The geometry of chaos: Dynamics of a nonlinear second-order difference equation

Abstract

The nonlinear second-order difference equationx _n+1=ax_n(1-x_n−1), where 0≦x _nX≦1 anda ≧1, is examined from varying points of view, analytical, numerical and geometrical. An analytic expression is obtained for an invariant attracting curveC _∞ (a) in phase space, which becomes the central object of study. This basic curve, which replaces the simple parabolic shape typical of many analogous first-order models, may have a complicated geometrical structure. As the parametera increases,C _∞(a) undergoes transformations characterized by the dynamical descriptions: stable node→stable focus→stable limit cycle →chaotic attractor. Although the limited characterization ofchaos by the appearance of nonperiodic solutions and solutions of arbitrarily large period is relied upon, this appears to be only a simplified approximation of the real behavior of solutions. Trajectories (x _n, x_n+1),n=0,1,…, are calculated using the related nonlinear planar mapT _a(x,y)=(y,ay(1−x)), and regions of persistence and escape are described for characteristic values ofa. The study of persistence, of even more fundamental interest than the associated problems of periodicity and stability, receives special attention. We introduce a geometrical model, similar in many respects to that for the well-known analoguex _n+1=ax_n(1−x _n), but having several new and important features. It appears that as the parametera increases in the chaotic regime there are infinitely many intermittent bursts of increase in the probability that any initial point (x ₀, x₁) will persist in the unit square under successive iterations of the mappingT _a, an unexpected property that should be of interest for applications. A discussion of the applicability of these results to population dynamics theory is given, and it is suggested that such equations might find useful application to problems in developmental biology as well.