Axisymmetric disc galaxies with smooth density profiles have a core region whose gravitational potential is close to that of an axisymmetric harmonic oscillator. A tidal interaction adds a perturbing potential to the core, which is nearly that of a slowly varying triaxial harmonic oscillator. We argue that even a weak perturbation of this form can dramatically affect the orbits in a core-sized region; loop orbits become box-like and nearly circular orbits become elliptic with secularly growing ellipticity. We refer to this as 'core-instability'. By considering in some detail a set of model problems, we argue how due to the core-instability, even a weak tidal forcing (~0.3 per cent of the core gravity) leads to strong, transient, spiral density waves in any thin, 'cold' disc component of galactic cores. We also argue that the self gravity of the disc is unlikely to interfere with the production of such a density wave as long as the disc contributes <~ 10 per cent of the core gravity (and so 10 per cent core mass). We postulate that many galactic cores do contain such a thin disc made of molecular clouds and young stars. Molecular cloud collisions will be greatly enhanced on the spiral density hill due to orbit crowdings/crossings. We suggest that most of the molecular cloud's gas will then be converted to massive stars, in a few nuclear rotation time-scales, leading to a 'nuclear starburst'. Observing core potentials of starburst galaxies with companions to be 'soft' and seeing tightly wound circum-nuclear spirals of enhanced star formation in the cores, will vindicate this model.