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We investigate a new class of $\eta$-deformed $AdS_5 \times T^{1,1}$backgrounds produced by $r$-matrices that satisfy the modified classicalYang-Baxter equation [Jour. High Ener. Phys. 03 (2022) 094]. We examine theclassical phase space of these (semi)classical strings by numerically studyingthe dynamics of the string sigma models over this deformed background, and wecompute several chaos signals. These involve figuring out the Poincar'e sectionand computing the Lyapunov exponents. In the (semi)classical limit, we discoverevidence that supports a non-integrable phase space dynamics.

Chaotic string dynamics in Bosonic $\eta$-deformed $AdS_5 \times T^{ 1,1}$ background