  We study the relationship between the standard or extended thermodynamic
phase structure of various AdS black holes and the Lyapunov exponents
associated with the null and time-like geodesics. We consider dyonic, Bardeen,
Gauss-Bonnet, and Lorentz-symmetry breaking massive gravity black holes and
calculate the Lyapunov exponents of massless and massive particles in unstable
circular geodesics close to the black hole. We find that the thermal profile of
the Lyapunov exponents exhibits distinct behaviour in the small and large black
hole phases and can encompass certain aspects of the van der Waals type
small/large black hole phase transition. We further analyse the properties of
Lyapunov exponents as an order parameter and find that its critical exponent is
$1/2$, near the critical point for all black holes considered here.
