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I am trying to understand the following article:
A. Lahellec, S. Hallegatte, J.-Y. Grandpeix, P. Dumas and S. Blanco, Europhys. Lett. 81, 60001 (2008).
In the beginning, it was quite easy to follow:

Consider a state-space model driven by some external forcing $\mathbf{h}(t)$:
$\partial_t\mathbf{x}(t)= \mathbf{g}(\mathbf{x}(t),\mathbf{h}(t)) $, $\mathbf{x}(t=0)=\mathbf{x}_0$ (1) where $\mathbf{x}$ is the state space vector.
Once a reference trajectory $T^{ref}$ is followed, perturbations to (1) are solutions to the tangent linear system:
$\partial_t\delta\mathbf{x}(t)=A(t)\delta\mathbf{x}(t)+B(t)\delta\mathbf{h}(t) $, $t\in\left[0,T\right]$ (2)
In (2) $\delta\mathbf{x}$ is the deviation of the perturbed system from $T^{ref}$ caused by perturbation $\delta\mathbf{h}(t)$ of the reference forcing function. $A(t)=\partial_x\mathbf{g}(t) $ and B(t)=$\partial_h\mathbf{g}(t)$ are the Jacobian matrices.

To create a feedback loop in the system:
A scalar variable $\varphi$ is added to the system, with $\varphi = f(x) $ i.e. a function of the state variable.
The primitive model (1) is modified to be sensitive to $\varphi$ in such a way that original and modified models are mathematically equivalent:
$\partial_t\mathbf{x}(t)=\mathbf{g'}(\mathbf{x}(t),\varphi(t),\mathbf{h}(t)) =\mathbf{g}(\mathbf{x}(t),\mathbf{h}(t)) $.
A scalar perturbation $u(t)$ is applied to $\varphi$ only: $\varphi (t) = f(x) + u(t)$. The system will respond to that change within the feedback loop: perturbation $u$ on $\varphi$ $\rightarrow$ selected mechanisms $\rightarrow$ rest of the system $\rightarrow$ response $\delta\varphi $.
enter image description here

The opening of the loop (figure on the left) means preventing the system from responding on $\varphi$.

(From here on i don't understand)
Following the preceding procedure, the original tangent system (2) is modified into the following two equations:
$\partial_t\delta\mathbf{x}(t)=A^b(t)\delta\mathbf{x}(t)+\left|b(t)\right\rangle\delta\varphi(t) $ $\; \; \; \; $ $ \delta\varphi(t)=\left\langle c(t)\right|\delta\mathbf{x}(t)+u(t) $ (3)
The column and row matrices $\left|b(t)\right\rangle = \partial_{\varphi}\mathbf{g'} $ and$ \left\langle c(t)\right| = \partial_xf $ formalize the feedback loop. For (3) to be equivalent to (2) the equality : $A(t)=A^b(t)+\left|b\right\rangle\left\langle c\right| (t) $ (4) must hold. System (3) takes the form of (2) when $\delta\varphi$ has been eliminated.

My questions are:
1. How is (2) modified into (3) ? I am not familiar with these notations and i don't get why the term $A^b $ appears and $B(t) $ disappears. O_O
2. Consequently, explain why (4) is the condition for (2) and (3) to be equivalent.
3. It seems like (3) is the equations for closed loop feedback. To open the loop we set $ \left\langle c \right|$ to 0 and disallow the system perturbation to influence $\varphi$. Is that correct ?

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2 Answers

Thank you for your interest in this article. Let me mention a supplementary document that you may have missed: http:www.lmd.jussieu.fr/Zoom/doc/LorenzGains_E.ps.gz

It shows how the linear feedback approach can give insights on even chaotic systems.

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thank you! Chaos was my last semester's subject :D –  Lorenz Chaos Jun 12 '12 at 3:49
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  1. It was just the chain rule with confusing notation:
    $d\mathbf{g'} = \frac{\partial \mathbf{g'}}{\partial\mathbf{x}}d\mathbf{x} + \frac{\partial \mathbf{g'}}{\partial\varphi}d\varphi$
    where $\mathbf{x}$ and $\varphi $ are function of t. $\mathbf{g'}(\mathbf{x},\varphi) = \partial_t\mathbf{x} $ and $d\mathbf{g'} = \partial_t\delta \mathbf{x} $. Assume no change in external forcing $\delta\mathbf{h}(\mathbf{t})=0 $
    $ \frac{\partial \mathbf{g'}}{\partial\mathbf{x}} = A^b $ which is just a weird way to denote the Jacobian matrix $\partial_\mathbf{x}\mathbf{g'} $
    $ \frac{\partial \mathbf{g'}}{\partial\varphi} = \left | b \right \rangle $ and $\frac{\partial f}{\partial\mathbf{x}} = \left\langle c \right | $

  2. Assume that the system is autonomous (no external forcing ) it follows immediately from 1.

  3. Yes.
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