Alain Goriely

The Kovalevskaya exponents are sets of exponents that can be associated with a given nonlinear vector field. They correspond to the Fuchs' indices of the linearized vector field around particular scale invariant solutions. They were used by S.Kovalevskaya to prove the single-valuedness of the classical cases of integrability of the rigid body motion. In this paper, a history of the discovery and multiple re-discoveries of the Kovalevskaya exponents is given together with the modern use of Kovalevskaya exponents in integrability theory and nonlinear dynamics.

The Kirchhoff analogy for elastic rods establishes the equivalence between the solutions of the classical spinning top and the stationary solutions of the Kirchhoff model for thin elastic rods with circular cross-sections. In this paper the Kirchhoff analogy is further generalized to show that the classical Kovalevskaya solution for the rigid body problem is formally equivalent to the solution of the Kirchhoff model for thin elastic rod with anisotropic cross-sections (elastic strips). These Kovalevskaya rods are completely integrable and are part of a family of integrable travelling waves solutions for the rod (Kovalevskaya waves). The analysis of homoclinic twistless Kovalevskaya rod reveals the existence of a three parameter family of solutions corresponding to the Steklov and Bobylev integrable case of the rigid body problem. Furthermore, the existence of these integrable solutions is discussed in conjunction with recent results on the stability of strips.

The analysis of complex-time singularities has proved to be the most useful tool for the analysis of integrable systems. Here, we demonstrate its use in the analysis of chaotic dynamics. First, we show that the Melnikov vector, which gives an estimate of the splitting distance between invariant manifolds, can be given explicitly in terms of local solutions around the complex-time singularities. Second, in the case of exponentially small splitting of invariant manifolds, we obtain sufficient conditions on the vector field for the Melnikov theory to be applicable. These conditions can be obtained algorithmically from the singularity analysis.