This paper investigates an autonomous predator-prey system of difference equations with three equilibrium points and exhibits chaos in the sense of Li-Yorke in the positive equilibrium point. Numerical simulations are presented to illustrate our results.

We investigate a discrete counterpart of planar dynamical system of nonlinear differential equations induced by kinetic differential equations for a two-species chemical reaction. Chemical reactions exhibit a wide range of dynamical behavior. We show how the theoretical analysis provides insight into the potential behavior of chemical reaction systems, determining the areas of parametric space which indicate scenarios for local stability, then for one type of bifurcation co-dimension one and one type of bifurcation co-dimension two. Precisely, we prove the existence of period-doubling bifurcation and 1:2 resonance bifurcation also, by using the center manifold theorem and the technique of normal forms. All mathematical investigations are illustrated with numerical examples, bifurcation diagrams, Lyapunov exponents and phase portraits.

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.

: We investigate the global asymptotic stability of the difference equation of the form

<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;:::

T. Wanner We investigate global dynamics of the equation xn+1=xn−12bxnxn−1+cxn−12+f,n=0,1,2,…, where the parameters b,c, and f are nonnegative numbers with condition b + c > 0,f ≠ 0 and the initial conditions x−1,x0 are arbitrary nonnegative numbers such that x−1+x0>0. We obtain precise characterization of basins of attraction of all attractors of this equation and describe the dynamics in terms of bifurcations of period‐two solutions. Copyright © 2015 John Wiley & Sons, Ltd.