restart;

Eqn1:=(c^2+c)^2+c=0;

S:=[solve(Eqn1,c)];

Digits:=20;

c:=evalf(S[2]);

Q:=x->x^2+c;

Q3:=Q(Q(Q(x)));

B1:=plot([Q(x),Q3,x],x=-2..2,y=-2..2):

Orbit:=[[0,0],[0,c],[c,c],[c,c^2+c],[c^2+c,c^2+c],[c^2+c,0]];

A1:=plot(Orbit,color=black):

with(plots):

display({A1,B1});

Center the orbit about the point the text refers to as alpha. Find alpha hat:

hstep:=0.1; int1a:=c^2+c-hstep; intb:=c^2+c+hstep;

B2:=plot([Q3,x],x=int1a..intb):

S2:=[fsolve(Q3-x=0,x,1.2..1.4)]; #Find the fixed point- the second is c^2+c

S3:=[fsolve(Q3=S2[1],x,1.2..1.4)];

Orbit2:=[[S3[1],S3[1]],[S3[2],S3[1]],[S3[2],S3[2]]];

A2:=plot(Orbit2,color=black):

alpha:=S3[1]; alphahat:=S3[2];

display({A2,B2});

Now look at the same information about the origin:

hstep:=0.25; int1a:=-hstep; intb:=hstep;

B3:=plot([Q3,x],x=int1a..intb):

S2:=[fsolve(Q3-x=0,x,-0.2..0.2)]; #Find the fixed point- the second is c^2+c

S3:=[fsolve(Q3=S2[1],x,-0.2..0.2)];

Orbit2:=[[S3[1],S3[1]],[S3[2],S3[1]],[S3[2],S3[2]]];

A3:=plot(Orbit2,color=black):

beta:=S3[1]; betahat:=S3[2];

display({A3,B3});

Now look at the same info about the point gamma

hstep:=0.01; int1a:=c-2*hstep; intb:=c+2*hstep;

B4:=plot([Q3,x],x=int1a..intb):

S2:=[fsolve(Q3-x=0,x,int1a..intb)]; #Find the fixed point- the second is c^2+c

S3:=[fsolve(Q3=S2[2],x,int1a..intb)];

Orbit2:=[[S3[2],S3[2]],[S3[1],S3[2]],[S3[1],S3[1]]];

A4:=plot(Orbit2,color=black):

display({A4,B4});

ggammahat:=S3[1]; ggamma:=S3[2];

Check the mappings numerically:

Q(alpha)=beta, Q(beta)=gamma Q(gamma)=alpha (repelling three cycle)

Q(alpha hat)=beta hat Q(beta hat)=gamma Q(gamma hat)=alpha hat

Q(alphahat)-betahat; Q(betahat)-ggamma; Q(ggammahat)-alphahat;

Q(alpha)-beta; Q(beta)-ggamma; Q(ggamma)-alpha;