The spin-1 Ising model is examined on the Bethe lattice (BL) by considering the effects of nearest and next-nearest neighbor bilinear (J) and biquadratic (K) exchange interactions on a honeycomb lattice. The exact recursion relations (ERR) are employed in the double-shell approximation to obtain the order-parameters. The phase diagrams of the model are obtained from the thermal analysis of the order-parameters. The phase transition lines separating the different phase regions, i.e. ferromagnetic (FM), antiferromagnetic (AFM), paramagnetic (PM) and chaotic, are calculated. The chaotic phase regions are especially interesting which is usually observed when the system presents phase transitions from AFM to FM phases.