Open trajectories

It is proven that an initial value problem of an autonomous system verifying some conditions has exactly one open continuously di€erentiable trajectory on R, whose limit set is not one point. The proof of the result is based upon continuation methods. 1. Preliminaries Di€erential equations play a key role in nature and technology because they control their processes. In time-dependent processes a particular solution of an ordinary di€erential equation system can go up to in®nity while time tends towards a ®nite number. Such dangerous e€ects where the process breaks down after a ®nite time are called blowing-up e€ects. The reason can be that the system loses too much of its initial energy by dissipation or that it is unstable. Engineers are interested in avoiding such e€ects. Therefore it is interesting to know the behaviour of the solution of such a system, when its variable is large enough. This variable can be considered to be time. In this paper priory estimates are given to prove that an initial value problem of an autonomous ordinary di€erential equation system has a global open continuously di€erentiable solution, whose limit sets are not one point. Priory conditions are