  Consider the following system of double coupled Schr\"odinger equations
arising from Bose-Einstein condensates etc.,
  \begin{equation*}
  \left\{\begin{array}{l}
  -\Delta u + u =\mu_1 u^3 + \beta uv^2- \kappa v,
  -\Delta v + v =\mu_2 v^3 + \beta u^2v- \kappa u,
  u\neq0, v\neq0\ \hbox{and}\ u, v\in H^1(\R^N),
  \end{array}
  \right.
  \end{equation*}where $\mu_1, \mu_2$ are positive and fixed, $\kappa$ and
$\beta$ are linear and nonlinear coupling parameters respectively. We first use
critical point theory and Liouville type theorem to prove some existence and
nonexistence results on the positive solutions of this system. Then using the
positive and non-degenerate solution of the scalar equation
$-\Delta\omega+\omega=\omega^3$, $\omega\in H_r^1(\R^N)$, we construct a
synchronized solution branch to prove that for $\beta$ in certain range and
fixed, there exist a series of bifurcations in product space $\R\times
H^1_r(\R^N)\times H^1_r(\R^N)$ with parameter $\kappa$.
