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Aerial manipulators, composed of multirotors and robotic arms, have a structure and function highly reminiscent of avian species. This paper studies the tracking control problem for aerial manipulators. This paper studies the tracking control problem for aerial manipulators. We propose an avian-inspired aerial manipulation system, which includes an avian-inspired robotic arm design, a Recursive Newton-Euler (RNE) method-based nonlinear flight controller, and a coordinated controller with two modes. Compared to existing methods, our proposed approach offers several attractive features. First, the morphological characteristics of avian species are used to determine the size proportion of the multirotor and the robotic arm in the aerial manipulator. Second, the dynamic coupling of the aerial manipulator is addressed by the RNE-based flight controller and a dual-mode coordinated controller. Specifically, under our proposed algorithm, the aerial manipulator can stabilize the end-effector's pose, similar to avian head stabilization. The proposed approach is verified through three numerical experiments. The results show that even when the quadcopter is disturbed by different forces, the position error of the end-effector achieves millimeter-level accuracy, and the attitude error remains within 1 degree. The limitation of this work is not considering aggressive manipulation like that seen in birds. Addressing this through future studies that explore real-world experiments will be a key direction for research.