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In this paper we study the integrability of the Hamiltonian system associated with the fourth Painlevé equation. We prove that one two‐parametric family of this Hamiltonian system is not integrable in the sense of the Liouville–Arnold theorem. Computing explicitly the Stokes matrices and the formal invariants of the second variational equations, we deduce that the connected component of the unit element of the corresponding differential Galois group is not Abelian. Thus the Morales–Ramis–Simó theory leads to a nonintegrable result. Moreover, combining the new result with our previous one we establish that for all values of the parameters for which the PIV$P_{IV}$ equation has a particular rational solution the corresponding Hamiltonian system is not integrable by meromorphic first integrals which are rational in t. [ABSTRACT FROM AUTHOR]

We study the effect of the unfolding of a reducible double confluent Heun equation from the point of view of the Stokes phenomenon. We introduce a small complex parameter ε that splits together the non-resonant singular points x = 0 and x = ∞ into four different Fuchsian singularities x L = − ε , x R = ε , and x L L = − 1 / ε , x R R = 1 / ε , respectively. The perturbed equation is a symmetric general Heun equation and its general solution depends analytically on ε . Then we prove that when the perturbed equation has exactly two resonant singularities of different type, all the Stokes matrices of the initial double confluent Heun equation are realized as a limit of the upper-triangular parts of the monodromy matrices of the perturbed equation when ε → 0 . To establish this result we combine a direct computation with a theoretical approach. [ABSTRACT FROM AUTHOR]

A Correction to this paper has been published: https://doi.org/10.1007/s10883-021-09575-w [ABSTRACT FROM AUTHOR]

We consider an initial problem (IP) for a linear neutral system with distributed delays and derivatives in Caputo's sense of incommensurate order, with different kinds of initial functions. In the case when the initial functions are with bounded variation, it is proven that this IP has a unique solution. The Krasnoselskii's fixed point theorem, a very appropriate tool, is used to prove the existence of solutions in the case of the neutral systems. As a corollary of this result, we obtain the existence and uniqueness of a fundamental matrix for the homogeneous system. In the general case, without additional assumptions of boundedness type, it is established that the existence and uniqueness of a fundamental matrix lead existence and uniqueness of a resolvent kernel and vice versa. Furthermore, an explicit formula describing the relationship between the fundamental matrix and the resolvent kernel is proven in the general case too. On the base of the existence and uniqueness of a resolvent kernel, necessary and sufficient conditions for the stability of the zero solution of the homogeneous system are established. Finally, it is considered a well-known economics model to describe the dynamics of the wealth of nations and comment on the possibilities of application of the obtained results for the considered systems, which include as partial case the considered model. [ABSTRACT FROM AUTHOR]