Related papers: On rational systems in the plane. I. Riccati Cases
Dynamics of the delay rational difference equation $\displaystyle{z_{n+1}=\frac{\alpha+\beta z_{n-k}}{\gamma - z_{n}}}$ with complex parameters $\alpha$, $\beta$, $\gamma$ and arbitrary complex initial conditions is investigated. Existence…
The dynamics of the second order rational difference equation $\displaystyle{z_{n+1}=\frac{\alpha + \beta z_{n}+ \gamma z_{n-1}}{A + B z_n + C z_{n-1}}}$ with complex parameters and arbitrary complex initial conditions is investigated. In…
We obtain the solution of the fourth order difference equation $$ x_{n+1}=\frac{ \alpha x_{n-3}}{A+B x_{n-1}x_{n-3}}$$ with the initial conditions; $x_{-3}=d,$ $x_{-2}=c,$ $x_{-1}=b,$ and $x_{0}=a$ are arbitrary nonzero real numbers,…
We derive some analytic closed-form solutions for a class of Riccati equation y'(x)-\lambda_0(x)y(x)\pm y^2(x)=\pm s_0(x), where \lambda_0(x), s_0(x) are C^{\infty}-functions. We show that if \delta_n=\lambda_n s_{n-1}-\lambda_{n-1}s_n=0,…
The dynamics of the second order rational difference equation in the title with complex parameters and arbitrary complex initial conditions is investigated. Two associated difference equations are also studied. The solutions in the complex…
The asymptotic behavior (such as convergence to an equilibrium, convergence to a 2-cycle, and divergence to infinity) of solutions of the following multi-parameter, rational, second order difference equation x_{n+1} =(ax_{n}^3+…
The dynamics of the family of maps $\displaystyle{f_{\alpha, \beta, \gamma, \delta}(z)=\frac{\alpha z + \beta}{\gamma z^2 +\delta z}}$ in complex plane is investigated computationally. This dynamical system $z_{n+1}=f_{\alpha, \beta,…
In this note we show the the system of difference equations $$ x_{n+1}=\dfrac{ay_{n-2}x_{n-1}y_n+bx_{n-1}y_{n-2}+cy_{n-2}+d}{y_{n-2}x_{n-1}y_n},$$ $$y_{n+1}=\dfrac{ax_{n-2}y_{n-1}x_n+by_{n-1}x_{n-2}+cx_{n-2}+d}{x_{n-2}y_{n-1}x_n},$$ where…
\begin{abstract} In this paper, we consider the following system of difference equations \begin{equation*} x_{n+1}=\alpha+\dfrac{y_{n}^p}{y_{n-2}^p},\ y_{n+1}=\alpha+ \dfrac{x_{n}^q}{x_{n-2}^q}, \ n=0, 1, 2, ... \end{equation*} where…
In the Euclidean plane ${\bf{E}}^2$, fix four pairwise distinct points \begin{equation*} \label{eqA} \begin{array}{ccc} A=(a_1,a_2),\ B=(b_1,b_2),\ C=(c_1,c_2),\ D=(d_1,d_2), \end{array} \end{equation*} together with four non-zero real…
A generalization of the already studied transformations of the linear differential equation into a system of the first order equations is given. The proposed transformation gives possibility to get new forms of the N-dimensional system of…
In this work we solve in closed form the system of difference equations \begin{equation*} x_{n+1}=\dfrac{ay_nx_{n-1}+bx_{n-1}+c}{y_nx_{n-1}},\; y_{n+1}=\dfrac{ax_ny_{n-1}+by_{n-1}+c}{x_ny_{n-1}},\;n=0,1,..., \end{equation*} where the…
The Riccati equation method is used for study the oscillatory and non oscillatory behavior of solutions of systems of two first order linear two by two dimensional matrix differential equations. An integral and an interval oscillatory…
When $A$ and $B$ are subsets of the integers in $[1,X]$ and $[1,Y]$ respectively, with $|A| \geq \alpha X$ and $|B| \geq \beta X$, we show that the number of rational numbers expressible as $a/b$ with $(a,b)$ in $A \times B$ is $\gg (\alpha…
In this paper the asymptotic stability of equilibria and periodic points of the following higher order rational difference Equation x_{n+1} =(alpha x_{n-k})/(1+x_{n}...x_{n-k}), k>=1, n=0,1,... is studied where the parameters ?alpha, betta,…
For any irrational $\alpha > 0$ and any initial value $z_{-1} \in \mathbb{C}$, we define a sequence of complex numbers $(z_n)_{n=0}^{\infty}$ as follows: $z_n$ is $z_{n-1} + e^{2 \pi i \alpha n}$ or $z_{n-1} - e^{2 \pi i \alpha n}$,…
We give a complete description of the qualitative behavior of the second-order rational difference equation #166. We also establish the boundedness character for the rational system in the plane #(8,30).
This work presents a newly renovated approach to the analysis of second-order Riccati equations from the point of view of the theory of Lie systems. We show that these equations can be mapped into Lie systems through certain Legendre…
It is proved that the members of the Riccati hierarchy, the so-called Riccati chain equations, can be considered as particular cases of projective Riccati equations, which greatly simplifies the study of the Riccati hierarchy. This also…
Using the Chiellini condition for integrability we derive explicit solutions for a generalized system of Riccati equations $\ddot{x}+\alpha x^{2n+1}\dot{x}+x^{4n+3}=0$ by reduction to the first-order Abel equation assuming the parameter…