# Autonomous Systems and Stability

• Lancen
In summary: So, I assumed the initial condition to be y(0)=x(0)+1 and solved for x. I got x=-1.5 so the system is stable.

#### Lancen

I have a problem regarding the equations dx/dt=x-xy and dy/dt=y+2xy. I need to find the critical points of this system and denote if they are stable or asymptotic or whatever. I flipped through the section on this and you can find the critical points by setting dx/dt and dy/dt to zero and solving for x and y. This gives (0,0) and (1,-1/2). But you still don't know what their nature is, so I tried to solve the system. I did this by dividing one by the other and getting dy/dx=y+2xy/x-xy. However I am stuck trying to solve this problem. I can't find a integrating factor for this in terms or either x or y (we haven't learned anything beyond that) in order to make it exact. How else can I solve this? I gather I can turn this into a matrix and solve it somehow by I am not sure.

Lancen said:
I have a problem regarding the equations dx/dt=x-xy and dy/dt=y+2xy. I need to find the critical points of this system and denote if they are stable or asymptotic or whatever. I flipped through the section on this and you can find the critical points by setting dx/dt and dy/dt to zero and solving for x and y. This gives (0,0) and (1,-1/2). But you still don't know what their nature is, so I tried to solve the system. I did this by dividing one by the other and getting dy/dx=y+2xy/x-xy. However I am stuck trying to solve this problem. I can't find a integrating factor for this in terms or either x or y (we haven't learned anything beyond that) in order to make it exact. How else can I solve this? I gather I can turn this into a matrix and solve it somehow by I am not sure.

Did you try looking at the eigenvalues of the linearized system at the critical points?

I can't find a integrating factor for this in terms or either x or y
Did you notice it's separable? (Oh, and you really should be using parentheses where appropriate. 1+2/2+3 is 5, not 3/5)

But as LeBrad suggests, you should be able to figure out the answer without solving the equation. I suggest you do that first, and then solve to check your answer!

Recheck your equilibrium points! x= 1, y= -1/2 do NOT make dx/dt and dy/dt 0.
As Hurkyl and LeBrad suggested, you don't need to solve the system to determine if it is stable. In fact, that's the whole point! Most non-linear systems can't be solved that easily. The stability can be determined by looking at the "linearized" system at each equilibrium point. (That's what LeBrad was talking about: non-linear systems don't have eigen-values.)

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I been busy the last few days and haven't had a chance to take a look at these forums. How do I find the linearized system? I took another look at the equation and realized it is indeed separable!

## 1. What are autonomous systems?

Autonomous systems are systems that can operate independently without external control or influence. They are often used in fields such as robotics, artificial intelligence, and control theory.

## 2. How do autonomous systems maintain stability?

Autonomous systems maintain stability through various control mechanisms, such as feedback control, predictive control, and adaptive control. These control methods allow the system to continuously monitor and adjust its behavior to achieve a desired outcome and avoid instability.

## 3. What are the benefits of using autonomous systems?

There are several benefits of using autonomous systems, including increased efficiency, improved accuracy, and reduced human error. Autonomous systems also have the potential to operate in hazardous or remote environments, reducing the risk to human operators.

## 4. What challenges do autonomous systems face in maintaining stability?

One of the main challenges for autonomous systems in maintaining stability is uncertainty. This can be caused by external factors such as changing environments or unexpected events. Additionally, the complexity of autonomous systems can make it difficult to predict and control their behavior.

## 5. How can the stability of autonomous systems be tested and evaluated?

The stability of autonomous systems can be tested and evaluated through simulation, where different scenarios and inputs can be applied to assess the performance of the system. Additionally, real-world testing and validation can also be used to evaluate the stability of autonomous systems in practical applications.