# What is the difference between non-linear oscillator and non-uniform oscillator?

The equation for a uniform oscillator is: $$\dot\theta = \omega$$ which has a solution of $$\theta(t) = \omega t +\theta_0$$.

For a non-uniform oscillator, the equation is: $$\dot\theta= \omega - a$$ where $$a(\theta) \ne 0$$.

These definitions are from Strogatz's book. Now I don't understand the difference between non-uniform and non-linear. Isn't a non-uniform oscillator is also a kind of non-linear oscillator? What is the difference between them?

• $\dot \theta = \omega$ has the solution $\theta(t) = \omega t + \theta_0$. Jan 31 at 5:55

On a more fundamental level the difference is between a linear differential equation, homogeneous differential equation, and an equation with constant coefficients.

Linear (ordinary) differential equation
If we take an equation $$\mathcal{L}x(t)=f(t),$$ where $$\mathcal{L}$$ is an arbitrary differential operator, then it is linear if the differential operator is linear, i.e., if for any functions $$x(t),y(t)$$ and any coefficients $$a,b$$, the action of the operator on the linear combination of these functions, $$z(t)=ax(t)+by(t)$$ is just a linear combination of its action on each of the functions $$\mathcal{L}z(t)=a\mathcal{L}x(t)+b\mathcal{L}y(t).$$

Homogeneous differential equation
Further, we can write a linear operator as $$\mathcal{L}=\sum_n a_n(t)\frac{d^n}{dt^n},$$ so that the corresponding linear equation can be written as $$\sum_n a_n(t)\frac{d^n x(t)}{dt^n}=f(t).$$ If the right-hand-side of thsi equation is zero, i.e., if we have only the derivative terms, then the equation is called homogeneous, otherwise - inhomogeneous.

Differential equation with constant coefficients
Finally, if the coefficients of the equation, $$\{a_n(t)\}$$ are constant (i.e., independent on $$t$$), it is called an equation with constant coefficients.

Oscillator
Oscillator is generally described by differential equation $$\ddot{x}(t)+\omega_0^2x(t)=0\Leftrightarrow \mathcal{L}=\frac{d^2}{dt^2}+\omega_0^2, \mathcal{L}x(t)=f(t)$$ This is a homogeneous linear differential equation with constant coefficients. Adding a driving force makes it inhomogeneous. (Perhaps, this is what is meant by non-uniform, although it may also refer to non-constant coefficients.)

Without referring to the source material, the differential equation $$\theta = \omega - a(\theta)$$ is linear if $$a(\theta)$$ is a linear relationship, i.e. $$a(\theta) = k \theta$$, with $$k$$ constant. It's only nonlinear if $$a(\theta)$$ is a nonlinear relation.

• So an oscillator can be non-uniform without being non-linear and vice-versa? Jan 31 at 6:02
• Btw, in the example of the book, they used $a(\theta)=a \sin \theta$. So, it seems that is non-uniform and also non-linear. Jan 31 at 6:04
• By the definition you give, yes. I'm going by the definition of "nonlinear" for differential equations and dynamic systems, though, and sometimes terminology can vary from discipline to discipline. I think this'll adhere to the standard, though. Jan 31 at 6:07

lets look at those two examples:

$$\dot\phi+\sin(\phi)=\omega$$

the solution

this is non uniform oscillator and also non linear oscillator

and $$\dot\phi=\omega$$

the solution $$~\phi(t)=\omega\,t$$