Along any solution curve that asymptotically approaches the saddle, either in forward or backward time, the value of \( H \) must be the same as at the saddle. Find a point on each of the 3 solution curves, colored black in the plot above, that asymptotically approach the saddle in either backwards or forwards time or both. Answers are \( x, y \) coordinates as a vector. **Phase Portrait and Differential Equations** In the nonlinear system of differential equations: \[ \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} y \\ x^2 - x \end{bmatrix} \] The quantity \[ H(x, y) = \frac{1}{2}y^2 + \frac{1}{2}x^2 - \frac{1}{3}x^3 \] is constant along solution curves. **Graph Explanation** The graph illustrates the phase portrait of the given differential system. Key features include: - **Arrows (Vector Field):** Red arrows depict the direction of the vector field, indicating the flow direction within the phase space. - **Solution Curves:** The blue curves represent the trajectories or solution curves of the system. These curves show how the system evolves over time. - **Critical Points:** The graph highlights a center (blue point) and a saddle point (red point). The center is surrounded by closed orbits, while the saddle point has trajectories that approach and veer away. - **Conservation of H(x, y):** The function \( H(x, y) \) remains constant on each blue solution curve, indicating conserved quantities in the system. This visualization provides a geometric interpretation of the system's behavior, showcasing stability and the nature of the critical points.
Along any solution curve that asymptotically approaches the saddle, either in forward or backward time, the value of \( H \) must be the same as at the saddle. Find a point on each of the 3 solution curves, colored black in the plot above, that asymptotically approach the saddle in either backwards or forwards time or both. Answers are \( x, y \) coordinates as a vector. **Phase Portrait and Differential Equations** In the nonlinear system of differential equations: \[ \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} y \\ x^2 - x \end{bmatrix} \] The quantity \[ H(x, y) = \frac{1}{2}y^2 + \frac{1}{2}x^2 - \frac{1}{3}x^3 \] is constant along solution curves. **Graph Explanation** The graph illustrates the phase portrait of the given differential system. Key features include: - **Arrows (Vector Field):** Red arrows depict the direction of the vector field, indicating the flow direction within the phase space. - **Solution Curves:** The blue curves represent the trajectories or solution curves of the system. These curves show how the system evolves over time. - **Critical Points:** The graph highlights a center (blue point) and a saddle point (red point). The center is surrounded by closed orbits, while the saddle point has trajectories that approach and veer away. - **Conservation of H(x, y):** The function \( H(x, y) \) remains constant on each blue solution curve, indicating conserved quantities in the system. This visualization provides a geometric interpretation of the system's behavior, showcasing stability and the nature of the critical points.
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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Transcribed Image Text:Along any solution curve that asymptotically approaches the saddle, either in forward or backward time, the value of \( H \) must be the same as at the saddle. Find a point on each of the 3 solution curves, colored black in the plot above, that asymptotically approach the saddle in either backwards or forwards time or both. Answers are \( x, y \) coordinates as a vector.
![**Phase Portrait and Differential Equations**
In the nonlinear system of differential equations:
\[
\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} y \\ x^2 - x \end{bmatrix}
\]
The quantity
\[ H(x, y) = \frac{1}{2}y^2 + \frac{1}{2}x^2 - \frac{1}{3}x^3 \]
is constant along solution curves.
**Graph Explanation**
The graph illustrates the phase portrait of the given differential system. Key features include:
- **Arrows (Vector Field):** Red arrows depict the direction of the vector field, indicating the flow direction within the phase space.
- **Solution Curves:** The blue curves represent the trajectories or solution curves of the system. These curves show how the system evolves over time.
- **Critical Points:** The graph highlights a center (blue point) and a saddle point (red point). The center is surrounded by closed orbits, while the saddle point has trajectories that approach and veer away.
- **Conservation of H(x, y):** The function \( H(x, y) \) remains constant on each blue solution curve, indicating conserved quantities in the system.
This visualization provides a geometric interpretation of the system's behavior, showcasing stability and the nature of the critical points.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd75c580-e1ed-40e9-aa84-8dd389134e22%2Fca165f98-2a41-409b-b07a-278897392c4a%2Fdjvqzt_processed.png&w=3840&q=75)
Transcribed Image Text:**Phase Portrait and Differential Equations**
In the nonlinear system of differential equations:
\[
\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} y \\ x^2 - x \end{bmatrix}
\]
The quantity
\[ H(x, y) = \frac{1}{2}y^2 + \frac{1}{2}x^2 - \frac{1}{3}x^3 \]
is constant along solution curves.
**Graph Explanation**
The graph illustrates the phase portrait of the given differential system. Key features include:
- **Arrows (Vector Field):** Red arrows depict the direction of the vector field, indicating the flow direction within the phase space.
- **Solution Curves:** The blue curves represent the trajectories or solution curves of the system. These curves show how the system evolves over time.
- **Critical Points:** The graph highlights a center (blue point) and a saddle point (red point). The center is surrounded by closed orbits, while the saddle point has trajectories that approach and veer away.
- **Conservation of H(x, y):** The function \( H(x, y) \) remains constant on each blue solution curve, indicating conserved quantities in the system.
This visualization provides a geometric interpretation of the system's behavior, showcasing stability and the nature of the critical points.
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