Numerical simulation of chaotic dynamical systems by the method of differential quadrature

In this paper, the differential quadrature (DQ) method is employed to solve some nonlinear chaotic systems of ordinary differential equations (ODEs). Here, the method is applied to chaotic Lorenz, Chen, Genesio and Rössler systems. The first three chaotic systems are described by three-dimensional systems of ODEs while the last hyperchaotic system is a four-dimensional system of ODEs. It is found that the DQ method is unconditionally stable in solving first-order ODEs. But, care should be taken to choose a time step when applying the DQ method to nonlinear chaotic systems. Similar to all conventional unconditionally stable time integration schemes, the unconditionally stable DQ time integration scheme may also be possible to produce inaccurate results for nonlinear chaotic systems with an inappropriately too large time step sizes. Numerical comparisons are made between the DQ method and the conventional fourth-order Runge–Kutta method (RK4). It is revealed that the DQ method can produce better accuracy than the RK4 using larger time step sizes.