We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in three-dimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames.

We incorporate the new theory of equivariant moving frames for Lie pseudo-groups into Vessiot’s method of group foliation of differential equations. The automorphic system is replaced by a set of reconstruction equations on the pseudo-group jets. The result is a completely algorithmic and symbolic procedure for finding both invariant and non-invariant solutions of differential equations admitti...

In random differential equations with bounded noise minimal forward invariant (MFI) sets play a central role since they support stationary measures. We study the stability and possible bifurcations of MFI sets. In dimensions 1 and 2 we classify all minimal forward invariant sets and their codimension one bifurcations in bounded noise random differential equations.

When a sample material is subjected to a linear temperature program, the heat flow rate into the sample i s proportional to its instantaneous specific heat. By recording this heat flow rate as a function of temperature, and comparing it with the heat flow rate into a standard material under the same conditions, the specific heat of the sample i s determined as a fundion of temperature. The meth...

This paper deals with the stability and observer design for Lur’e systems with multivalued nonlinearities, which are not necessarily monotone or time-invariant. Such differential inclusions model the motion of state trajectories which are constrained to evolve inside time-varying non-convex sets. Using Lyapunov-based analysis, sufficient conditions are proposed for local stability in such syste...

This is the second paper on invariant hyperfunction solutions of invariant linear differential equations on the vector space of n × n real symmetric matrices. In the preceding paper [22], we proved that every invariant hyperfunction solution is expressed as a linear combination of Laurent expansion coefficients of the complex power of the determinant function with respect to the parameter. Fund...

We present the package SADE (Symmetry Analysis of Differential Equations) for the determination of symmetries and related properties of systems of differential equations. The main methods implemented are: Lie, non classical, Lie-Bäcklund and potential symmetries, invariant solutions, first-integrals, Nöther theorem for both discrete and continuous systems, solution of ordinary differential equa...

Generalised symmetries and point symmetries coincide for systems of first-order ordinary differential equations and are infinite in number. Systems of linear first-order ordinary differential equations possess a generalised rescaling symmetry. For the system of first-order ordinary differential equations corresponding to the time-dependent linear oscillator the invariant of this symmetry has th...

Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudostable and pseudo-unstable manifolds for a class of random partial differential equations and stochastic partial differential equations is shown. Unlike the invarian...

We show that the algebra of equi-affine differential invariants of a suitably generic surface S ⊂ R is entirely generated by the third order Pick invariant via invariant differentiation. The proof is based on the new, equivariant approach to the method of moving frames. The goal of this paper is to prove that, in three-dimensional equi-affine geometry, all higher order differential invariants o...