**2.1. Introduction**

DDEs are differential equations in which the derivatives of some unknown functions at present time are dependent on the values of the functions at previous times. In real-world systems, delay is very often encountered in many practical systems, such as control systems [18], lasers, traffic models [19], metal cutting, epidemiology, neuroscience, population dynamics [20], and chemical kinetics [21]. Recent theoretical and computational advancements in DDEs reveal that DDEs are capable of generating rich and plausible dynamics with realistic parameter values. Naturally, occurrence of complex dynamics is often generated by well-formulated DDE models. This is simply due to the fact that a DDE operates on an infinite-dimensional space consisting of continuous functions that accommodate high-dimensional dynamics.

For example, the Lotka–Volterra predator prey model [22] with crowding effect does not produce sustainable oscillatory solutions that describe population cycles. However, the Nicholson's blowflies model [23] can generate rich and complex dynamics. Delayed fractional differential equations (DFDEs) are correspondingly used to describe dynamical systems [23]. In recent years, DFDEs begin to arouse the attention of many researchers [7, 19, 20, 25–27]. Simulating these equations is an important technique in the research, and accordingly, finding effective numerical methods for the DFDEs is a necessary process. Several methods based on Caputo or Riemann–Liouville definitions [28] have been proposed and analyzed. For instance, based on the predictor–corrector scheme, Diethelm et al. introduced the ABM algorithm [29– 31] and mean while some error analysis was presented to improve the numerical accuracy. In 2011, Bhalekar and Daftardar-Gejji [25] have been extended the ABM algorithm to solve DDEs of fractional order and presented numerical illustrations to demonstrate utility of the method. In 2012, Ma et al. [32] have presented the numerical solution of a VOF financial system which is calculated by using the ABM method.

This section presents class of numerical method for solving the variable-order fractional nonlinear delay differential equations (VOFDDEs). The main aim of this part is to study VOFDDEs numerically.
