Can analyze using Perturbation methods. In particular:

• Poincaré- Lindstedt method. Letting dependent variable $x$ depend on the independent variable $t$ via terms of all orders i.e. $t$, $\epsilon t$, $\epsilon^2 t$, etc. However, these term are forced to only appear together, with the form: $t+ \epsilon \omega_1 t + \epsilon^2 \omega_2 t +...$. That is, we just expand the frequency as a perturbation series. A consequence is that constants of integration are not allowed to depend on $t$ at any order.
• Method of Multiple Scales. We allow dependent variable $x$ depend on the independent variable $t$ via terms of all orders i.e. $t$, $\epsilon t$, $\epsilon^2 t$, etc., without any constraint. A consequence is that we can treat these as independent variables, or scales. "Constants" of integration can now depend on them (i.e. on the slower scales than the scale corresponding to the order considered)
• Krylov-Bogoliubov Method of Averaging. Assumes that solution, $x(t)$ has the form that it has when $\epsilon=0$ but the constants of integration can now change with time. As we have two arbitrary time-dependent functions (in the case of a second order ode), we are clearly underdetermining the problem. Therefore Krylov and Bogoliubov added the constraint that the time derivative of $x(t)$, i.e. $\dot{x}(t)$, also hase the same form as when $\epsilon=0$ with the same constants of integration upgraded to the same time-dependent functions.

For example (for the method of averaging), if $x(t)=a \cos{(t+\theta)}$ is the $\epsilon=0$ solution, then we require:

• $x(t)=a(t) \cos{(t+\theta(t))}$, and
• $\dot{x}(t)=-a(t) \sin{(t+\theta(t))}$

Duffing oscillator

Van der Pol oscillator. Paper about it's periodic solutions Apply method of multiple scales.

Relaxation oscillations and transition layers

As an example consider the van der Pol eq. with the nonlinear term very large ($\Lambda \gg1$), instead of very small.

We introduce a variable $y$ s.t. eq. becomes $y'=-x$. One also shows that $x$ evolves to a state of quasi-equilibrium (very quickly on time scale $1/\Lambda^2$) given by a curve on y-x plane. Then it moves along that curve, and one finds that the system must do jumps that are also very fast (on time scale $1/\Lambda^2$ again) periodically. See plot... Well... I'm ommiting many details. See starting from page 11 on notes

Synchronization and coupled oscillators

Kuramoto model

Lecture notes on nonlinear vibrations

Books:

Nayfeh

Hayashi