text: In this paper we solve the problem of finding a trajectory that shows that a given hybrid dynamical system with deterministic evolution leaves a given set of states considered to be safe. The algorithm combines local with global search for achieving both efficiency and global convergence. In local search, it exploits derivatives for efficient computation. Unlike other methods for falsification of hybrid systems with deterministic evolution, we do not restrict our search to trajectories of a certain bounded length but search for error trajectories of arbitrary length.
  @cite_0: Virtually all methods for global optimization consist of two phases: a global phase, aimed at thorough exploration of the feasible region or subsets of the feasible region where it is known the global optimum will be found, and a local phase aimed at locally improving the approximation to some local optima. Often these two phases are blended into the same algorithm, which automatically switches between exploration and refinement. In this paper, some methods will be reviewed in which the two phases are well separated. A formal definition of two-phase methods will be given and some of the characteristics of these methods will be outlined.
summary: Our algorithm can be viewed as an adaptation of the Best Start two-phase method for global optimization @cite_0 to our context.