I know how to linearize a nonlinear system around equilibrium point (with Taylor series). There are lots of example on the internet about it. However, I didn't find a simple explication about linearization around a trajectory. Do you have a explanation or an example ?
Thank you
Say, you have a nonlinear equation $\dot y =f(t,y)$ (here $y$ and $f$ can be vector-valued). To linearize around a trajectory $y_0$, write $y = y_0+z$, thinking of $z$ as small. Then the ODE becomes $$ \dot y_0+\dot z = f(t,y_0+z) \approx f(t,y_0)+f_y(t,y_0) z \tag{1} $$ where $f_y$ is the partial derivative of $f$ in the second argument. Simplify (1) and forget the "approximate" part: $$ \dot z = f_y(t,y_0) z \tag{2} $$ This is a linear equation for $z$, with variable coefficient $a(t)=f_y(t,y_0(t))$.