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The eigenvalues of the matrix A are given below. Match them to the appropriate phase portraits of the system X'(t) = = AX(t). X1,2 = -1, -2 X1,2 = 3, -1 X1,2 A₁,2 = ±i X1,2 1 3 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 -0.8 -1 -1 A B C D -0.8 1 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 -0.8 -1 -1 Phase Portrait -0.6 -0.4 -0.2 0 0.2 0.4 0.6 x(t) Phase Portrait = = ±2i -0.8 -0.6 -0.4 -0.2 0 x(t) 0.8 1 0.2 0.4 0.6 0.8 1 2 4 1 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 -0.8 -1 -1 -0.8 -0.6 -0.4 0.8 0.6 0.4 0.2 -0.2 -0.4% -0.6 -0.8 -1 -1 -0.8 Phase Portrait -0.2 -0.6 -0.4 0.2 0.4 0.6 0 x(t) Phase Portrait -0.2 0 x(t) 0.2 0.4 0.8 0.6 1 0.8 1

The eigenvalues of the matrix A are given below. Match them to the appropriate phase portraits of the system X'(t) = = AX(t). X1,2 = -1, -2 X1,2 = 3, -1 X1,2 A₁,2 = ±i X1,2 1 3 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 -0.8 -1 -1 A B C D -0.8 1 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 -0.8 -1 -1 Phase Portrait -0.6 -0.4 -0.2 0 0.2 0.4 0.6 x(t) Phase Portrait = = ±2i -0.8 -0.6 -0.4 -0.2 0 x(t) 0.8 1 0.2 0.4 0.6 0.8 1 2 4 1 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 -0.8 -1 -1 -0.8 -0.6 -0.4 0.8 0.6 0.4 0.2 -0.2 -0.4% -0.6 -0.8 -1 -1 -0.8 Phase Portrait -0.2 -0.6 -0.4 0.2 0.4 0.6 0 x(t) Phase Portrait -0.2 0 x(t) 0.2 0.4 0.8 0.6 1 0.8 1

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: Matching Eigenvalues with Phase Portraits** The goal of this exercise is to match the given eigenvalues of the matrix \( A \) to the appropriate phase portraits of the system \(\dot{\mathbf{X}}(t) = A \mathbf{X}(t)\). ### Eigenvalues: - **A**: \(\lambda_{1,2} = -1, -2\) - **B**: \(\lambda_{1,2} = 3, -1\) - **C**: \(\lambda_{1,2} = \pm 2i\) - **D**: \(\lambda_{1,2} = -\frac{1}{2} \pm i\) ### Phase Portraits: 1. **Diagram 1**: Displays a spiral shape converging towards the origin, indicating that it represents a system with complex eigenvalues with negative real parts. 2. **Diagram 2**: Shows a saddle point pattern, characterized by trajectories diverging along one axis and converging along another, suggesting one positive and one negative real eigenvalue. 3. **Diagram 3**: Features a node pattern with trajectories diverging away from the origin, reflecting positive real eigenvalues. 4. **Diagram 4**: Depicts concentric closed orbits around the origin, typical of purely imaginary eigenvalues. **Matching:** - **A** \(\lambda_{1,2} = -1, -2\) matches with Diagram 1. - **B** \(\lambda_{1,2} = 3, -1\) matches with Diagram 3. - **C** \(\lambda_{1,2} = \pm 2i\) matches with Diagram 4. - **D** \(\lambda_{1,2} = -\frac{1}{2} \pm i\) matches with Diagram 2. These portraits help visualize the behavior of the system's solutions over time, with the phase portraits providing insight into the stability and type of equilibrium points.
Transcribed Image Text:**Title: Matching Eigenvalues with Phase Portraits** The goal of this exercise is to match the given eigenvalues of the matrix \( A \) to the appropriate phase portraits of the system \(\dot{\mathbf{X}}(t) = A \mathbf{X}(t)\). ### Eigenvalues: - **A**: \(\lambda_{1,2} = -1, -2\) - **B**: \(\lambda_{1,2} = 3, -1\) - **C**: \(\lambda_{1,2} = \pm 2i\) - **D**: \(\lambda_{1,2} = -\frac{1}{2} \pm i\) ### Phase Portraits: 1. **Diagram 1**: Displays a spiral shape converging towards the origin, indicating that it represents a system with complex eigenvalues with negative real parts. 2. **Diagram 2**: Shows a saddle point pattern, characterized by trajectories diverging along one axis and converging along another, suggesting one positive and one negative real eigenvalue. 3. **Diagram 3**: Features a node pattern with trajectories diverging away from the origin, reflecting positive real eigenvalues. 4. **Diagram 4**: Depicts concentric closed orbits around the origin, typical of purely imaginary eigenvalues. **Matching:** - **A** \(\lambda_{1,2} = -1, -2\) matches with Diagram 1. - **B** \(\lambda_{1,2} = 3, -1\) matches with Diagram 3. - **C** \(\lambda_{1,2} = \pm 2i\) matches with Diagram 4. - **D** \(\lambda_{1,2} = -\frac{1}{2} \pm i\) matches with Diagram 2. These portraits help visualize the behavior of the system's solutions over time, with the phase portraits providing insight into the stability and type of equilibrium points.
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