Find all the equilibrium points for the system below. At each equilibrium point, linearize the system and classify whether the equilibrium is small signal stable or unstable by computing the eigen values at each equilibrium point dx, :-5x,+2x,*x2 dt dx2 = -2x,+x, *X, dt
Find all the equilibrium points for the system below. At each equilibrium point, linearize the system and classify whether the equilibrium is small signal stable or unstable by computing the eigen values at each equilibrium point dx, :-5x,+2x,*x2 dt dx2 = -2x,+x, *X, dt
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Finding Equilibrium Points and Stability**
To analyze the stability of a dynamical system, we first need to find the equilibrium points and then assess their stability by linearizing the system and evaluating the eigenvalues at each equilibrium point.
**System of Differential Equations:**
\[
\frac{dx_1}{dt} = -5x_1 + 2x_1x_2
\]
\[
\frac{dx_2}{dt} = -2x_2 + x_1x_2
\]
### Steps to Analyze the System:
1. **Identify Equilibrium Points:**
- Set \(\frac{dx_1}{dt} = 0\) and \(\frac{dx_2}{dt} = 0\).
- Solve the system of equations to find equilibrium points \((x_1, x_2)\).
2. **Linearize the System:**
- For each equilibrium point, calculate the Jacobian matrix of the system.
- Evaluate the Jacobian at the equilibrium points.
3. **Compute Eigenvalues:**
- Determine the eigenvalues of the Jacobian matrix at each equilibrium point.
4. **Classify Stability:**
- If all the eigenvalues have negative real parts, the equilibrium is small signal stable.
- If any eigenvalue has a positive real part, the equilibrium is unstable.
This analysis helps to determine the behavior of the system near the equilibrium points, providing insight into the system's dynamics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb5312a15-860e-407c-ae9e-8729c86c8316%2F688c43f6-5149-4ce0-a622-034f660f3afa%2Fzb8c787_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding Equilibrium Points and Stability**
To analyze the stability of a dynamical system, we first need to find the equilibrium points and then assess their stability by linearizing the system and evaluating the eigenvalues at each equilibrium point.
**System of Differential Equations:**
\[
\frac{dx_1}{dt} = -5x_1 + 2x_1x_2
\]
\[
\frac{dx_2}{dt} = -2x_2 + x_1x_2
\]
### Steps to Analyze the System:
1. **Identify Equilibrium Points:**
- Set \(\frac{dx_1}{dt} = 0\) and \(\frac{dx_2}{dt} = 0\).
- Solve the system of equations to find equilibrium points \((x_1, x_2)\).
2. **Linearize the System:**
- For each equilibrium point, calculate the Jacobian matrix of the system.
- Evaluate the Jacobian at the equilibrium points.
3. **Compute Eigenvalues:**
- Determine the eigenvalues of the Jacobian matrix at each equilibrium point.
4. **Classify Stability:**
- If all the eigenvalues have negative real parts, the equilibrium is small signal stable.
- If any eigenvalue has a positive real part, the equilibrium is unstable.
This analysis helps to determine the behavior of the system near the equilibrium points, providing insight into the system's dynamics.
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