Linear Stability Conditions for a First Order 4-Dimensional Discrete Dynamic

Abstract

MotivationsA standard method used to determine the linear stability of a first order discrete dynamic equates to determining whether or not the eigenvalues of the Jacobian evaluated at the equilibrium all have magnitude less than one. This is often quite algebraically cumbersome in many mathematical models where the elements of the Jacobian are expressed as a combination of many different parameters. In those cases the direct calculation of the eigenvalues is not at all practical and the Gershgorin theorem is too harsh as it only guarantees linear stability if all the discs happen to lie inside the unit circle. In the case of a nonlinear 2-dimensional discrete dynamical system one can calculate the trace, tr (J), and determinant, det (J), of the Jacobian evaluated at the equilibrium and if use the set of inequalities.( ) ( ) ( ) 1, 1det J tr J det J < < + that describe Thompson's [1] stability triangle. This set of two in equalities is necessary and sufficient for linear stability of the 2-dimensional discrete dynamic, that is, satisfying both conditions equates to both eigenvalues of the 2×2 Jacobian having magnitude less than 1. Linear stability conditions of a first order 3-dimensional discrete dynamic have been derived [2]. In the case of a non-linear 3-dimensional discrete dynamical system one can calculate the trace, tr (J), sum of principle minors, Σ M i (J), and determinant, det(J), of the Jacobian evaluated at the equilibrium and if use the set of two inequalitiesThis set of two inequalities is necessary and sufficient for linear stability of the 3-dimensional discrete dynamic, that is, satisfying both conditions equates to all of the eigenvalues of the 3×3 Jacobian having magnitude less than 1. Note that the |det(J)|<1 condition in the 3-dimensional case [2] is always satisfied if the above two inequalities are satisfied. These conditions proved very useful in my dynamic evolutionary game theory research and generated much interest as a tool for other researchers as well. The 3-dimensional conditions have aided in researchers in economics [3,4], and finance [5]. Could an analogous approach be effective with a 4-dimensional discrete dynamic? As in the 3-dimensional case necessary and sufficient conditions are needed that equate to linear stability. These conditions will be derived in terms of the coefficient of the characteristic equation of the Jacobian such as the determinant, trace, principle sum of minors of the Jacobian, and sum of the determinants of the principal 2×2 submatrices of the Jacobian. These four elements are the magnitudes of the coefficients of the characteristic polynomial [6]. Stability ConditionsLinear stability conditions for a first order 4-dimensional discrete dynamic, F x y z w = are equivalent to the necessary and sufficient conditions that the Jacobian, J, evaluated at the equilibrium has eigenvalues of magnitude less than one. These conditions will be expressed in terms of the determinant of the Jacobian, det(J), trace of the Jacobian, tr(J), sum of the determinants of the p...