# Harmonic oscillator: Why not chaotic?

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• greypilgrim
In summary, the conversation explores the concept of chaotic motion in relation to a harmonic oscillator. The speaker questions why the movement of a harmonic oscillator is not considered chaotic, given the unpredictable nature of the phase over time. They also discuss whether labeling the motion as chaotic changes its physics, and whether a pendulum can be considered chaotic. The conversation concludes with the mention of nonlinear oscillators, such as the Duffing and van der Pol, which can display chaotic motion.
greypilgrim
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?

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.

## 1. Why is the harmonic oscillator not chaotic?

The harmonic oscillator is a type of system that exhibits periodic motion, meaning that it repeats the same pattern over and over again. This is due to the fact that the restoring force in the system is directly proportional to the displacement from equilibrium, resulting in a linear relationship between the two. Chaotic systems, on the other hand, have a highly sensitive dependence on initial conditions and exhibit unpredictable behavior. Since the harmonic oscillator has a stable, predictable motion, it is not considered chaotic.

## 2. Can a harmonic oscillator become chaotic?

In theory, yes, a harmonic oscillator can become chaotic if certain conditions are met. These conditions include introducing nonlinearity into the system, changing the parameters of the system, or adding external forces. However, in most practical cases, the harmonic oscillator remains stable and does not become chaotic.

## 3. What is the difference between a harmonic oscillator and a chaotic oscillator?

The main difference between a harmonic oscillator and a chaotic oscillator is in their behavior. A harmonic oscillator exhibits regular, periodic motion, while a chaotic oscillator exhibits unpredictable and irregular behavior. This is due to the fact that a harmonic oscillator is based on linear equations, while a chaotic oscillator is based on nonlinear equations.

## 4. How is the stability of a harmonic oscillator determined?

The stability of a harmonic oscillator is determined by the relationship between the restoring force and the displacement from equilibrium. If the restoring force is directly proportional to the displacement, the system is considered stable. If the restoring force is not directly proportional, the system may exhibit chaotic behavior.

## 5. Can a harmonic oscillator be used to model real-world systems?

Yes, the harmonic oscillator is a commonly used model for various physical systems, such as pendulums, springs, and electrical circuits. However, it should be noted that these real-world systems may exhibit nonlinear behavior and may not perfectly follow the equations of a simple harmonic oscillator. In these cases, the harmonic oscillator model can still provide a useful approximation, but may not accurately predict all aspects of the system's behavior.

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