Stability of two 3 × 3 close‐to‐cyclic systems of exponential difference equations

In this paper, we study the stability of the zero equilibria of the following systems of difference equations: xn+1=a1xn+b1yne−xn,yn+1=a2yn+b2zne−yn,zn+1=a3zn+b3xne−zn, xn+1=a1yn+b1xne−yn,yn+1=a2zn+b2yne−zn,zn+1=a3xn+b3zne−xn, where a1, a2, a3, b1, b2, and b3 are real constants, and the initial valu...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences 2018-11, Vol.41 (17), p.7936-7948
Main Authors: Mylona, Chrysoula, Psarros, Nikolaos, Papaschinopoulos, Garyfalos, Schinas, Christos J.
Format: Article
Language:English
Subjects:
centre manifold
Centre manifold theory
Difference equations
discrete dynamics
dynamical systems
Eigenvalues
Mathematical analysis
population dynamics
Real numbers
Stability
systems of difference equations
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Summary:In this paper, we study the stability of the zero equilibria of the following systems of difference equations: xn+1=a1xn+b1yne−xn,yn+1=a2yn+b2zne−yn,zn+1=a3zn+b3xne−zn, xn+1=a1yn+b1xne−yn,yn+1=a2zn+b2yne−zn,zn+1=a3xn+b3zne−xn, where a1, a2, a3, b1, b2, and b3 are real constants, and the initial values x0, y0, and z0 are real numbers. We study the stability of those systems in the special case when one of the eigenvalues of the coefficient matrix of the corresponding linearized systems is equal to −1 and the remaining eigenvalues have absolute value less than 1, using centre manifold theory.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.5256