documentclass[11pt]{article}
usepackage{amsmath}
usepackage{amssymb}
usepackage{subfigure}

usepackage[font=footnotesize]{caption}
usepackage{pgfplots}
usepackage{tikz}

% Scriptsize axis style.
pgfplotsset{every axis/.append style={tick label style={/pgf/number format/fixed},font=scriptsize,ylabel near ticks,xlabel near ticks,grid=major}}

begin{document}
begin{figure}
    begin{tikzpicture}[samples=200,smooth]
        begin{scope}
            clip(-4,-1) rectangle (4,4);
            draw plot[domain=0:360] ({cos(x)*sqrt(20/(sin(2*x)+2))},{sin(x)*sqrt(20/(sin(2*x)+2))});
            draw plot[domain=0:360] ({cos(x)*sqrt(16/(sin(2*x)+2))},{sin(x)*sqrt(16/(sin(2*x)+2))});
            draw plot[domain=0:360] ({cos(x)*sqrt(12/(sin(2*x)+2))},{sin(x)*sqrt(12/(sin(2*x)+2))});
            draw plot[domain=0:360] ({cos(x)*sqrt(8/(sin(2*x)+2))},{sin(x)*sqrt(8/(sin(2*x)+2))});
            draw plot[domain=0:360] ({cos(x)*sqrt(4/(sin(2*x)+2))},{sin(x)*sqrt(4/(sin(2*x)+2))});
            draw plot[domain=0:360] ({cos(x)*sqrt(1/(sin(2*x)+2))},{sin(x)*sqrt(1/(sin(2*x)+2))});
            draw plot[domain=0:360] ({cos(x)*sqrt(0.0625/(sin(2*x)+2))},{sin(x)*sqrt(0.0625/(sin(2*x)+2))});

            draw[->,blue,ultra thick] (-2,3.65) to (-1.93,3);
            draw[->,blue,ultra thick] (-1.93,3) to (-1.75,2.4);
            draw[->,blue,ultra thick] (-1.75,2.4) to (-1.5,1.8);
            draw[-`enter code here`>,blue,ultra thick] (-1.5,1.8) to (-1.15,1.3);
            node at (-1.4,3.8){scriptsize $w[0]$};
            node at (-1.2,3.2){scriptsize $w[1]$};
            node at (-1.05,2.6){scriptsize $w[2]$};
            node at (-0.8,2){scriptsize $w[3]$};
            node at (-0.6,1.4){scriptsize $w[4]$};
        end{scope}
    end{tikzpicture}
end{figure}
end{document}