Gfrktyl

Assumed we have some kind of squared 3D graph like this one:

Minimum Working Example (MWE):

documentclass{standalone}

usepackage{tikz, pgfplots}



begin{document}



begin{tikzpicture}

   begin{axis}[samples=20]

      addplot3[surf, domain=-2:2] {-x^2-y^2};

   end{axis}

end{tikzpicture}



end{document}

Screenshot of the result:

How can I replace the current graph with some 4th degree polynomial formula in both directions x and y, e.g. -1/3*x^4+x^2 and -1/3*y^4+y^2?

Draft of the desired result

In the end it should look like that:

The draft is more or less a circular volcano with a crater in the middle, I hope you can imagine. :-)

I don't know why, but several approaches with...

addplot3[surf, domain=-2:2] {(-1/3*y^4+y^2)*(-1/3*y^4+y^2)};

... do not show up like I expected.

Something of this sort? (Up to a sign this is a so-called Mexican hat potential.)

documentclass[tikz,border=3.14mm]{standalone}

usepackage{pgfplots}

pgfplotsset{compat=1.16}



begin{document}



begin{tikzpicture}

   begin{axis}[samples=20,zmin=0,zmax=1]

      addplot3[surf, domain=0:1,domain y=0:360,z buffer=sort]

       ({x*cos(y)},{x*sin(y)},{3*(0.5*x^2-x^4)+0.5});

   end{axis}

end{tikzpicture}

end{document}

Or, if you do

 addplot3[surf, domain=-1:1] {(x^2+y^2)-0.5*(x^2+y^2)^2};

you'll get

Yes, it is possible to extend the plot to the axes, but it is not as straightforward as one may think (or I am missing something obvious).

documentclass[tikz,border=3.14mm]{standalone}

usepackage{pgfplots}

pgfplotsset{compat=1.16}



begin{document}



begin{tikzpicture}[declare function={f(x,y)=(x^2+y^2)-0.5*(x^2+y^2)^2;}]

   begin{axis}[samples=20,zmin=0,zmax=1,xmin=-1,xmax=1,ymin=-1,ymax=1]

    % clip (-1,-1,{f(-1,-1)}) -- (1,-1,{f(1,-1)}) -- (1,1,{f(1,1)})

    % --  (1,1,1) -- (-1,-1,1); % not needed

    addplot3[surf, domain=-2:2,samples=50,point meta={max(f(x,y),0)}] {f(x,y)};   

   end{axis}

end{tikzpicture}

end{document}

One way to obtain a more shallow local minimum is to increase the plot range.

documentclass[tikz,border=3.14mm]{standalone}

usepackage{pgfplots}

pgfplotsset{compat=1.16}



begin{document}

begin{tikzpicture}[declare function={f(x,y)=(x^2+y^2)-0.5*(x^2+y^2)^2;}]

pgfmathsetmacro{myxmax}{2}

pgfmathsetmacro{myzmin}{f(myxmax,0)}

   begin{axis}[samples=20,zmin=myzmin,zmax=1,xmin=-myxmax,xmax=myxmax,ymin=-myxmax,ymax=myxmax]

    addplot3[surf, domain=-2:2,samples=50,point meta={max(f(x,y),myzmin)}] {f(x,y)};   

   end{axis}

end{tikzpicture}

end{document}