​
c=Array[cc,{2,2}];​​f[a_]:=0.5(a[[1,1]]^2+a[[1,2]]^2-a[[1,1]]^2a[[1,2]]-a[[1,2]]^2a[[1,1]]);f1[a_]:=a[[1,1]]-a[[1,1]]a[[1,2]]-0.5a[[1,2]]^2;f2[a_]:=a[[1,2]]-a[[1,1]]a[[1,2]]-0.5a[[1,1]]^2;
sol=NDSolve[{c[[1,1]]'[t]c[[1,1]][t]-c[[1,1]][t]c[[1,2]][t]-0.5c[[1,2]][t]^2,c[[1,1]][0]0.1,c[[1,2]]'[t]c[[1,2]][t]-c[[1,1]][t]c[[1,2]][t]-0.5c[[1,1]][t]^2,c[[1,2]][0]0.1},{c[[1,1]][t],c[[1,2]][t]},{t,0,1}]
{{cc[1,1][t]InterpolatingFunction[{{0.,1.}},<>][t],cc[1,2][t]InterpolatingFunction[{{0.,1.}},<>][t]}}
sol2=NDSolve[{c[[1,1]]'[t]f1[c],c[[1,1]][0]0.1,c[[1,2]]'f2[c],c[[1,2]][0]0.1},{c[[1,1]][t],c[[1,2]][t]},{t,0,1}]
NDSolve::dvnoarg:The function cc[1,1] appears with no arguments.

NDSolve[{
′
cc[1,1]
[t]cc[1,1]-cc[1,1]cc[1,2]-0.5
2
cc[1,2]
,cc[1,1][0]0.1,
′
cc[1,2]
-0.5
2
cc[1,1]
+cc[1,2]-cc[1,1]cc[1,2],cc[1,2][0]0.1},{cc[1,1][t],cc[1,2][t]},{t,0,1}]
sol3=NDSolve[{c[[1,1]]'[t]Derivative[{{1,0},{0,0}}][f][c],c[[1,1]][0]0.1,c[[1,2]]'[t]Derivative[{{0,1},{0,0}}][f][c],c[[1,2]][0]0.1},{c[[1,1]][t],c[[1,2]][t]},{t,0,1}]
NDSolve::dvnoarg:The function cc[1,1] appears with no arguments.

NDSolve[{
′
cc[1,1]
[t]0.5(2cc[1,1]-2cc[1,1]cc[1,2]-
2
cc[1,2]
),cc[1,1][0]0.1,
′
cc[1,2]
[t]0.5(2cc[1,1]-2cc[1,1]cc[1,2]-
2
cc[1,2]
),cc[1,2][0]0.1},{cc[1,1][t],cc[1,2][t]},{t,0,1}]
Derivative[{{1,0},{0,0}}][f][c]
0.5(2cc[1,1]-2cc[1,1]cc[1,2]-
2
cc[1,2]
)