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By using the genus properties, we establish some criteria for the second-order p(t)-Laplacian system $$ \frac{d}{dt}\big(|\dot{u}(t)|^{p(t)-2}\dot{u}(t)\big)-a(t)|u(t)|^{p(t)-2}u(t) +\nabla W(t, u(t))=0 $$ to have at least one, and infinitely many homoclinic orbits. where $t\in {\mathbb{R}},\; u\in {\mathbb{R}}^{N}$, $p(t)\in C(\mathbb{R},\mathbb{R})$ and $p(t)>1$, $a\in C({\mathbb{R}}, {\mathbb{R}})$ and $W\in C^{1}({\mathbb{R}}\times {\mathbb{R}}^{N}, {\mathbb{R}})$ may not be periodic in t.