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A construction of the bi-Hamiltonian structures for integrable systems onregular time scales is presented. The trace functional on an algebra of$\delta$-pseudo-differential operators, valid on an arbitrary regular timescale, is introduced. The linear Poisson tensors and the related Hamiltoniansare derived. The quadratic Poisson tensors is given by the use of the recursionoperators of the Lax hierarchies. The theory is illustrated by$\Delta$-differential counterparts of Ablowitz-Kaup-Newell-Segur and Kaup-Broerhierarchies.

Bi-Hamiltonian structures for integrable systems on regular time scales