Implicit one-dimensional discrete maps and their connection with existence problem of chaotic dynamics in 3-D systems of differential equations (Q440795)

The present paper is a continuation of the work [Appl. Math. Comput. 218, No. 8, 4546--4566 (2011; Zbl 1247.34069)].NEWLINENEWLINEIn this paper, the following autonomous chaotic system is considered NEWLINE\[NEWLINE\begin{cases} \dot{x}(t)= a_1 x(t) + a_{11} y^2(t) + a_{12}y(t) z(t) + a_{22} z^2(t),\\ \dot{x}(t)= b_1 y(t)+ c_1 z(t) + bx(t)y (t),\\ \dot{z}(t)= c_1(t)+ b_1 z(t) + cx(t)z(t), \end{cases}NEWLINE\]NEWLINE NEWLINEwhere \(a_1\), \(a_{11}\), \(a_{12}\),\dots, \(c_1\), \(b_1\), \(c\) are coefficients. Each equation of the system has a quadratic term.NEWLINENEWLINEFor some values of the coefficients, the system can exhibit Lorenz-type attractors. Basic properties of the dynamical system have been analyzed by the means of Lyapunov exponents and Poincaré map. This analysis verifies the existence of chaotic dynamics in the system. The author gives a detailed investigation of the characteristics of this system.