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.
