We study the interplay between a particular marginal deformation of N = 4
super Yang-Mills theory, the β deformation, and integrability in the holographic setting.
Using modern methods of analytic non-integrability of Hamiltonian systems, we find that,
when the β parameter takes imaginary values, classical string trajectories on the dual
background become non-integrable. We expect the same to be true for generic complex β
parameter. By exhibiting the Poincar´e sections and phase space trajectories for the generic
complex β case, we provide numerical evidence of strong sensitivity to initial conditions.
Our findings agree with expectations from weak coupling that the complex β deformation
is non-integrable and provide a rigorous argument beyond the trial and error approach to