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I am attempting to work through "Shahverdiev, Sivaprakasam, and Shore (2002) Lag synchronization in time-delayed systems", but I'm missing something basic up front.

The problem is to take a unidirectionally coupled dynamical system,

$\dot{x} =-\alpha x + (m_1+m_3) \sin x_{\tau_1}$

$\dot{y} =-\alpha y + m_2 \sin y_{\tau_1} + m_3 \sin x_{\tau_2}$

where subscript $\tau$ denotes a time-delay, e.g. $x_{\tau_1} = x(t-\tau_1)$, and determine its stability manifold. The synchronization error is defined to be $\Delta = x_{\tau_2 - \tau_1} - y$, but we are interested in the error dynamics, $\dot{\Delta}$, which the authors list as

$\dot{\Delta}=-\alpha \Delta + m_2 \cos x_{\tau_1} \Delta_{\tau_1}$

My question is about this last step. To differentiate $\Delta$, it seems to me that one could use the expressions for $\dot{x}$ and $\dot{y}$ directly, i.e.

$\dot{\Delta} = \dot{x}(t-\tau_2+\tau_1) - \dot{y}(t)$.

Doing that, however, leaves me with

$\dot{\Delta} = -\alpha \Delta + m_2 \left( \sin x_{\tau_2} - \sin y_{\tau_1} \right)$

I feel as though I'm missing a simple step here. Could anyone point me in the right direction?

Here's a second example, this time from "Senthilkumar, Kurths, and Lakshmanan (2009) Stability of synchronization in coupled time-delay systems using Krasovskii-Lyapunov theory":

The system:

$\dot{x} = -a x(t) + b f(x(t-\tau))$

$\dot{y} = -a y(t) + b f(y(t-\tau)) + K(t)(x(t)-y(t))$

The synchronization error (for small values, the authors state):

$\dot{\Delta} = -(a + K(t))\Delta + b f^\prime(y(t-\tau))\Delta(t-\tau)$

Again, I am unsure of the source of the $f^\prime$ term and why that term isn't simply $b \left( f(x(t-\tau)) - f(y(t-\tau)) \right)$. In both examples, it seems a derivative is being taken, but I don't see why that would be.

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Did you read Krasovskii book on equations with delay, KrasovskiÄ­, N. N. Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay. Translated by J. L. Brenner Stanford University Press, Stanford, Calif. 1963 vi+188 pp. –  Mark Sapir May 24 '13 at 23:34
    
I've only read several papers on the topic, but I'll try to find the book. Thank you for replying. –  lomendil May 25 '13 at 0:16
    
You are 100% correct, of course, but your equations are exact and theirs are just linear approximations good for small perturbations. You have seen the formula $f(x)-f(y)\approx f'(y)(x-y)$ (or $f'(x)(x-y)$, if you prefer that one) before, haven't you? Here it is in action. –  fedja May 25 '13 at 3:18
    
fedja, that's the undergraduate-type thing that I was forgetting! Thank you. I consider that the answer. –  lomendil May 25 '13 at 4:13
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