Turn on thread page Beta

Dynamical system f(x)=(x^4)sin(1/x). How to determine the stability watch

    • Thread Starter
    Offline

    6
    ReputationRep:
    Dynamical systemf(x)=(x^4)sin(1/x) (for x not equal to zero) and f(x)=0 (for x=0). How to find equilibrium points, and determining the stability of each equilibrium point?

    I found the equilibrium points x=0 and x=1/(kπ), where k is integer. 

 f′(x)=4(x^3)sin(1/x)−(x^4)cos(1/x).

    When k is 'even', f′(x=1/k*pi)=1/(pi^4)
    which is positive, therefore not asymptotically stable.
    And when k is 'odd', f′(x=1/k*pi)=minus1/(pi^4)
    which is negative, therefore asymptotically stable.

    But f′(x=0)=0, which is not hyperbolic. And this is the part that I can not solve.


    Thanks in advance.
    Offline

    14
    ReputationRep:
    For a continuous dynamical system, it's pretty strange to have a function rather than an ODE... but anyway. Generally non-hyperbolic equilibria are a pain, but in a one-dimensional case like this, you can use the higher-order derivative test: https://en.wikipedia.org/wiki/Deriva...erivative_test
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: November 10, 2016
The home of Results and Clearing

2,891

people online now

1,567,000

students helped last year

University open days

  1. Sheffield Hallam University
    City Campus Undergraduate
    Tue, 21 Aug '18
  2. Bournemouth University
    Clearing Open Day Undergraduate
    Wed, 22 Aug '18
  3. University of Buckingham
    Postgraduate Open Evening Postgraduate
    Thu, 23 Aug '18
Poll
A-level students - how do you feel about your results?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.