Consider the first order Hamiltonian system \(\overset{.}{z}={\mathcal J}H_{z}(t,z) \) where \(z=(p,q)\in {\mathbb R}^{2N},{\mathcal J} =\left( \begin{matrix} 0 & -I \\ I & 0 \end{matrix} \right) \) and \(H\in C^{1}({\mathbb R}\times {\mathbb R}^{2N},{\mathbb R}) \) has the form \( H\left( t,z\right)=\frac{1}{2}L\left( t\right) z\cdot z+R\left( t,z\right) \) with \(L\left( t\right) \) a continuous symmetric \(2N\times 2N\) matrix-valued function, \(R_{z}\left( t,z\right) =o\left(| z| \right) \) as \(z\rightarrow 0\) and asymptotically linear as \(| z| \rightarrow \infty \). A solution \(z\) of this system is a homoclinic orbit if \(z\neq 0\) and \(z\left( t\right) \rightarrow 0\) as \(| t| \rightarrow \infty \). The existence and multiplicity of homoclinic orbits is studied without assuming periodicity conditions.