Controlling complex tasks in robotic systems, such as circular motion for cleaning or following curvy lines, can be dealt with using nonlinear vector fields. This article introduces a novel approach called the rotational obstacle avoidance method (ROAM) for adapting the initial dynamics when obstacles partially occlude the workspace. ROAM presents a closed-form solution that effectively avoids star-shaped obstacles in spaces of arbitrary dimensions by rotating the initial dynamics toward the tangent space. The algorithm enables navigation within obstacle hulls and can be customized to actively move away from surfaces while guaranteeing the presence of only a single saddle point on the boundary of each obstacle. We introduce a sequence of mappings to extend the approach for general nonlinear dynamics. Moreover, ROAM extends its capabilities to handle multiobstacle environments and provides the ability to constrain dynamics within a safe tube. By utilizing weighted vector-tree summation, we successfully navigate around general concave obstacles represented as a tree-of-stars. Through experimental evaluation, ROAM demonstrates superior performance in minimizing occurrences of local minima and maintaining similarity to the initial dynamics, outperforming existing approaches in multiobstacle simulations. Due to its simplicity, the proposed method is highly reactive and can be applied effectively in dynamic environments. This was demonstrated during the collision-free navigation of a 7-degree-of-freedom robot arm around dynamic obstacles.