Lorenz chaotic system generated from Shimizu-Morioka system or Finance system: Differential geometric approach

Abstract

The problem of how to generate the Lorenz attractor from several nonlinear control systems is investigated in this paper. To be more precise, the conversions from the controlled Shimizu-Morioka system and the controlled Finance system to the Lorenz system are achieved by using the differential geometric control theory. For each case a scalar control input and a state transformation are proposed. The main approach of this paper is to convert all of those three-order systems into so called lower triangular forms which all have the same first two equations. Thus converting the controlled Shimizu-Morioka system or the controlled Finance system into the Lorenz attractor is feasible by choosing an appropriate scalar control input in the third equation of each of the two control systems. To this end, firstly, in order to use the tools of the differential geometry we construct a controlled Lorenz system by treating the vector field of the Lorenz attractor as the drift vector field and treating a linear vector field with three parameters as an input vector field. When those parameters are selected in a special manner, the conditions under which the controlled Lorenz system can be equivalently transformed into the lower triangular form are satisfied. Secondly, a state transformation, through which the controlled Lorenz system can be described as a lower triangular form, is obtained by a method like Gaussian elimination instead of solving three complicated partial differential equations. Employing several partial state transformations, choosing those three parameters and setting a scalar control input, we can reduce the equations of the controlled Lorenz system into its simplest lower triangular form. Thirdly, through two state transformations designed for the controlled Shimizu-Morioka system and the controlled Finance system respectively, the two control systems are converted into their lower triangular forms which are both similar to that of the Lorenz system in a way aforementioned. A smooth scalar controller is given to achieve the anti-control from the controlled Shimizu-Morioka system to the Lorenz attractor while another non-smooth scalar controller is designed to realize the generalized synchronization from the controlled Finance system to the Lorenz system no matter what the initial values of the two systems are. Finally, two numerical simulations demonstrate the control schemes designed in this paper.