Linear stability conditions for a first order \(n\)-dimensional mapping
Stability is one of the most important concepts in dynamical systems. It is well known that a fixed point of a discrete dynamical system is locally asymptotically stable if all the eigenvalues of the Jacobian matrix, evaluated at the fixed point, have absolute value less than one. However, in the most cases, one is not able to compute the eigenvalues. The challenge here is to provide stability conditions without knowing the eigenvalues of the Jacobian matrix. According to this, the author's main aim is to present an alternative way to compute the coefficients of a characteristic polynomial of a matrix via its trace, determinant and the sum of the minors. The second section of the paper contains notations and preliminaries. In the third section the author revisits the characteristic polynomial of a matrix and determines its coefficients for future purposes. The main results are contained in the fourth section, where the necessary and sufficient conditions for the local stability of a first-order discrete dynamical system are obtained. Here, \(2\)-, \(3\)-, \(4\)-, \(5\)- and \(n\)-dimensional maps are considered in detail. Finally, the last section contains some illustrative examples: the Cournot Duopoly game model, the delay logistic model and the Ricker competition model of \(4\) species.
scientific article; zbMATH DE number 7345483