In this paper, we study gradient-like flows without heteroclinic intersections on an \(n\)-sphere up to topological conjugacy. We prove that such a flow is completely defined by a bicolor tree corresponding to a skeleton formed by codimension one separatrices. Moreover, we show that such a tree is a complete invariant for these flows with respect to the topological equivalence also. This result implies that for these flows with the same (up to a change of coordinates) partitions into trajectories, the partitions for elements, composing isotopies connecting time-one shifts of these flows with the identity map, also coincide. This phenomenon strongly contrasts with the situation for flows with periodic orbits and connections, where one class of equivalence contains continuum classes of conjugacy. In addition, we realize every connected bicolor tree by a gradient-like flow without heteroclinic intersections on the \(n\)-sphere. In addition, we present a linear-time algorithm on the number of vertices for distinguishing these trees.

在本文中，我们研究了\（n \）上没有非斜交点的类梯度流-球形，直到拓扑共轭。我们证明了这样的流完全是由双色树定义的，该双色树对应于由codimension一分离所形成的骨架。此外，我们还表明，就拓扑等价而言，这样的树对于这些流是完全不变的。该结果意味着，对于这些具有相同（直到坐标发生变化）的划分为轨迹的流，元素的划分（这些同位素将这些流的一个时间偏移与同一性图联系起来）也是重合的。这种现象与具有周期性轨道和连接的流动的情况形成鲜明对比，在流动和轨道中，一类等价包含连续的共轭类。另外，我们通过类似梯度的流来实现每个连通的双色树，而在树上没有异质相交\（n \）-球形。此外，我们提出了一种用于区分这些树的顶点数量的线性时间算法。