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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
Find and classify all fixed points as a function of r.
![**Analyzing the Long-Term Behavior of a Mathematical Map**
This section explores the long-term dynamics of the map given by the equation:
\[ x_{n+1} = \frac{rx_n}{1 + x_n^2} \]
where \( r > 0 \).
**Introduction**
The equation describes a recursive sequence where each term \( x_{n+1} \) is determined by the preceding term \( x_n \). The parameter \( r \) is a positive constant that influences the behavior of the sequence over time.
**Objective**
To understand how the sequence behaves as \( n \) approaches infinity, we investigate various values of \( r \) and initial conditions for \( x_0 \). Specifically, we'll consider:
- Whether the sequence converges to a stable value.
- The conditions under which the sequence remains bounded.
- Any periodic or chaotic behavior exhibited by the sequence.
**Discussion**
The fraction \(\frac{rx_n}{1 + x_n^2}\) suggests that the values of the sequence are influenced both by scaling with \( r \) and moderated by the presence of \( x_n^2 \) in the denominator. This can lead to interesting dynamics, such as saturation effects where large values of \( x_n \) reduce the change in \( x_{n+1} \).
**Conclusion**
By analyzing this map for different parameters, one can gain insights into systems that exhibit similar nonlinear behaviors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88e1e2e4-888b-4182-8c02-fd46dda7f6b1%2F8f92259e-67c4-4597-97b2-6a35120761b2%2Fz8jhwv7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Analyzing the Long-Term Behavior of a Mathematical Map**
This section explores the long-term dynamics of the map given by the equation:
\[ x_{n+1} = \frac{rx_n}{1 + x_n^2} \]
where \( r > 0 \).
**Introduction**
The equation describes a recursive sequence where each term \( x_{n+1} \) is determined by the preceding term \( x_n \). The parameter \( r \) is a positive constant that influences the behavior of the sequence over time.
**Objective**
To understand how the sequence behaves as \( n \) approaches infinity, we investigate various values of \( r \) and initial conditions for \( x_0 \). Specifically, we'll consider:
- Whether the sequence converges to a stable value.
- The conditions under which the sequence remains bounded.
- Any periodic or chaotic behavior exhibited by the sequence.
**Discussion**
The fraction \(\frac{rx_n}{1 + x_n^2}\) suggests that the values of the sequence are influenced both by scaling with \( r \) and moderated by the presence of \( x_n^2 \) in the denominator. This can lead to interesting dynamics, such as saturation effects where large values of \( x_n \) reduce the change in \( x_{n+1} \).
**Conclusion**
By analyzing this map for different parameters, one can gain insights into systems that exhibit similar nonlinear behaviors.
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