Time-fixed geometry of 2nd order ODEs

Point classification of 2nd order ODEs: Tresse classification revisited and beyond

In 1896 Tresse gave a complete description of relative differential invariants for the pseudogroup action of point transformations on the 2nd order ODEs. The purpose of this paper is to review, in light of modern geometric approach to PDEs, this classification and also discuss the role of absolute invariants and the equivalence problem.

Numerical methods for the 2nd moment of stochastic ODEs

Numerical methods for stochastic ordinary differential equations typically estimate moments of the solution from sampled paths. Instead, in this paper we directly target the deterministic equation satisfied by the first and second moments. For the canonical examples with additive noise (Ornstein–Uhlenbeck process) or multiplicative noise (geometric Brownian motion) we derive these deterministic...

The Geometry of a pair of second-order ODEs and Euclidean spaces

Paraconformal geometry of nth order ODEs, and exotic holonomy in dimension four

We characterise nth order ODEs for which the space of solutions M is equipped with a particular paraconformal structure in the sense of [2], that is a splitting of the tangent email [email protected] email [email protected]

Verified High-order Integration of Daes and Higher-order Odes

Within the framework of Taylor models, no fundamental difference exists between the antiderivation and the more standard elementary operations. Indeed, a Taylor model for the antiderivative of another Taylor model is straightforward to compute and trivially satisfies inclusion monotonicity. This observation leads to the possibility of treating implicit ODEs and, more importantly, DAEs within a ...

Point equivalence of second-order ODEs: Maximal invariant classification order

We show that the local equivalence problem of second-order ordinary differential equations under point transformations is completely characterized by differential invariants of order at most 10 and that this upper bound is sharp. We also demonstrate that, modulo Cartan duality and point transformations, the Painlevé–I equation can be characterized as the simplest second-order ordinary different...