An incommensurate fractional discrete macroeconomic system: Bifurcation, chaos, and complexity

Abstract

This study proposes a novel fractional discrete-time macroeconomic system with incommensurate order. The dynamical behavior of the proposed macroeconomic model is investigated analytically and numerically. In particular, the zero equilibrium point stability is investigated to demonstrate that the discrete macroeconomic system exhibits chaotic behavior. Through using bifurcation diagrams, phase attractors, the maximum Lyapunov exponent and the 0–1 test, we verified that chaos exists in the new model with incommensurate fractional orders. Additionally, a complexity analysis is carried out utilizing the approximation entropy (ApEn) and C 0 complexity to prove that chaos exists. Finally, the main findings of this study are presented using numerical simulations.

Hidden multistability of fractional discrete non-equilibrium point memristor based map

“…It is clear that, for c ≠ 0, the memristor-based map (19) has no equilibrium points. Thus, according to [51], all attractors generated by the discrete memristor-based map (19) We can see that as β increases, the dynamical behavior of the map (19) has evolved from periodic to chaotic through a period-doubling bifurcation.…”

“…It is clear that, for c ≠ 0, the memristor-based map (19) has no equilibrium points. Thus, according to [51], all attractors generated by the discrete memristor-based map (19) We can see that as β increases, the dynamical behavior of the map (19) has evolved from periodic to chaotic through a period-doubling bifurcation. In addition, to illustrate the presence of chaos and hyperchaos on the memristor map (19), hidden multistability attractors for β = 3.6 and with different initial conditions are depicted in figure 3.…”

“…In this study, by using the Caputo difference operator we expand the integer-order non-equilibrium point (NEP) map to produce the fractional-order discrete non-equilibrium point (DNEP) map based on memristors. The first-order difference of the memristor map (19) is denoted by the following formula:…”

“…Discrete fractional calculus has drawn the interest of a great number of researchers during the last several years [8][9][10][11][12], and they have been increasingly interested in its potential applications in neural networks, secure communication, biology, and other domains [13][14][15]. Recently numerous different dynamics including chaos, hyperchaos and coexisting attractors in fractional-order systems have been explored [16][17][18][19][20][21][22]. For example, the coexisting chaos and hyperchaos in the fractional-order discrete SIE epidemic model have been analyzed in [23].…”

A new two-dimensional fractional discrete rational map: chaos and complexity

“…This recent topic has reached its peak of publications lately by proposing several fractional discrete-time operators, and stability analysis, transforms, and many theoretical results including [19][20][21]. This has contributed in proposing further fractional-order chaotic maps like [22][23][24][25][26][27][28] coupled with several control schemes and implementations like [29][30][31][32][33]. For instance in [22], Wu and Baleanu proposed a discrete fractional-order logistic map in the left Caputo discrete deltas sense.…”