Consider an analytic or polynomial autonomous system of differential equations having two rest points (possibly, coinciding). The authors search for trajectories connecting these rest points. The method is based on introduction of two different logarithmic changes of time on the positive and the negative semi-axis, after which the search for a homoclinic or heteroclinic trajectory is reduced to solving an equation for the corresponding scaling factors. As examples, the authors consider the stretch-twist-fold flow, the Nagumo system and the mathematical pendulum.