In this paper we present a local dynamics and investigate the global behavior of the following system of difference equations$x_{n+1}=ax_{n}^{3}+by_{n}^{3}$ $y_{n+1}=Ax_{n}^{3}+By_{n}^{3}$ $n\in\mathbb{N}_0$ with non-negative parameters and initial conditions $x_{0}$ and $y_{0}$ that are real numbers. We establish the relations for local stability of equilibriums and necessary and sufficient conditions for the existence of period-two solution(s). We then use this result to give global behavior results for special ranges of parameters and determine the basins of attraction of all equilibrium points.

In this paper, we observed the ordinary differential equation (ODE) system and determined the equilibrium points. To characterize them, we used the existing theory developed to visualize the behavior of the system. We describe the bifurcation that appears, which is characteristic of higher-dimensional systems, that is when a fixed point loses its stability without colliding with other points. Although it is difficult to determine the whole series of bifurcations that lead to chaos, we can say that it is a common opinion that it is precisely the Hopf bifurcation that leads to chaos when it comes to situations that occur in applications. Here, subcritical and supercritical bifurcation occurs, and we can say that subcritical bifurcation represents a much more dramatic situation and is potentially more dangerous than supercritical bifurcation, technically speaking. Namely, bifurcations or trajectories jump to a distant attractor, which can be a fixed point, limit cycle, infinity, or in spaces with three or more dimensions, a foreign attractor.

In this paper we observed the global dynamics and the occurrence of a certain bifurcation for the corresponding values of a certain rational difference equation of the second order with analyzed quadratic terms. The analysis of the local stability of the unique equilibrium point, as well as the unique periodic solution of period two, was performed in detail. The constraint of the equations on both sides for the corresponding values of the parameters is proved and on this basis the global stability is analyzed. The existence of Neimark-Sacker bifurcation with respect to the arrangement of equilibrium points has been proven. Thus, the basins of attraction have been determined in full for all the positive values of the parameters and all the positive initial conditions.

In this paper we proved the existence and local stability of prime period-two solutions for the equation 𝐱𝐧􀬾𝟏 􀵌 𝛂𝐱𝐧 𝟐 􀬾𝛃𝐱𝐧􀬾𝛄𝐱𝐧􀰷𝟏 𝐀𝐱𝐧 𝟐 􀬾𝐁𝐱𝐧􀬾𝐂𝐱𝐧􀰷𝟏 , for certain values of parameters ,,,A,B,C0, where ++>0 , A+B+C>0, and where the initial conditions x₋₁, x₀>0 are arbitrary real numbers such that at least one is strictly positive. For the obtained periodic solutions, it is possible to be locally asymptotically stable, saddle points or nonhyperbolic points. The existence of repeller points is not possible.

In this paper, a polynomial system of plane differential equations is observed. The stability of the non-hyperbolic equilibrium point was analyzed using normal forms. The nonlinear part of the differential equation system is simplified to the maximum. Two nonlinear transformations were used to simplify first the quadratic and then the cubic part of the system of equations.

In this paper we will look at the one system of ODE and analyze it. We aim to determine the points of equilibrium; examine their character and establish the existence of a bifurcation for the corresponding parameter value. A detailed analysis of local stability was performed for all values of the given parameter. For a certain value of the parameter, the existence of supercritical Hopf bifurcation of the observed system of differential equations has been proved. Also, the existence of a limit cycle that is always stable has been proved.