Development of a Lyapunov Exponent Based Chaos Diagram in the Parameter Plane of Logistic Map

Abstract

A one dimensional model of population growth called logistic map can be used as platform for introducing beginners to the phenomenon of chaos and nonlinear dynamics. Despite the simplicity of logistic map, it has been used with success for the introduction of fixed point attractor, periodic and aperiodic responses, sensitivity to initial conditions, return map and bifurcation diagram. This understanding motivated the present study to develop two dimensional chaos diagram for a one dimensional logistic map as a way of introducing the beginners to the theories of fractals and chaos. Simulation of unsteady solutions, steady solutions and its corresponding Lyapunov exponent characterisation of logistic map were effected for selection of drive parameters for various grid resolutions, constant step size and at one grid point a time from 0.3 and 0.5 initial conditions. Validation were made of graphical results of parameter versus Lyapunov exponent and the parabola-attractors. The Lyapunov exponent characterisation results were grouped into three classifications: divergence, periodic and chaotic based on average Lyapunov value. There is qualitative agreement of validation results. The total number of grid points with divergence or periodic or chaotic response increases exponentially with increasing resolutions. The zoomed parameters counterpart has average 0%, 60% and 40% of divergent, periodic and chaotic results respectively across resolutions. The corresponding chaos diagram exhibited fractal structures by its layers of order within chaos as can be found in the bifurcation diagrams of nonlinear dynamical systems.