We investigate the local and global character of the unique equilibrium point and boundedness of the solutions of certain homogeneous fractional difference equation with quadratic terms. Also, we consider Neimark–Sacker bifurcations and give the asymptotic approximation of the invariant curve.

<jats:p>We investigate the global asymptotic stability of the following second order rational difference equation of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mi>B</mml:mi><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mi>b</mml:mi><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:msubsup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfenced></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0,1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo></mml:math> where the parameters <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>B</mml:mi></mml:mrow></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>F</mml:mi></mml:mrow></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:math> and initial conditions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mrow><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math> are positive real numbers. The map associated with this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the parametric space. In some cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability. Also, we show that considered equation exhibits the Naimark-Sacker bifurcation resulting in the existence of the locally stable periodic solution of unknown period.</jats:p>

We investigate the global asymptotic stability and Naimark-Sacker bifurcation of the dierence equation xn+1 = F bxnxn 1 +cx 2 1 +f ; n = 0; 1;:::

AbstractWe investigate the global asymptotic behavior of solutions of the following anti-competitive system of rational difference equations: xn+1=γ1ynA1+xn,yn+1=β2xnA2+B2xn+yn,n=0,1,…, where the parameters γ1, β2, A1, A2 and B2 are positive numbers and the initial conditions (x0,y0) are arbitrary nonnegative numbers. We find the basins of attraction of all attractors of this system, which are the equilibrium points and period-two solutions.MSC:39A10, 39A11.

We investigate the global asymptotic behavior of solutions of the following anti-competitive system of rational difference equations: xn+1=γ1ynA1+xn,yn+1=β2xnA2+B2xn+yn,n=0,1,…, where the parameters γ1, β2, A1, A2 and B2 are positive numbers and the initial conditions (x0,y0) are arbitrary nonnegative numbers. We find the basins of attraction of all attractors of this system, which are the equilibrium points and period-two solutions. MSC:39A10, 39A11.