The geometric setting for systems of ODEs comprises, in particular, the construction of a canonical nonlinear connection. This connection induces another geometric objects that serve as efficient tools for the study of the system. There are various approaches to construct such connections, some of which are mentioned in the paper. With reference to the inspiring work of Kosambi, the authors propose a uniform setting both for systems of second and higher order.
The system of ODEs of order \(k+1\) on a smooth manifold \(M\) is represented by a semispray on \(T^kM\), the bundle of \(k\)-velocities of \(M\). The canonical nonlinear connection on \(T^kM\) is determined by the requirement of compatibility with the tangent structure (which is the tensor field of type \((1,1)\) describing the flag of vertical subbundles of the tangent bundle of \(T^kM\)). The point symmetries, newtonoid vector fields, and first-order variations of the system are expressed in terms of the canonical nonlinear connection, respectively, in terms of the induced dynamical covariant derivative and the Jacobi endomorphism. The components of the Jacobi endomorphism form the basic invariants of the system of ODEs and they are explicitly described as well. As an application it is shown that these components correspond to the well-known Wünschmann-type invariants associated to equations of third and fourth order. Another application for systems of equations describing the biharmonic curves on Riemannian manifolds is discussed in detail.