We investigate the local stability and the global asymptotic stability of the difference equationxn+1=αxn2+βxnxn-1+γxn-1/Axn2+Bxnxn-1+Cxn-1,n=0,1,…with nonnegative parameters and initial conditions such thatAxn2+Bxnxn-1+Cxn-1>0, for alln≥0. We obtain the local stability of the equilibrium for all values of parameters and give some global asymptotic stability results for some values of the parameters. We also obtain global dynamics in the special case, whereβ=B=0, in which case we show that such equation exhibits a global period doubling bifurcation.
In this paper we will present the Julia set and the global behavior of a
quadratic second order difference equation of type xn+1 = axnxn-1 + ax2n-1
+ bxn-1 where a > 0 and 0 ? b < 1 with non-negative initial conditions.
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.
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.
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