Background: the dynamics and the tangent linear model

Let the dynamics be given by

$$\frac{d\mathbf{x}}{dt} = \mathbf{F}(\mathbf{x}(t),t) .$$ (1)

Let us denote the trajectory starting at point $\mathbf{x_0}$ at $t=0$ by $\mathbf{z}(t;\mathbf{x_0})$, i.e., $\mathbf{z}(0;\mathbf{x_0}) = \mathbf{x_0}$. Then the evolution of a tangent vector $\mathbf{y_0}$ at $\mathbf{x_0}$ is given by the tangent linear model (TLM)

$$\frac{d\mathbf{y}}{dt} = \left.\frac{\partial \mathbf{F}(\mathbf{... ...thbf{z}(t;\mathbf{x_0})} \mathbf{y}(t) ,\qquad \mathbf{y}(0) = \mathbf{y_0} .$$ (2)

Thus the tangent linear model depends on the trajectory around which we are linearizing. Now, we can be a little sloppy and say that small perturbations around the trajectory also evolve according to the above TLM. Of course, the error introduced by this sloppiness will decrease as the initial perturbations are made smaller or the tangent linear model is integrated for shorter time.

In principle, the TLM is integrated very simply because it is a linear equation and the solution can be written as

$$\mathbf{y}(t) = \mathbf{M}(t,t_0;\mathbf{x_0}) \mathbf{y_0} ,$$ (3)

where the linear operator (matrix) $\mathbf{M}(t,t_0;\mathbf{x_0})$ depends on the trajectory (here denoted through the initial condition $\mathbf{x_0}$). This operator will henceforth be written simply as $\mathbf{M}$.