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Problem 4: Suppose we have a one-dimensional system given by * = f(x), with f sufficiently smooth to ensure unique solutions are known to exist for some time (e.g. f is at least C¹ ). (a) Is it possible for this system to have precisely two stable fixed points and no others? Explain your reasoning, and give as clear a proof as you can. (b) Is it possible for this system to have precisely two unstable fixed points and no others? Explain your reasoning, and give as clear a proof as you can.

Problem 4: Suppose we have a one-dimensional system given by * = f(x), with f sufficiently smooth to ensure unique solutions are known to exist for some time (e.g. f is at least C¹ ). (a) Is it possible for this system to have precisely two stable fixed points and no others? Explain your reasoning, and give as clear a proof as you can. (b) Is it possible for this system to have precisely two unstable fixed points and no others? Explain your reasoning, and give as clear a proof as you can.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 16EQ
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Problem 4: Suppose we have a one-dimensional system given by * = f(x), with f sufficiently smooth to ensure unique solutions are known to exist for some time (e.g. f is at least C¹ ). (a) Is it possible for this system to have precisely two stable fixed points and no others? Explain your reasoning, and give as clear a proof as you can. (b) Is it possible for this system to have precisely two unstable fixed points and no others? Explain your reasoning, and give as clear a proof as you can.
Transcribed Image Text:Problem 4: Suppose we have a one-dimensional system given by * = f(x), with f sufficiently smooth to ensure unique solutions are known to exist for some time (e.g. f is at least C¹ ). (a) Is it possible for this system to have precisely two stable fixed points and no others? Explain your reasoning, and give as clear a proof as you can. (b) Is it possible for this system to have precisely two unstable fixed points and no others? Explain your reasoning, and give as clear a proof as you can.
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