Integrability analysis of chaotic and hyperchaotic finance systems

Abstract

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 equationṡ x = z + (y − a)x,ẏ = 1 − by − x 2 ,ż = − x − cz. The second system, called a hyperchaotic finance model, is defined bẏ x = z + (y − a)x + u, z = − x − cz,ẏ = 1 − by − x 2 , u = − dx y − ku. In both models, (a, b, c, d, k) are real positive parameters. In order to present the complexity of these systems Poincaré 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 := 1 + d(a + d − c) > 0. We give analytic proofs of these facts analyzing properties of the differential Galois groups of variational equations along cer-The research has been supported by the National Science Centre of Poland under Grants DEC

Comment on ,,On the integrability of 2D Hamiltonian systems with variable Gaussian curvature” by A. A. Elmandouh

“…where the quantities λ 1 and λ 2 are postulated in (16). The proof of Theorem 2 consists of the direct application of the Kimura theorem 3 to the obtained Riemann P-equation (25) with the differences of the exponents (26). Indeed, the first four elements given in item (i) of the list (17) were deduced from Case A of Theorem 3, while the remaining entries of (17) come from Case B of this theorem.…”

“…This approach is based on the analysis of the differential Galois group of variational equations of a considered system along a certain particular solution. The Morales-Ramis theory has been used recently for study integrability of various important physical and astronomical systems, see, e.g., papers [23][24][25][26][27][28][29][30][31][32][33]. We also mention review paper [34], in which certain examples can be found.…”

Collective correlations, dynamics, and behavioural inconsistencies of the cryptocurrency market over time

“…In the nonlinear dynamics community, such market dynamics have been studied with a range of methodologies, including chaotic systems [ 12 – 14 ], clustering [ 15 , 16 ], sample entropy [ 17 , 18 ], and principal components analysis [ 19 – 21 ]. Various asset classes have attracted interest, including equities [ 22 ], fixed income [ 23 ], and foreign exchange [ 24 ].…”

Transition to hyperchaos and rare large-intensity pulses in Zeeman laser

“…Rössler [10] and defined as hyperchaos. Hyperchaos was later observed in a variety of systems with dimension larger than 3, such as the semiconductor superlattices [11], electronic circuits [12][13][14], coupled oscillators [15][16][17], finance systems [18], radiophysical generator [19], interacting gas bubbles in a liquid [20], and two element nonlinear chimney model [21]. It has also been reported in lasers [22][23][24][25].…”