Iterations of a cubic map on a one-dimensional simplex

In this paper, we present a comprehensive analysis of the dynamics of a cubic operator defined by a non-stochastic matrix (containing at least one negative entry) on the one-dimensional simplex. This operator depends on two parameters. We identify all fixed points and some periodic points, and under certain conditions on the parameters, we describe the limit points. Furthermore, we determine a region in the parameter space where the corresponding operator exhibits chaotic behavior.

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