The Numerical Connection between Map and its Differential Equation: Logistic and Other Systems

Abstract

Abstract The logistic map, in this article, is used to show the propagation of the numerical error for numerically solved logistic ODE's. When this equation is solved numerically (discretization procedure) the solution shows a chaotic behavior by varying the control parameter (integration step-size). However, the logistic ODE dx/dt = (k − 1)x − kx 2 also has an analytical solution, i.e., not presenting the signature of chaos. The connection between the two limits, analytical solution and chaos, allows us to establish a region of acceptability for the numerical error. Therefore the limits of the numerical process can be found in between the limits of the analytical and chaotic solutions. This study can be extended to any other ODE or map. For example, there is a similar numerical connection between the chaotic bi-dimensional Cat Map with its respective integrable ODE, which has an analytical solution. Our discussion applied to those particular systems is generic, that is, the solutions obtained from the numerical discretization can be more instable than those obtained from the continuous procedure. The important point is to find a compromise between the step-size and the propagation length in order to obtain the best numerical solution since the instabilities can be unavoidable.