@Zeineb  My answer is incomplete and only applies to case b = 0 (I just did not notice this number b). k=0 will also be the solution for this case. But in the general case, for an arbitrary b, the answer will be much more complicated and will depend on  b . I'm working on it now and soon my answer above will be updated.

@tizozadoxo  This linear system has the unique solution  (0, 0, 0) . But maybe you are confusing the dimension of the solution space with the number of solutions. From a geometric point of view, every solution of the system is a point in the space  R^3 . But a point has dimension  0 .

@Carl Love  I understood the dots above the variables in the first equation of the system as derivatives.

See this thread 

@Lali_miani   Because  u(1)=sin(u(0)), u(2)=sin(u(1))=sin(sin(u(0)))  and so on.

@Lali_miani  For example  (sin@@3)(x)  means  sin(sin(sin(x))) :

expand((sin@@3)(x));

                     sin(sin(sin(x)))

@Lali_miani   But  sin(1.5) = 0.9974949866  not  0.84

@vanzzy  I changed the colors a bit to make them all different. Some graphics in separate ranges merge. For example, red with yellow and black with blue.

Download file2_(1).mw

@Chouette  It seems that a simple procedure called  Intersect  solves your problem. The procedure uses  evala  command, which apparently reliably recognizes the coincidence of algebraic numbers. Numbers can be given both explicitly (in radicals) and implicitly (as  RootOf  placeholder). So in the last example below it is shown that each root of the first equation is also the root of the second equation. The procedure works as follows: first, remove the duplicate elements (if any) from the first set, and then compare each element of the first set with each element of the second set, and so we get their intersection.

restart;
Intersect:=proc(S1::set,S2::set)
local T, m:=nops(S1), n:=nops(S2), SS1, m1;
T:={seq(seq(`if`(evala(S1[i]-S1[j])=0,S1[i],NULL), j=i+1..m), i=1..m-1)};
SS1:=S1 minus T;
m1:=nops(SS1);
{seq(seq(`if`(evala(SS1[i]-S2[j])=0,SS1[i],NULL), j=i..n), i=1..m1)};
end proc:

Examples of use:

Intersect({sqrt(6)},{sqrt(2)*sqrt(3)});
                       
Sol1 := {solve( R^2-sqrt(2)*R-1 )};
Sol2 := {solve( (sqrt(3)+1)*R^2 = sqrt(3)-1 )};
Intersect(Sol1,Sol2);
         
Intersect({sqrt(7)+1}, {6/sqrt(8-2*sqrt(7))});
                     
S1:={solve(x^17+2*x^11+4*x^5-1=0)}:
S2:={solve(x^23-x^6-8*x^5+2=0)}:
Intersect(S1,S2);
   

Download Intersect.mw

@Chouette  Use  rationalize(simplify(...))  for algebraic numbers:

Sol1 := {solve( R^2-sqrt(2)*R-1 )};
Sol2 := {solve( (sqrt(3)+1)*R^2 = sqrt(3)-1 )};
rationalize(simplify(Sol1));
rationalize(simplify(Sol2));

@JSalisbury  Should be  Point[i] := ...

@vv  I did not understand why it happened. Now I did  restart;  and got the same result as sand's one:

`-`(a, b);
                           a - b

@sand15  In Maple 2018.3 (worksheet mode, Standard GUI)  I get  that  Matrix(3, 3, `.`) = Matrix(3, 3, `*`)

Matrix(3, 3, `.`);
Matrix(3, 3, `*`);
                            

@Teep  You can do this automatically as follows

MaplePrimes_network_labels_new.mw

@Teep   See help on the  GraphTheory:-RelabelVertices  command.