The rank of a sequence as an indicator of chaos in discrete nonlinear dynamical systems

An alternative technique for clocking the convergence of iterative chaotic maps is proposed in this paper. It is based on the concept of the Hankel rank of a solution of the discrete nonlinear dynamical system. Computation and visualization of pseudoranks in the space of system’s parameters and initial conditions provides the insight into the fractal nature of the dynamical attractor and reveals the stable, the unstable manifold and the convergence properties of the system. All these manifolds are produced by a simple and a straightforward computational rule and are intertwined in one figure. On the other hand, the computation of ranks of subsequences of solutions helps to identify and assess the sensitivity of the system to initial conditions and can be used as a simple and effective numerical tool for qualitative investigation of discrete iterative maps.

Research Highlights
► The H-rank technique is proposed to clock the convergence of iterative maps.
► Stable, unstable manifold and the manifold of non-asymptotic convergence intertwine.
► The sensitivity to initial conditions is assessed by the H-rank method.