Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
**Title: Finding Equilibrium Points and Stability Analysis** **Objective:** To find all the equilibrium points for the given dynamic system and determine whether each point is small signal stable or unstable by linearizing the system at these points. **System Equations:** 1. \(\frac{dx_1}{dt} = -x_1 + 0.5 \cdot x_1 \cdot x_2\) 2. \(\frac{dx_2}{dt} = -x_2 + 0.5 \cdot x_1 \cdot x_2\) **Procedure:** 1. **Finding Equilibrium Points:** - Set \(\frac{dx_1}{dt} = 0\) and \(\frac{dx_2}{dt} = 0\) to solve for \(x_1\) and \(x_2\). 2. **Stability Analysis:** - At each equilibrium point, determine the Jacobian matrix of the system. - Evaluate the eigenvalues of the Jacobian matrix. - Classify the equilibrium point as stable, unstable, or saddle point based on the eigenvalues: - If all eigenvalues have negative real parts, the equilibrium is stable. - If at least one eigenvalue has a positive real part, the equilibrium is unstable. **Note:** - No graphs or diagrams are included in this description.
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Transcribed Image Text:**Title: Finding Equilibrium Points and Stability Analysis** **Objective:** To find all the equilibrium points for the given dynamic system and determine whether each point is small signal stable or unstable by linearizing the system at these points. **System Equations:** 1. \(\frac{dx_1}{dt} = -x_1 + 0.5 \cdot x_1 \cdot x_2\) 2. \(\frac{dx_2}{dt} = -x_2 + 0.5 \cdot x_1 \cdot x_2\) **Procedure:** 1. **Finding Equilibrium Points:** - Set \(\frac{dx_1}{dt} = 0\) and \(\frac{dx_2}{dt} = 0\) to solve for \(x_1\) and \(x_2\). 2. **Stability Analysis:** - At each equilibrium point, determine the Jacobian matrix of the system. - Evaluate the eigenvalues of the Jacobian matrix. - Classify the equilibrium point as stable, unstable, or saddle point based on the eigenvalues: - If all eigenvalues have negative real parts, the equilibrium is stable. - If at least one eigenvalue has a positive real part, the equilibrium is unstable. **Note:** - No graphs or diagrams are included in this description.
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