Abstract
This paper deals with the analytical representation of bifurcations of each 3D discrete dynamics depending on the set of bifurcation parameters. The procedure of bifurcation analysis proposed in this paper represents the 3D elaboration and specification of the general algorithm of then-dimensional linear bifurcation analysis proposed by the author earlier. It is proven that 3D domain of asymptotic stability (attraction) of the fixed point for a given 3D discrete dynamics is bounded by three critical bifurcation surfaces: the divergence, flip and flutter surfaces. The analytical construction of these surfaces is achieved with the help of classical Routh–Hurvitz conditions of asymptotic stability. As an application the adjustment process proposed by T. Puu for the Cournot oligopoly model is considered in detail.
Highlights
The purpose of this paper is to construct the 3D analytical representation of the general procedure of linear bifurcation analysis developed by Sonis (1993, 1997)
Jacobi matrix of the linear approximation of the dynamics equals to 1; theflip surface corresponding to the existence of the eigenvalue 1, and theflutter surface corresponding to the pair of complex conjugated eigenvalues with absolute values equal to 1
First step of the analysis includes the construction of the Jacobi matrix of a linear approximation of the non-linear dynamics (2.1): OF, OF, OFt
Summary
The purpose of this paper is to construct the 3D analytical representation of the general procedure of linear bifurcation analysis developed by Sonis (1993, 1997). It will be proven further that the domain of attraction of the fixed point of 3D discrete dynamics is bounded by three critical bifurcation surfaces: the divergence surface corresponding to the case in which one of the eigenvalues of the Jacobi matrix of the linear approximation of the dynamics equals to 1; theflip surface corresponding to the existence of the eigenvalue 1, and theflutter surface corresponding to the pair of complex conjugated eigenvalues with absolute values equal to 1. The crossing of these surfaces by the movement of the fixed point will generate the plethora of all possible bifurcation phenomena
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