  In the spatial circular restricted three-body problem librational motion
around the Lagrangian points $L_4$ and $L_5$ exists up to high inclinations of
the massless object. We report the results of numerical investigations on the
stability of this librational motion for systematic variations in the
inclination of the Trojan and the mass-ratio $\mu$ of the two massive bodies.
We show that stable motion prevails for inclinations below 60 degrees and
mass-ratios $\mu < 0.04$ for typical integration times up to $10^6$ revolution
periods. At even higher mass-ratios -- beyond the critical mass-ratio for the
planar case -- stable orbits were found to exist for up to $10^7$ periods at
moderate inclinations. We extracted the librational frequencies on a grid in
the parameter space from the Fourier spectra and traced their variation.
Several resonances between the short and long period librational frequency as
well as the vertical frequency lie inside the investigated region. The
application of the Lyapunov characteristic indicator and spectral number
methods also reveals the chaotic regions. This simple model is equally
applicable in the Solar system for low mass ratios for Trojans of the planets,
as well as to Trojan-type exoplanets in binary star systems at high
mass-ratios.
