Harmonic oscillator: Why not chaotic?

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Discussion Overview

The discussion centers on the nature of harmonic oscillators and their classification regarding chaotic behavior. Participants explore whether the inherent properties of harmonic oscillators, particularly in relation to phase uncertainty and angular frequency, could lead to chaotic motion, and they compare this with other types of oscillators.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants assert that harmonic oscillators are not considered chaotic due to the linear build-up of phase error over time, which they argue does not meet the criteria for chaos.
  • One participant questions the relevance of labeling motion as chaotic, suggesting that it may not alter the underlying physics.
  • Another participant emphasizes that while harmonic oscillators do not exhibit chaotic behavior, nonlinear oscillators, such as the Duffing and van der Pol oscillators, can display chaos.
  • There is a repeated assertion regarding the predictability of phase modulo 2π and its implications for chaos, with some participants linking this to the broader definition of chaotic systems.

Areas of Agreement / Disagreement

Participants generally disagree on whether the characteristics of harmonic oscillators can be classified as chaotic, with some supporting the notion that they are not chaotic and others exploring the implications of phase uncertainty. The discussion remains unresolved regarding the classification of motion in different types of oscillators.

Contextual Notes

Participants reference specific definitions and properties of chaotic motion, indicating that the discussion is contingent on the definitions of chaos and the characteristics of different types of oscillators. The relationship between linear and nonlinear systems is also highlighted, but not fully resolved.

greypilgrim
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Hi.

As far as I know, the movement of a harmonic oscillator normally is not considered to be chaotic. Why not? Since the angular frequency can never be known to absolute precision, an error in the phase builds up. I can see that this build-up is only linear in time (if we assume the angular frequency to be constant), but since the phase only matters modulo 2π, φ(t) mod 2π is completely unpredictable time after some long enough time t. Isn't this exactly what chaos is about?
 
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Does labeling it "chaotic" or not change the physics in any way?
 
greypilgrim said:
Hi.

As far as I know, the movement of a harmonic oscillator normally is not considered to be chaotic. Why not? Since the angular frequency can never be known to absolute precision, an error in the phase builds up. I can see that this build-up is only linear in time (if we assume the angular frequency to be constant), but since the phase only matters modulo 2π, φ(t) mod 2π is completely unpredictable time after some long enough time t. Isn't this exactly what chaos is about?

Would you call the motion of a pendulum chaotic?
 
greypilgrim said:
Hi.

As far as I know, the movement of a harmonic oscillator normally is not considered to be chaotic. Why not? Since the angular frequency can never be known to absolute precision, an error in the phase builds up. I can see that this build-up is only linear in time (if we assume the angular frequency to be constant), but since the phase only matters modulo 2π, φ(t) mod 2π is completely unpredictable time after some long enough time t. Isn't this exactly what chaos is about?

Perhaps you are not being sufficiently specific here- a linear oscillator, whether driven, damped, or both, will never display chaotic motion. Chaotic motion has a fairly specific definition in terms of a geometrical approach to the solution of ordinary differential equations:

https://en.wikipedia.org/wiki/Attractor

Nonlinear oscillators can indeed display chaotic motion: Duffing and van der Pol oscillators are two common examples.
 

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