arXiv subjects

Sukanya Basu

Publications and source records attributed to Sukanya Basu.

2 recordsLinked to original sources

arXiv2008

On the Global Behavior of Solutions to a Planar System of Difference Equations

We establish the relation between local stability of equilibria and slopes of critical curves for a specific class of difference equations. We then use this result to give global behavior results for nonnegative solutions of the system of difference equations \begin{equation*} %\tag{LGIN} \begin{array}{rcl} x_{n+1} & = & \displaystyle \frac{b_1 x_n}{1+x_n+c_1 y_{n}} +h_1 y_{n+1} & = & \displaystyle \frac{b_2 y_n}{1+y_n+c_2 x_{n}} +h_2 \end{array} \quad n=0,1,..., \quad (x_0,y_0) \in [0,\infty)\times [0,\infty) \end{equation*} with positive parameters. In particular, we show that the system has between one and three equilibria, and that the number of equilibria determines global behavior as follows: if there is only one equilibrium, then it is globally asymptotically stable. If there are two equilibria, then one is a local attractor and the other one is nonhyperbolic. If there are three equilibria, then they are linearly ordered in the south-east ordering of the plane, and consist of a local attractor, a saddle point, and another local attractor. Finally, we give sufficient conditions for having a unique equilibrium.

math.DS↗

arXiv2008

Global Behavior of Solutions to Two Classes of Second Order Rational Difference Equations

For nonnegative real numbers $α$, $β$, $γ$, $A$, $B$ and $C$ such that $B+C>0$ and $α+β+γ>0$, the difference equation \begin{equation*} x_{n+1}=\displaystyle\frac{α+βx_{n}+γx_{n-1}}{A+B x_{n}+C x_{n-1}}, \quad n=0,1,2,... %, \quad x_{-1},x_{0}\in [0,\infty) \end{equation*} has a unique positive equilibrium. A proof is given here for the following statements: \medskip \noindent Theorem 1. {\it For every choice of positive parameters $α$, $β$, $γ$, $A$, $B$ and $C$, all solutions to the difference equation \begin{equation*} x_{n+1}=\displaystyle\frac{α+βx_{n}+γx_{n-1}}{A+B x_{n}+C x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1},x_{0}\in [0,\infty) \end{equation*} converge to the positive equilibrium or to a prime period-two solution.} \medskip \noindent Theorem 2. {\it For every choice of positive parameters $α$, $β$, $γ$, $A$, $B$ and $C$, all solutions to the difference equation \begin{equation*} x_{n+1}= \displaystyle\frac{α+βx_{n}+γx_{n-1}}{B x_{n}+C x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1},x_{0}\in (0,\infty) \end{equation*} converge to the positive equilibrium or to a prime period-two solution.}

math.DS↗