Algebraic approach for the reconstruction of Rössler system from the $x_{3}-$ variable

Abstract

In this paper we propose a simple method to identify the unknown parameters and to estimate the underlying variables from a given chaotic time series $\{x_{3}(t_{k})\}_{0}^{k=n}$ of the three-dimensional Rössler system (RS). The reconstruction of the RS from its $x_{3}-$ variable is known to be considerably more difficult than reconstruction from its two other variables. We show that the system is observable and algebraically identifiable with respect to the auxiliary output $\ln (x_{3})$, hence, a differential parameterization of the output and its time derivatives can be obtained. Based on these facts, we proceed to form an extended re-parameterized system (linear-in-the -parameters), which turns out to be invertible, allowing us to estimate the variables and missing parameters.