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I have the following system of equations

E1 = Sin[u[1]] + 24.99741937197641*(-0.2 + u[1]);
E2 = Sin[u[2]] + 24.99741937197641*(0.00006931712037649973*(-0.2 + u[1]) + 1.*(-2*u[1] + u[2]));
NSolve[{E1 == 0, E2 == 0}, {u[1], u[2]}]

Any suggestions?

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You can use FindRoot,

FindRoot[{E1 == 0, E2 == 0}, {u[1], 0}, {u[2], 0}]

{u[1] -> 0.192352, u[2] -> 0.370231}

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