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?
 A: 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.)
A: 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.
A: 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 $~\phi(t)=\omega\,t$

