) (r-s))/(r+s))((Sech[(Subscript[X, 1])])^2)+((2 (r^2)(s^2))/(((r+s))^2))-2(((r^2)(s^2)Tanh[(Subscript[X, 1])])/(((r+s))^2))

a11a=A11a

a22a=MapAt[FullSimplify,A22a,2]

a11b=MapAt[FullSimplify,A11b,2]

a22b=A22b

a11a  a  a22a  se liší jen o člen  ∓(((r^2)(s^2)Tanh[(Subscript[X, 1])])/(((r+s))^2)), podobně a11b  a  a22b  se liší jen o člen  ±2(((r^2)(s^2)Tanh[(Subscript[X, 1])])/(((r+s))^2))

poma=Simplify[(a22a-a11a)/2]

A=a11a+2(((r^2)(s^2)Tanh[(Subscript[X, 1])])/(((r+s))^2))

B=2(((r^2)(s^2)Tanh[(Subscript[X, 1])])/(((r+s))^2))

a12a/b=(C±E) (e^(∓2 (Subscript[X, 1])))  ,   a21a/b=(C∓E) (e^(±2 (Subscript[X, 1])))

Cc=((2 (r^3) s )/(((r+s))^2)) -((2 r s )/(r+s))

Ee=((2 (r^3) s  Tanh[(Subscript[X, 1])])/(((r+s))^2))

MatrixForm[{{A∓B,(Cc ±Ee)(E^(∓ 2(Subscript[X, 1])))},{(Cc ∓Ee) (E^(± 2(Subscript[X, 1]))),A±B}}]

J1a={{A-B,(Cc +Ee) (E^(-2 (Subscript[X, 1]))) },{(Cc -Ee) (E^(+2 (Subscript[X, 1]))),A+B}}

J1b={{A+B,(Cc -Ee)(E^(+2 (Subscript[X, 1]))) },{(Cc +Ee) (E^(-2 (Subscript[X, 1]))),A-B}}

FullSimplify[Det[J1a]]

Δ  =FullSimplify[A^2-B^2-Cc^2+Ee^2]

Musíme si determinant upravit sami. Použíjeme vytahy
 1+Cosh[2 (Subscript[X, 1])] = 2 ((Cosh[(Subscript[X, 1])])^2)
Sech[(Subscript[X, 1])]Cosh[(Subscript[X, 1])] = 1

FullSimplify[ 1+Cosh[2 (Subscript[X, 1])] ]

Sech[(Subscript[X, 1])]Cosh[(Subscript[X, 1])]

Δ=(((((2 (r^3)-2 (r^2) s+2(r-2 (r^2)+s) ((Cosh[ (Subscript[X, 1])])^2)))^2) ((Sech[(Subscript[X, 1])])^4))/(4 (((r+s))^2)))

Δ=((((( (r^2)(r-s)((Sech[(Subscript[X, 1])])^2)+(r+s)-2 (r^2)))^2) )/( (((r+s))^2)))

Δ=(((1+ (r^2)(((r-s))/((r+s)))((Sech[(Subscript[X, 1])])^2)-2 ((r^2)/((r+s)))))^2)

σ=2 A

diskriminant=FullSimplify[σ^2-4 Δ]

Zjistíme, kdy je diskriminant rovný 0, platí dis=16(( (r^2) (s^2))/(((r+s))^2))(A-(( (r^2) (s^2))/(((r+s))^2))), 
dis = 0 právě tehdy když A-(( (r^2) (s^2))/(((r+s))^2)) =0

A-(( (r^2) (s^2))/(((r+s))^2))

p2=FullSimplify[A-(( (r^2) (s^2))/(((r+s))^2))/.{s->((r (r Tanh[t]-2 t))/(r Tanh[t]+2 t)),(Subscript[X, 1])->t}]

CoefficientList[p2,r]

res=Solve[a r^2+b r+c==0,r]

res/.{a->(1/4),b->-1-t Coth[t]+2 t Csch[t] Sech[t],c->1-2 t Coth[t]+(t^2) ((Coth[t])^2)}

Simplify[%]

Simplify[Tanh[t]/(2 t)(2+2 t Coth[t]-4 t Csch[t] Sech[t])]

Upravíme:
(Sqrt[ t (-t Sech[t]^2+Tanh[t])])=t (Sqrt[Tanh[t]^2-1 + Tanh[t]/t])
r =(( 2t)/(Tanh[t]))[1-2 ((Sech[t])^2)+((Tanh[t] )/t)+2 Tanh[t] (Sqrt[Tanh[t]^2-1 + Tanh[t]/t])] = (( 2t)/(Tanh[t]))[1-2 ((Sech[t])^2)+((Tanh[t])/t) + 2 Tanh[t]  (Sqrt[Tanh[t]^2-1 + Tanh[t]/t])] =
 = (( 2t)/(Tanh[t])) (([Tanh[t]+(Sqrt[Tanh[t]^2-1 + Tanh[t]/t])])^2)

Nakreslíme křivku

r1[t_]=((2 t)/(Tanh[t]))(((Tanh[t]+(Sqrt[((Tanh[t])^2)+((Tanh[t])/t)-1])))^2)

ParametricPlot[ {r1[t],((r1[t](r1[t] Tanh[t]-2 t) )/((r1[t] Tanh[t]+2 t)))},{t,0.1,1}]

Předpokládejme, že diskriminant ⩾ 0 (vlastní čísla jsou reálná)

