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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 equilibrium points at each equilibrium = -x,+0.5x,*x2 dt dx2 = -x,+0.5x,*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 equilibrium points at each equilibrium = -x,+0.5x,*x2 dt dx2 = -x,+0.5x,*X, dt

Advanced Engineering Mathematics
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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**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.
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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