Hopf bifurcation and intermittent transition to hyperchaos in a novel strong four-dimensional hyperchaotic system

Abstract

This paper presents a new four-dimensional autonomous system having complex hyperchaotic dynamics. Basic properties of this new system are analyzed, and the complex dynamical behaviors are investigated by dynamical analysis approaches, such as time series, Lyapunov exponents' spectra, bifurcation diagram, phase portraits. Moreover, when this new system is hyperchaotic, its two positive Lyapunov exponents are much larger than those of hyperchaotic systems reported before, which implies the new system has strong hyperchaotic dynamics in itself. The Kaplan-Yorke dimension, Poincaré sections and the frequency spectra are also utilized to demonstrate the complexity of the hyperchaotic attractor. It is also observed that the system undergoes an intermittent transition from period directly to hyperchaos. The statistical analysis of the intermittency transition process reveals that the mean lifetime of laminar state between bursts obeys the power-law distribution. It is shown that in such four-dimensional continuous system, the occurrence of intermittency may indicate a transition from period to hyperchaos not only to chaos, which provides a possible route to hyperchaos. Besides, the local bifurcation in this system is analyzed and then a Hopf bifurcation is proved to occur when the appropriate bifurcation parameter passes the critical value.All the conditions of Hopf bifurcation are derived by applying center manifold theorem and Poincaré-Andronov-Hopf bifurcation theorem. Numerical simulation results show consistency with our theoretical analysis.

A Four-Dimensional Hyperchaotic Finance System and Its Control Problems

“…A dynamic model of finance has been reported in [29][30][31], which is composed of four subblocks, production, money, stock, and labor force, and expressed by four first-order differential equations. The model describes the time variations of four state variables: the interest rate , the investment demand , the price exponent , and the average profit margin .…”

“…Since this chaotic finance system is proposed, many works have been done for this finance system [29,30]. Recently, a novel hyperchaotic finance system [31] is presented in 2010 and some important results have been obtained. However, there are some limitations in the existing results.…”

Parameter Estimation for Dynamic Model of the Financial System

“…In the last few decades, much eff ort has been devoted to the theory of chaos control, primarily in the areas of unstable equilibrium points and unstable periodic solutions (Hubler, 1989). The methods which have been developed are in particular suitable for the case of chaos suppression in various chaotic systems (Wu, 2010;He, 2013;Chen, 2014).…”

Dynamic analysis and control of a new hyperchaotic finance system

“…Chaos in this system is not very obvious, and the largest Lyapunov exponent is 0.034432. It is found that the largest Lyapunov exponent presented in [15] is 29.79, but it is a strong four-dimensional hyperchaotic system which is not concerned with economy. …”