Hopf Bifurcation, Positively Invariant Set, and Physical Realization of a New Four‐Dimensional Hyperchaotic Financial System

Abstract

This paper introduces a new four-dimensional hyperchaotic financial system on the basis of an established three-dimensional nonlinear financial system and a dynamic model by adding a controller term to consider the effect of control on the system. In terms of the proposed financial system, the sufficient conditions for nonexistence of chaotic and hyperchaotic behaviors are derived theoretically. Then, the solutions of equilibria are obtained. For each equilibrium, its stability and existence of Hopf bifurcation are validated. Based on corresponding first Lyapunov coefficient of each equilibrium, the analytical proof of the existence of periodic solutions is given. The ultimate bound and positively invariant set for the financial system are obtained and estimated. There exists a stable periodic solution obtained near the unstable equilibrium point. Finally, the dynamic behaviors of the new system are explored from theoretical analysis by using the bifurcation diagrams and phase portraits. Moreover, the hyperchaotic financial system has been simulated using a specially designed electronic circuit and viewed on an oscilloscope, thereby confirming the results of the numerical integrations and its real contribution to engineering.

Integrability analysis of chaotic and hyperchaotic finance systems

“…where [X, Y, V, U ] T are the variations of [x, y, v, u] T and y = y(t) satisfies (24). The normal part of this system containing variables (X, V ) can be written as follows…”

“…Over the last years, hyperchaotic generators have been a topic of active search. Except Rössler's system, many hyperchaotic models have been found, both in theoretical and in practical ways; see, for instance, recent papers [21][22][23][24][25][26][27][28], and references therein.…”

“…Nowadays, in the era of computers, researchers have found chaotic behaviors in various known models and new ones with very complex dynamic are being proposed. For instance, let us mention the forced van der Pol model [31,32], the Kaldorian model [33,34], various types of the IS-LM model developed by Hick and Hansen [35][36][37], Goodwin's accelerator model [38,39], and recently, the Keynesian model [40], the Cournot-Puu system [41], the Huang-Li model [42,43] and its generalizations to 4D hyperchaotic systems [22,24,28], to cite just a few. We also highly recommend this year special issue, entitled Dynamics Models in Economics and Finance, published in [44].…”

A Financial Chaotic System Control Method Based on Intermittent Controller

“…Furthermore, the fractional Routh-Hurwitz conditions are used to control chaos in the proposed fractional-order system to its equilibria via the linear feedback control method. Kai et al [31] introduce a new four-dimensional hyperchaotic financial system on the basis of an established three-dimensional nonlinear financial system and a dynamic model by adding a controller term to consider the effect of control on the system. In [32], the system of integer and fractional differential equations is applied to model the financial system, and the control law is designed to synchronize two integer-order financial systems and two fractional-order financial systems.…”

Bifurcation Analysis and Stability Criterion for the Nonlinear Fractional‐Order Three‐Dimensional Financial System with Delay

“…Financial and economic problems have been intensively studied in the past years as the impact of the economic crisis on human society has become more and more serious . A lack of control over the financial system will likely cause greater losses.…”