A question about a nonlinear oscillator

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    Nonlinear Oscillator
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Discussion Overview

The discussion revolves around the behavior of a nonlinear oscillator described by the equation x''(t) + e (x'(t)^3) + x(t) = 0, particularly focusing on the existence of limit cycles and the effects of nonlinearity as the parameter e varies. Participants explore numerical integration methods and analytical approaches to understand the system's dynamics over time.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant reports obtaining a limit cycle through numerical integration and questions the implications of a negative energy derivative suggesting decay to zero.
  • Another participant mentions using a small value of e (1e-3) and observes oscillatory motion followed by decay, suggesting the system behaves almost linearly due to the small e.
  • A different participant used a larger value of e (0.3) and noted that the system faded out, indicating that the nonlinearity becomes significant over time.
  • Concerns are raised about the accuracy of the Runge-Kutta method for larger values of e and the potential for numerical methods to misrepresent the system's behavior.
  • There is a suggestion that the matrix exponential method may only capture the linear aspects of the system and not the full nonlinear behavior.
  • Some participants express uncertainty about the stability of the matrix A in the context of the e^At approach and its implications for observing limit cycles.

Areas of Agreement / Disagreement

Participants express differing views on the effects of the parameter e on the system's behavior, with some suggesting that small values lead to limit cycles while larger values reveal decay. The discussion remains unresolved regarding the implications of different numerical methods and their accuracy in capturing the system's dynamics.

Contextual Notes

Participants note limitations related to the choice of numerical methods, the dependence on the parameter e, and the potential for transient behaviors in the nonlinear system that may not be fully captured by linear approximations.

pendulum
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I numerically integrate the following nonlinear oscillator:
x''(t) + e (x'(t)^3) + x(t) = 0 , where e<<1
and what I get is a limit cycle.
The energy derivative appears to be negative , which means that
x(t) approaches zero while t approaches infinity.
I also used the analytical method of two-timing, and the first asymptotic term x0(t) does approach zero for large t.
( The algorithm for the numerical integration is Runge-Kutta4. It's unlikely to have written it incorrectly.)
So where am I wrong?
Is it possible there is a limit cycle after all?
 
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What value of e did you use?

I used 1e-3 and Matlab (stiff solver ode15s) showed oscillatory motion until approximately t=7000, then decay to zero.

What you've got is an almost linear system.

The eigenvalues (of the linear stability analysis) are imaginary - so you would expect limit cycle behaviour. However, the nonlinear part eventually drives your system.

(It is almost linear because e is small.)
 
I used e=0.3 which is not that small.
So I dropped the R-K4, and used a matrix of the form e^At for the numerical integration, and the system did fade out.
So I guessed that R-K4 was to accurate for this case. At least for the times 'I could reach'. You see I've been running the integration in matlab, but I've not learned to use ode's yet. So the procedure was very slow (even for t=0:200).
(I find it quite inconvenient that Matlab is so slow in loops.)

Anyway thanks. I think I get your point. The non-linearity reveals itself after a long time in the particular system.
 
This is completely off-topic, but when I'm confronted with a bottleneck in Matlab, I always write the critical parts in C and then call it as MEX file. This usually speeds up things by orders of magnitude.
 
Thank you Tantoblin.
 
pendulum said:
I used e=0.3 which is not that small.
So I dropped the R-K4, and used a matrix of the form e^At for the numerical integration, and the system did fade out.
...but looking at [tex]e^{At}[/tex] will only give you results for the linear system (+ something about transients of the full nonlinear system)
 
I am not sure whether I understand what you mean, but the A in the e^At wasn't stable (or linear if better).
 
pendulum said:
I am not sure whether I understand what you mean, but the A in the e^At wasn't stable (or linear if better).
You'll only see the limit cycle behaviour by looking at the matrix exponentials...

ie. From the purely complex eigenvalues of the linearised system given by A.

iie. if the nonlinearity kicks in (for any e), you won't see it.
 

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