Combining with bifurcation theory in dynamical systems, a nonlinear ( 3 + 1 ) -dimensional modified KdV-Zakharov-Kuznetsov equation is theoretically investigated. Firstly, based on different parameter values and discriminant in cubic equation, types of equilibria and their corresponding phase portraits are qualitatively analyzed ,respectively. Secondly, by the Jacobi elliptical functions, formulas of some bounded traveling wave solutions and homoclinic loops are formally obtained, extending results in literature. Finally, several multi-mode approximations are numerically presented via Hamiltonian method, which indicate periodicity of bounded traveling wave solutions.