(Subscript[λ, 1])= A +2 (( r s)/((r+s)))(Sqrt[A-(( (r^2) (s^2))/(((r+s))^2))])

(Subscript[λ, 2])= A -2 (( r s)/((r+s)))(Sqrt[A-(( (r^2) (s^2))/(((r+s))^2))])

Musíme zjistit, zda
-1< (Subscript[λ, 12]) <1

Pro oblast  mezi křivkami (Subscript[C, 1]) (Subscript[C, 2])  a pro r<(Subscript[g, r]) a s< (Subscript[g, s]) platí, že r/s(1+((s-r)/2)((Sech[(Subscript[X, 1])])^2))>1 ⟹-((2 (r^2))/(r+s))+(((r^2) (r-s) ((Sech[(Subscript[X, 1])])^2))/(r+s)) < - ((2 s r)/((r+s))) ⟹  (Subscript[λ, 1])<1
Platí 0 < (Subscript[λ, 2])< (Subscript[λ, 1]) ⟹   0< (Subscript[λ, 2])<(Subscript[λ, 1])<1 
Pro oblast  mezi křivkami (Subscript[C, 1]) (Subscript[C, 2])  a pro r>(Subscript[g, r]) a s> (Subscript[g, s]) platí, že r/s(1+((s-r)/2)((Sech[(Subscript[X, 1])])^2))<-1 ⟹-((2 (r^2))/(r+s))+(((r^2) (r-s) ((Sech[(Subscript[X, 1])])^2))/(r+s)) >  ((2 s r)/((r+s))) ⟹  (Subscript[λ, 2])>1
Platí 0 < (Subscript[λ, 2])< (Subscript[λ, 1]) ⟹   1< (Subscript[λ, 2])<(Subscript[λ, 1])

Teď se pokusím vyřešit tu komplexní oblast. Ptám se, kdy |(Subscript[λ, 1])| = 1, t.j.  |(Subscript[λ, 1])((|)^2) = 1

rov1=A^2+4r^2s^2/(r+s)^2(A-r^2s^2/(r+s)^2)==1/.{(Subscript[X, 1])->t,s->r(r Tanh[t]-2 t)/(r Tanh[t]+2 t)}

FullSimplify[(((2 t+r Tanh[t]))^2) (((r+((r (-2 t+r Tanh[t]))/(2 t+r Tanh[t]))))^2)]

FullSimplify[r+((r (-2 t+r Tanh[t]))/(2 t+r Tanh[t]))]

FullSimplify[r-((r (-2 t+r Tanh[t]))/(2 t+r Tanh[t]))]

rov1=((4 (r^4) (((-2 t+r Tanh[t]))^2) (1+(((r^4) (((-2 t+r Tanh[t]))^2))/( 4r^4 ((Tanh[t])^2)))-((2 (r^2)(2 t+r Tanh[t]))/(2 (r^2) Tanh[t]))+(((r^2) ((Sech[t])^2) (4 r t))/(2 (r^2) Tanh[t]))))/( 4r^4 ((Tanh[t])^2)))+(((1+((2 (r^4) (((-2 t+r Tanh[t]))^2))/( 4r^4 ((Tanh[t])^2)))-((2 (r^2)(2 t+r Tanh[t]))/(2 (r^2) Tanh[t]))+(((r^2) ((Sech[t])^2) (4 r t))/(2 (r^2) Tanh[t]))))^2)==1

rov1=((Coth[t])^2) (((-2 t+r Tanh[t]))^2) (1+2 (r^2) Csch[t] Sech[t]+(1/4) ((Coth[t])^2) (((-2 t+r Tanh[t]))^2)-Coth[t] (2 t+r Tanh[t]))+2 (2 r t  Csch[t] Sech[t]+(1/2) ((Coth[t])^2) (((-2 t+r Tanh[t]))^2)-Coth[t] (2 t+r Tanh[t]))+(((2 r t Csch[t] Sech[t]+(1/2) ((Coth[t])^2) (((-2 t+r Tanh[t]))^2)-Coth[t] (2 t+r Tanh[t])))^2)==0

rov2=Expand[rov1]

list=CoefficientList[2 r+3 (r^2)-2 (r^3)+((r^4)/2)-4 t Coth[t]-4 r t Coth[t]+4 (r^2) t Coth[t]-4 (r^3) t Coth[t]+12 (t^2) ((Coth[t])^2)+8 r (t^2) ((Coth[t])^2)+12 (r^2) (t^2) ((Coth[t])^2)-16 (t^3) ((Coth[t])^3)-16 r (t^3) ((Coth[t])^3)+8 (t^4) ((Coth[t])^4)-8 (r^3) t ((Csch[t])^2)-8 r (t^2) ((Csch[t])^2)-8 (r^2) (t^2) ((Csch[t])^2)+8 (r^2) (t^2) Coth[t] ((Csch[t])^2)+8 r (t^3) Coth[t] ((Csch[t])^2)+2 (r^4) Csch[t] Sech[t]+4 r t Csch[t] Sech[t]-4 (r^2) t Csch[t] Sech[t]+2 (r^3) t Csch[t] Sech[t]+4 (r^2) (t^2) ((Csch[t])^2) ((Sech[t])^2),r]

Simplify[list[[1]]]

Simplify[list[[2]]]

Simplify[list[[3]]]

Simplify[list[[4]]]

Simplify[list[[5]]]