In this paper, we study the stability of the zero equilibria of two close-to-symmetric systems of difference equations with exponential terms in the special case in which one of their eigenvalues is equal to −1 and the other eigenvalue has an absolute value of less than 1. In the present study, we use the approach of center manifold theory.

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.

No abstract

In this paper, we study the conditions under which the following close‐to‐symmetric system of difference equations with exponential terms: xn+1=a1ynb1+yn+c1xnek1−d1xn1+ek1−d1xn, yn+1=a2xnb2+xn+c2ynek2−d2yn1+ek2−d2yn where ai, bi, ci, di, and ki, for i=1,2, are real constants and the initial values x0 and y0 are real numbers, undergoes Neimark–Sacker, flip, and transcritical bifurcation. The analysis is conducted applying center manifold theory and the normal form bifurcation analysis.

In this paper, we study the stability of the zero equilibrium and the occurrence of flip bifurcation of the following system of difference equations: xn+1=a1ynb1+yn+c1xnek1−d1xn1+ek1−d1xn, yn+1=a2znb2+zn+c2ynek2−d2yn1+ek2−d2yn, zn+1=a3xnb3+xn+c3znek3−d3zn1+ek3−d3zn, where ai, bi, ci, di, and ki, for i=1, 2, and 3, are real constants and the initial values x0, y0, and z0 are real numbers. We study the stability of this system in the special case when one of the eigenvalues is equal to −1 and the remaining eigenvalues have absolute value less than 1, using the center manifold theory.