Chaos in Horndeski cosmologies

We analyze how a scalar field can affect the chaotic behaviour of homogeneous and isotropic Bianchi IX cosmologies. It is known that a massless, minimally coupled scalar field removes the chaos. However, in more general Horndeski theories, the situation is more complex. We find that in shift-symmetric $K$-essence theories, chaos persists if the scalar field contribution to the initial-value constraint is {\it subleading} compared to that of the anisotropies. In this case, solutions oscillate as they approach the singularity, just as in the vacuum case, and a similar behaviour is found when a non-minimal coupling is included. If the scalar field contribution is not subleading, then chaos is removed and the singularity is approached smoothly. An unusual and entirely new result appears when changing the sign in front of the scalar kinetic term, yielding the theory of a phantom scalar. If the scalar field is subleading, then solutions remain chaotic and oscillate when approaching the singularity, as before. However, if the scalar field is not subleading, solutions are also chaotic, but the spacetime singularity disappears, and the universe behaves as an apparently infinite sequence of anisotropic bounces. The spatial volume then oscillates within finite bounds, never reaching zero, while the amplitudes and positions of these oscillations appear completely random. To the best of our knowledge, this type of chaos has never been described.