Analysis of two variable Non-linear difference equation

[cuak2000]

Hey everyone,

My understanding of difference equation is very limited and, after consulting a few books, I decided to post this here to get some ideas or at least some references on where to look.

I have a symmetric first order difference equation system in two variables, X and Y.
Though convergence might be interesting too, my main question with this system is the following:
Given that initially X(0) > Y(0) under what parameter values will Y catch up (or not) on X?

Just to illustrate here is my system, though I'm not sure this will be the final form:

\begin{equation}
X_{t+1} = \frac{ d*e^{bX_t} (1+m)p_{sub}(a - \frac{e^{bX_t} }{e^{bX_t} + e^{bY_t} } )^{\lambda} \, X_t }{ (1+g+m)p_{r}( e^{bX_t} + e^{bY_t}) }
\end{equation}

and

\begin{equation}
Y_{t+1} = \frac{ d*e^{bY_t} (1+m)p_{sub}(a - \frac{e^{bY_t} }{e^{bY_t} + e^{bX_t} } )^{\lambda} \, Y_t }{ (1+g+m)p_{r}( e^{bY_t} + e^{bX_t}) }
\end{equation}

Where d , m, p_sub , a, b, g and \lambda are positive parameters (I'll probably fix 5 numerically and analyse the system varying \lambda and d ).

My first attempt was to use the ratio of change X_{t+1} / X_t .
If X(0) > Y(0) , then if X_{t+1} / X_t > Y_{t+1} / Y_t , Y will never catch up on X. On the other hand, if Y grows faster than X and the difference doesn't go to zero, then it might be reasonable to speak about convergence. I haven't figured out what the precise requirements would be.

Like I said, this system might not be exactly what I will use (though it will look similar), so any specific or general ideas are welcome.
Thanks!

c

 

[jvicens]

Hi there,

Your question is indeed interesting and complex. To fully understand the behavior of your system, you will need to analyze it using numerical methods or by solving it analytically. However, here are a few general ideas that might help you in your analysis:

1. Look at the equilibrium points of the system: To understand the behavior of your system, it's important to first identify the equilibrium points (or steady states) where the system is stable and doesn't change over time. In your case, the equilibrium points would be when X_{t+1} = X_t and Y_{t+1} = Y_t. This will give you an idea of the long-term behavior of your system.

2. Use linearization: If your system is nonlinear, you can use linearization techniques to approximate its behavior around the equilibrium points. This will help you analyze the stability of the system and identify any critical points that might affect the convergence of Y to X.

3. Vary the parameters: As you mentioned, varying the parameters \lambda and d can give you insights into the behavior of the system. You can also try varying other parameters such as b and m to see how they affect the convergence of Y to X.

4. Use bifurcation analysis: Bifurcation analysis is a powerful tool to study the behavior of nonlinear systems. By varying a parameter (such as \lambda or d) and observing how it affects the behavior of the system, you can identify any bifurcation points where the system changes its behavior drastically.

5. Consider different initial conditions: In addition to X(0) > Y(0), you can also consider other initial conditions to see how they affect the convergence of Y to X. This will give you a broader understanding of the system's behavior.

I hope these ideas help you in your analysis. Good luck!