Overdetermined SDE and other problems

[econslc]

Hello all,

I am working with an economic problem and have ended up with the following dynamical system which is overdetermined, Jacobian has zero determinant (so one zero eigenvalue?) and probably other problems. The system looks something like this:

dot{x1}=c1 x_2+c2 x_4/(x_2 x_1)
dot{x2}=x_3*(1-x_1)+c3
dot{x3}=c4 x_1
dot{x4}=c5 x_1

where c are parameters or constants. For overall stability first condition requires an x_1 s.t. dot{x_3}=dot{x_4}=0. Next even if that is not the case I still want to say something about how fast the system diverges. In any case if I assume the condition and get a fixed point for x_1, it means I can solve for x_3 from dot{x_2}=0 but then I end up with one equation dot{x_1} and two unknowns x_4 and x_2. Mathematically that can not be solved without further assumptions. Should/Can I do that numerically? Is there a formal people approach this problems?

I have been playing around with some data and I can get a feel about how fast the model diverges based on the choices for the parameters. My question is, can I can analyze in a formal way any qualitative features of such model, or is it simply badly specified and I should just forget about it? Any references you can direct me to would be very much appreciated. Finally I tried playing with this model in Mathematica with NDSolve but it gives me "overdetermined" error. Any way to get around that?
Many thanks!

 

[gato_]

It is overdetermined because third and fourth equations are proportional. Repeat the analysis removing the third equation, and setting $$x_{3}=c_{6}+\frac{c_{4}}{c_{5}} x_{4}$$
By the way, the equation is singular at $$x_{1}=0$$

 

[gato_]

Here is how you make the local analysis, after eliminating x3 from there. As x1=0 is singular, we will try $$x_{1}=\epsilon\rightarrow 0$$. The equilibrium points are
$$x_{1}=\epsilon , x_{4}=-c_{5}(c_{3}+c_{6})/c_{4} , x_{2}=\gamma/\epsilon^{1/2}$$

Where $$\gamma=\pm \sqrt(\frac{c_{2} c_{5}}{c_{1} c_{4}}(c_{3}+c_{6}))$$. After computing the Jacobian, the eigenvalue equation is given by:

$$\epsilon^{3/2}\Lambda^{3}-\gamma \Lambda^{2}-[\epsilon c_{2}c_{5}/(c_{1}\gamma)+2c_{1}c_{3}\epsilon^{3/2}]\Lambda-2c_{1}c_{4}\epsilon^{3/2}=0$$

For small values of $$\epsilon$$, you get the following approximation:

$$\Lambda_{1}\sim \gamma\epsilon^{-3/2}+O(\epsilon^{1/2})$$

$$\Lambda_{2,3}\sim \frac{c_{2}c_{5}}{2\gamma^{2}c_{1}}(-1\pm i)\epsilon^{1/2}$$

Showing instability in the direction corresponding to the first eigenvalue, for $$\gamma>0$$, and stability otherwise.

This needs some revising...

 

[econslc]

Gato, thanks a lot for taking the time to answer my post. I will work on your suggestions.

 

[blue_raver22]

I understand the frustration of encountering an overdetermined SDE and other problems in your research. It can be disheartening to hit a roadblock in your analysis or to have your model be deemed "badly specified." However, I want to assure you that these challenges are a natural part of the scientific process and can lead to valuable insights and discoveries.

To address your specific questions, there are a few things to consider. First, it is important to carefully examine your model and assumptions. Are there any simplifications or assumptions that can be made to make the system solvable? Can you introduce additional parameters to make the system less overdetermined? This may require some creative thinking and trial and error, but it could lead to a more manageable model.

In terms of your question about solving the system numerically, it is possible to do so but it may not provide the most accurate results. Numerical methods are useful for exploring the behavior of a system and making predictions, but they are not a substitute for analytical solutions. Therefore, if possible, it may be worth exploring analytical solutions or approximations before resorting to numerical methods.

As for analyzing qualitative features of your model, there are various techniques and tools available. One approach is to use bifurcation analysis, which allows you to study how the behavior of the system changes as parameters are varied. Another option is to use stability analysis, which can help determine the stability of fixed points in your system. Both of these methods require some mathematical background and may be challenging to implement, but they can provide valuable insights into the behavior of your model.

In terms of references, I would recommend looking into books or papers on dynamical systems and nonlinear dynamics. There are also various software packages and online resources available for conducting bifurcation and stability analysis.

Overall, I encourage you to continue exploring and analyzing your model, even if it may seem daunting at times. Scientific progress often involves facing and overcoming challenges, and I have no doubt that you will find a way to make progress with your research. Best of luck!