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Question: Anxious!! How to express the part of the unstable curve with a dashline. Hope someone can help me achieve this.

Question:Anxious!! How to express the part of the unstable curve with a dashline. Hope someone can help me achieve this.

ssllys 35 Maple
linestyle implicitplot plotting
August 23 2018
1

How to express the part of  unstable curve with a dashline.   Hope someone can help me achieve this.  

>  

(3/8)*alpha[2](t[1], t[2])*A(t[2])^3*exp(I*B(t[2]))*exp(-I*B(t[2]))+((1/2)*I)*alpha[1](t[1], t[2])*A(t[2])*exp(I*B(t[2]))*exp(-I*B(t[2]))+I*(diff(A(t[2]), t[2]))*exp(I*B(t[2]))*exp(-I*B(t[2]))-A(t[2])*(diff(B(t[2]), t[2]))*exp(I*B(t[2]))*exp(-I*B(t[2]))-(1/2)*f0*exp(I*sigma*t[2])*exp(-I*B(t[2]))

 (13)
> 

combine(%, 'exp');

(3/8)*alpha[2](t[1], t[2])*A(t[2])^3+((1/2)*I)*alpha[1](t[1], t[2])*A(t[2])+I*(diff(A(t[2]), t[2]))-A(t[2])*(diff(B(t[2]), t[2]))-(1/2)*f0*exp(I*sigma*t[2]-I*B(t[2]))

 (14)
> 

subs(I*B(t2)=I*sigma*t2-I*C(t2),B(t2)=sigma*t2-C(t2), %);

(3/8)*alpha[2](t[1], t[2])*A(t[2])^3+((1/2)*I)*alpha[1](t[1], t[2])*A(t[2])+I*(diff(A(t[2]), t[2]))-A(t[2])*(diff(sigma*t[2]-C(t[2]), t[2]))-(1/2)*f0*exp(I*sigma*t[2]-I*(sigma*t[2]-C(t[2])))

 (15)
> 

conds := combine(%, 'exp');

(3/8)*alpha[2](t[1], t[2])*A(t[2])^3+((1/2)*I)*alpha[1](t[1], t[2])*A(t[2])+I*(diff(A(t[2]), t[2]))-A(t[2])*(sigma-(diff(C(t[2]), t[2])))-(1/2)*f0*exp(I*sigma*t[2]-I*(sigma*t[2]-C(t[2])))

 (16)
> 

convert(conds, 'trig');

(3/8)*alpha[2](t[1], t[2])*A(t[2])^3+((1/2)*I)*alpha[1](t[1], t[2])*A(t[2])+I*(diff(A(t[2]), t[2]))-A(t[2])*(sigma-(diff(C(t[2]), t[2])))-(1/2)*f0*(cos(C(t[2]))+I*sin(C(t[2])))

 (17)
> 

cond1 := collect(coeff(%, I, 0), [diff(A(t2), t2), cos(C(t2))]);

(3/8)*alpha[2](t[1], t[2])*A(t[2])^3-A(t[2])*(sigma-(diff(C(t[2]), t[2])))-(1/2)*f0*cos(C(t[2]))

 (18)
> 

cond2 := collect(coeff(`%%`, I, 1), [diff(A(t2), t2), sin(C(t2)), cos(C(t2))]);

(1/2)*alpha[1](t[1], t[2])*A(t[2])+diff(A(t[2]), t[2])-(1/2)*f0*sin(C(t[2]))

 (19)
> 

A(t2):= A; C(t2):=C; cond1 = 0; cond2 =0;

A

  

C

  

(3/8)*alpha[2](t[1], t[2])*A^3-A*sigma-(1/2)*f0*cos(C) = 0

  

(1/2)*alpha[1](t[1], t[2])*A-(1/2)*f0*sin(C) = 0

 (20)
> 

remove(has, cond1, cos)^2+remove(has, cond2, sin)^2 = simplify(select(has, cond1, cos)^2+select(has, cond2, sin)^2, 'trig');

((3/8)*alpha[2](t[1], t[2])*A^3-A*sigma)^2+(1/4)*alpha[1](t[1], t[2])^2*A^2 = (1/4)*f0^2

 (21)
> 

af_eq:=%;

((3/8)*alpha[2](t[1], t[2])*A^3-A*sigma)^2+(1/4)*alpha[1](t[1], t[2])^2*A^2 = (1/4)*f0^2

