Fixed points of multivariable calculus

Question

I have discrete case.

$z=1-x-y$;

$x=a_1(x^{2}+2yz)+a_2(z^{2}+2xy)+a_3(y^{2}+2xz)$;

$y=b_1(x^{2}+2yz)+b_2(z^{2}+2xy)+b_3(y^{2}+2xz)$;

$z=c_1(x^{2}+2yz)+c_2(z^{2}+2xy)+c_3(y^{2}+2xz)$;

where $a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3, $ are arbitrary which value less than 1.

To find the behavior of fixed point, i can't use jacobian because at point (0,0,0), jacobian gives me input of all 0, which is not useful.

Thus, i try for Hessian at point (0,0,0). What kind of properties of the eigenvalues of Hessian to see the dynamical system? Because if in jacobian, modulus of eigenvalues less than 1 gives me convergence of fixed point. Then i'm asking for properties of Hessian's eigenvalues to see the dynamical system of my fixed points.

Thanks for helping.

Answer 1

If the differential vanishes, then the fixed point is called superattracting and all the points of a certain neighborhood converge under iteration to the fixed point.