Integrability analysis of chaotic and hyperchaotic finance systems

In this paper, we consider two chaotic finance models recently studied in the literature. The first one, introduced by Huang and Li, has a form of three first-order nonlinear differential equations (x) over dot = z + (y - a) x, (y) over dot = 1 - by - x (2), (z) over dot = -x - cz. The second system, called a hyperchaotic finance model, is defined by (x) over dot = z + (y - a) x + u, (y) over dot = 1- by -x(2), (z) over dot = -x -cz, (u) over dot = - dxy - ku. In both models, (a, b, c, d, k) are real positive parameters. In order to present the complexity of these systems Poincare cross sections, bifurcation diagrams, Lyapunov exponents spectrum and the Kaplan-Yorke dimension have been calculated. Moreover, we show that the Huang-Li system is not integrable in a class of functions meromorphic in variables (x, y, z), for all real values of parameters (a, b, c), while the hyperchaotic system is not integrable in the case when k = c and Delta := 1 + d(a + d - c) > 0. We give analytic proofs of these facts analyzing properties of the differential Galois groups of variational equations along certain particular solutions. On the other hand, we show that for certain sets of the parameters (a, b, c, d, k), when Delta <= 0, the hyperchaotic system possesses a polynomial first integral, which can be used to reduce the dimension of the system by one.