On a system of nonlinear differential equations for the model of totally connected traffic

In the paper the qualitative properties solutions of the system non-linear equations, describing one-way movement of particles chain on aline with follower velocity defined by some function of distance from theleader, are researched. In the case when the given function is the velocityof the first particle (leader) in the chain, the model is called a model ofleader. If the given function is the velocity of the last particle (outsider),the model is called a model of “shepherd”.
The sufficient conditions for the existence of the chain with the givenconstraints on the velocity and acceleration are obtained.