How can I find linear functions that will tend to equilibrium in a given system?

[Mynock]

Hey, I'm trying to find two linear functions, f(r) and h(r) for the following system:

dp/dt = A*f(r)

dr/dt = -B*h(r)

where A and B are constants greater than zero. I'm trying to find linear functions that will tend to equillibrium, and also where

limit df(r)/dt = 0
r->0

I have been trying various linear functions and have been unable to come up with a solution. Is there a solution? and if so, what would be one? It's probalby something simple that I'm overlooking. Any help would be appreciated. Thanks.

 

[leach]

The linear (homogeneous) functions in one variable, 'r' in this case, have the general form:

$$f(r) \ = \ \alpha\cdot r,\quad g(r) \ = \ \beta\cdot r, \qquad \alpha,\beta\in\mathbb{R}$$

So this system can be trivially solved, since the variables are separated. On one hand, you have:

$$\dot{r}\ = \ -B\beta\, r$$

so

$$r(t) \ = \ r_0e^{-B\beta t}$$

The condition for stability is obviously $$\beta > 0$$.

On the other hand:

$$\dot{p}\quad = \quad A\alpha\, r \quad = \quad A\alpha r_0 e^{-B\beta t}$$

Then,

$$p(t) \quad = \quad \gamma \ - \ \frac{A\alpha r_0}{B\beta} e^{-B\beta t}$$

where $$\gamma$$ is an integration constant. This means that, as long as $$\beta > 0$$, your system will be stable, with equilibrium point $$p = \gamma, r = 0$$.

 

[tacman]

Hi there,

Finding linear functions that will tend to equilibrium in this system is definitely possible. Let's start by looking at the first equation:

dp/dt = A*f(r)

To find a linear function that will tend to equilibrium, we need to find a function f(r) that will make the derivative of p, dp/dt, equal to 0. This means that the slope of the function must be 0 at all points, which will result in a horizontal line. Since we want the function to tend to equilibrium, this horizontal line must also intersect with the y-axis at the equilibrium point.

One way to achieve this is to have f(r) be a constant function that is equal to the equilibrium point. So our first linear function could be:

f(r) = c, where c is the equilibrium point.

Now let's look at the second equation:

dr/dt = -B*h(r)

Similarly, to find a linear function that will tend to equilibrium, we need to find a function h(r) that will make the derivative of r, dr/dt, equal to 0. Again, this means that the slope of the function must be 0 at all points, resulting in a horizontal line that intersects with the y-axis at the equilibrium point.

Following the same logic as before, we can have h(r) be a constant function equal to the equilibrium point. So our second linear function could be:

h(r) = c, where c is the equilibrium point.

Now, to satisfy the condition of the limit of df(r)/dt = 0 as r approaches 0, we can choose any value for c as long as it is a positive constant. This will ensure that the slope of both functions is 0 at all points, including at r = 0.

So to summarize, our two linear functions for this system would be:

f(r) = c, where c is the equilibrium point

h(r) = c, where c is a positive constant

I hope this helps and makes sense. Let me know if you have any further questions. Good luck!