 (22)
> 

 alpha[1]:=0.0087;alpha[2]:=2.5871;f0:=f[0]

0.87e-2

  

2.5871

  

f[0]

 (23)
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##
> 

 

> 

 

> 

e:='e':    f0:='f0':   alpha[1]:='alpha[1]':   alpha[2]:='alpha[2]':
> 

cond1;

(3/8)*alpha[2](t[1], t[2])*A^3-A*sigma-(1/2)*f0*cos(C)

 (24)
> 

cond2;

(1/2)*alpha[1](t[1], t[2])*A-(1/2)*f0*sin(C)

 (25)

``

> 

linalg[jacobian]([-cond2,-cond1/A],[A,C]);

Matrix(2, 2, {(1, 1) = -(1/2)*alpha[1](t[1], t[2]), (1, 2) = (1/2)*f0*cos(C), (2, 1) = -((9/8)*alpha[2](t[1], t[2])*A^2-sigma)/A+((3/8)*alpha[2](t[1], t[2])*A^3-A*sigma-(1/2)*f0*cos(C))/A^2, (2, 2) = -(1/2)*f0*sin(C)/A})

 (26)
> 

 

``

> 

subs(cos(C)=solve(cond1,cos(C)),sin(C)=solve(cond2,sin(C)),%);

Matrix(2, 2, {(1, 1) = -(1/2)*alpha[1](t[1], t[2]), (1, 2) = (1/8)*A*(3*alpha[2](t[1], t[2])*A^2-8*sigma), (2, 1) = -((9/8)*alpha[2](t[1], t[2])*A^2-sigma)/A+((3/8)*alpha[2](t[1], t[2])*A^3-A*sigma-(1/8)*A*(3*alpha[2](t[1], t[2])*A^2-8*sigma))/A^2, (2, 2) = -(1/2)*alpha[1](t[1], t[2])})

 (27)

``

> 

map(simplify,%):

``

> 

linalg[charpoly](%,lambda):

``

> 

p:=collect(%,lambda);

(1/4)*alpha[1](t[1], t[2])^2+lambda*alpha[1](t[1], t[2])+lambda^2+(27/64)*A^4*alpha[2](t[1], t[2])^2-(3/2)*A^2*alpha[2](t[1], t[2])*sigma+sigma^2

 (28)
> 

 

``

> 

coeff(p,lambda,0);

(1/4)*alpha[1](t[1], t[2])^2+(27/64)*A^4*alpha[2](t[1], t[2])^2-(3/2)*A^2*alpha[2](t[1], t[2])*sigma+sigma^2

 (29)
> 

 

``

> 

stab_cond:=%;

(1/4)*alpha[1](t[1], t[2])^2+(27/64)*A^4*alpha[2](t[1], t[2])^2-(3/2)*A^2*alpha[2](t[1], t[2])*sigma+sigma^2

 (30)
> 

 

``

> 

af_eq;

((3/8)*alpha[2](t[1], t[2])*A^3-A*sigma)^2+(1/4)*alpha[1](t[1], t[2])^2*A^2 = (1/4)*f0^2

 (31)

``

> 

with(plots, implicitplot,implicitplot3d):
> 

 

> 

 f0 :=0.1; alpha[1]:=0.0087;alpha[2]:=2.5871;

.1

  

0.87e-2

  

2.5871

 (32)
> 

 

``

> 

p1:=implicitplot(stab_cond,sigma = -1.2 .. 2, A = 0 .. 2,  numpoints = 20000, axes = box, axesfont=[SYMBOL, 14],labels = [sigma, A], labelfont = [SYMBOL, 16],color="Green",tickmarks=[9,12],thickness=2,linestyle=dash):

``

> 

with(plots,textplot):    with(plots,display):

``

> 

ps1:=textplot([0.75, 0.65, "Unstable region"], 'align' = {'above', 'right'},rotation = 0.6, font=[Times,bold,12]):

``

> 

pu1:=implicitplot(af_eq,sigma = -1.2 .. 2, A = 0 .. 2,  numpoints = 20000, axes = box, axesfont=[SYMBOL, 14],labels = [sigma, A], labelfont = [SYMBOL, 16],color="red",tickmarks=[9,12],thickness=2):
> 

pp1:=display({p1,ps1,pu1});

 
> 

Like this picture.

 


 

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