ht) Consider the discrete-time dynamical system Xr+1 - What is the equilibrium for this system? What is the updating function rule f(x)? What is the derivative of the updating function? Is the equilibrium stable, unstable, or neither? 0.57x, +0.
ht) Consider the discrete-time dynamical system Xr+1 - What is the equilibrium for this system? What is the updating function rule f(x)? What is the derivative of the updating function? Is the equilibrium stable, unstable, or neither? 0.57x, +0.
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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Question
![## Discrete-Time Dynamical System
**Problem Statement:**
Consider the discrete-time dynamical system defined by the equation:
\[ x_{t+1} = 0.57x_t + 0. \]
### Questions:
1. **What is the equilibrium for this system?**
- [Text box for answer]
2. **What is the updating function rule \( f(x) \)?**
- [Text box for answer]
3. **What is the derivative of the updating function?**
- [Text box for answer]
4. **Is the equilibrium stable, unstable, or neither?**
- [Text box for answer]
### Analysis:
- **Equilibrium**: Determine the fixed point where \( x_{t+1} = x_t \).
- **Updating Function**: Identify \( f(x) = 0.57x + 0 \).
- **Derivative**: Calculate the derivative \( f'(x) = 0.57 \).
- **Stability**: Evaluate whether the fixed point is stable based on the magnitude of the derivative. If \(|f'(x)| < 1\), the equilibrium is stable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F58cffe43-d701-4c21-9740-08fe8d98ee79%2F7e830cd3-a2c9-4921-8566-0820a793078c%2Fzkirmze_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Discrete-Time Dynamical System
**Problem Statement:**
Consider the discrete-time dynamical system defined by the equation:
\[ x_{t+1} = 0.57x_t + 0. \]
### Questions:
1. **What is the equilibrium for this system?**
- [Text box for answer]
2. **What is the updating function rule \( f(x) \)?**
- [Text box for answer]
3. **What is the derivative of the updating function?**
- [Text box for answer]
4. **Is the equilibrium stable, unstable, or neither?**
- [Text box for answer]
### Analysis:
- **Equilibrium**: Determine the fixed point where \( x_{t+1} = x_t \).
- **Updating Function**: Identify \( f(x) = 0.57x + 0 \).
- **Derivative**: Calculate the derivative \( f'(x) = 0.57 \).
- **Stability**: Evaluate whether the fixed point is stable based on the magnitude of the derivative. If \(|f'(x)| < 1\), the equilibrium is stable.
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