Testing Stability of Linear System Fixed Points

• coverband
In summary, the conversation is about classifying fixed points in a linear system and determining their stability. The attempt at a solution involves finding the fixed point at dy/dx = 0/0 and questioning how to test for stability. The solution may involve looking at the eigenvalues of the coefficient matrix on the right side.
coverband

Homework Statement

Classify the fixed points of the following linear system and state whether they are stable or unstable

$$\dot{x}=x + y$$
$$\dot{y} = x + 3y$$

The Attempt at a Solution

Fixed point at dy/dx = 0/0. Therefore fixed point = (0,0)

How does one test for stability?

Thanks

coverband said:

Homework Statement

Classify the fixed points of the following linear system and state whether they are stable or unstable

$$\dot{x}=x + y$$
$$\dot{y} = x + 3y$$

The Attempt at a Solution

Fixed point at dy/dx = 0/0. Therefore fixed point = (0,0)

How does one test for stability?

Thanks
0/0 is not a number, so how can you say that there is a fixed point at dy/dx = 0/0?

How does one test for stability?

coverband said:
How does one test for stability?

Doesn't your text have a test for stability which involves looking at the eigenvalues of the coefficient matrix on the right side?

1. What is the purpose of testing stability of linear system fixed points?

The purpose of testing stability of linear system fixed points is to determine whether a system will tend towards a stable state or an unstable state over time. This is important in understanding the behavior of a system and predicting its future outcomes.

2. How is the stability of linear system fixed points tested?

The stability of linear system fixed points is tested by analyzing the eigenvalues of the system's Jacobian matrix. If all eigenvalues are negative, the fixed point is stable. If at least one eigenvalue is positive, the fixed point is unstable.

3. What is the significance of a fixed point being stable or unstable?

A stable fixed point indicates that the system will tend towards a steady state, while an unstable fixed point indicates that the system will not reach a steady state and may exhibit chaotic behavior.

4. Can the stability of linear system fixed points change over time?

Yes, the stability of linear system fixed points can change over time. A fixed point that was initially stable may become unstable due to changes in the system's parameters or external influences. It is important to regularly re-evaluate the stability of fixed points in a system.

5. How is the stability of linear system fixed points used in real-world applications?

The stability of linear system fixed points is used in a variety of fields, including engineering, economics, and ecology. It can be used to predict the long-term behavior of a system, design control strategies, and identify critical points in a system that may lead to instability.

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