# Harmonic oscillator: Why not chaotic?

• I
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?

## Answers and Replies

Dale
Mentor
2020 Award
Does labeling it "chaotic" or not change the physics in any way?

PeroK
Science Advisor
Homework Helper
Gold Member
2020 Award
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?

Andy Resnick
Science Advisor
Education Advisor
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.