of a particular type of piecewise linear time-delay systems from the local sequential synchronization as the coupling strength is increased from zero to that of the hyperchaotic regimes. It is worth to emphasize that the hyperchaotic attractors of the piecewise linear time-delay systems are characterized by much more complex topological properties than the chaotic attractor of the Mackey-Glass time-delay system studied in  [@suresh10] and so is of considerable significance in estimating the onset of GPS through sequential synchronization.

\[sec:level2a\]Linear array of piecewise linear time-delay systems
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Now, we consider an array of unidirectionally coupled piecewise linear time-delay systems with open end boundary conditions and with parameter mismatches. Linear stability and bifurcation analysis of this system has been well studied in Ref. [@lakshmanan10; @dvsijbc2005]. Many kind of synchronizations and their transitions have been reported in the coupled piecewise linear time-delay systems [@senthilkumar05; @senthilkumar07; @lakshmanan10]. The array of unidirectionally coupled piecewise linear first-order delay differential equations is represented as

$$\begin{aligned}
\dot{x}_1(t)&=&-\beta x_1(t)+ \alpha_{1} f(x_{1}(t-\tau)),  \\
\dot{x}_i(t)&=&-\beta x_i(t)+ \alpha_{i} f(x_{i}(t-\tau))+ \varepsilon(x_{i-1}(t)-x_{i}(t)), \quad i = 2, 3,\cdots, N,\end{aligned}$$

\[eq\_1\]

where $\alpha, \beta$ are the system parameters, $\tau$ is the time-delay and $\varepsilon$ is the coupling strength. We use the open end boundary conditions $x_{0}=x_{1}$ 