# Multi-Switching Combination Synchronization of Three Fractional-Order Delayed Systems
## 1 Introduction  
Fractional calculus has attracted researchers from various fields due to fractional dimensions widely existing in nature and engineering fields [1; 2; 3]. Compared to the integer-order dynamical systems, the fractional-order counterparts can exhibit more complex dynamical behaviors. Some of the researches on integer-order dynamical systems can be generalized to fractional-order dynamical systems. Fractional-order dynamical systems have been widely investigated, such as synchronization [4], identification [5], stabilization [6] and approximate entropy analysis [7; 8]. Time-delay is a frequently encountered phenomenon in real applications, such as physical, communication, economical, pneumatic and biological systems [9]. Introducing time-delay into a system can enrich its dynamic characteristics and describe a real-life phenomenon more precisely. Thus, the fractional-order delayed differential equation (FDDE) is becoming a hot topic for scientists and engineers, and it has many theoretical and practical applications [10]. Nowadays, the chaotic behavior and synchronization of FDDE attract intensive research interests. In [11], [PERSON] et al. introduced the fractional-order delayed Liu system. The fractional-order delayed financial system was presented in [12], and hybrid projective synchronization between the aforementioned two systems was achieved in [13]. The fractional-order delayed Chen system was proposed in [14], while its adaptive synchronization was investigated in [15]. The fractional-order delayed porous media was proposed in [16]. In [17], a fractional-order delayed Newton-Leipnik system was taken as an example to present intermittent synchronizing delayed fractional nonlinear system. Due to its wide applications in secure communication, synchronization of fractional-order delayed chaotic systems are extensively investigated [18; 19; 20; 21]. However, all the above-mentioned and other synchronization schemes are in traditional drive-response ways, which only have unique drive and response systems. Recently, [PERSON] et al. [22] extended the traditional drive-response synchronization to combination synchronization, which has two drive systems and one response system. Compared to the drive-response synchronization, combination synchronization has strongeranti-decode and anti-attack abilities in image encryption and secure communication, in which origin messages are split into two parts and each part can be embedded into two separate drive systems. There are many works on combination synchronization [23; 24; 25; 26]. To further strengthen the security in secure communication, [PERSON] et al. [27] proposed multi-switching combination synchronization scheme, in which the two drive systems are synchronized with the response system in different states. Based on the nonlinear control technique, [PERSON] [28] studied multi-switching combination synchronization of three non-identical chaotic systems. [PERSON] [29] investigated adaptive multi-switching combination synchronization among three non-identical chaotic systems. In [30], [PERSON] et al. proposed globally exponential multi-switching combination synchronization scheme, and applied it to secure communications. Multi-switching combination synchronization was applied in encrypted audio communication in [31]. The previous work on multi-switching combination synchronization schemes are based on integral-order chaotic systems. To improve the security in these synchronization schemes, based on fractional-order chaotic systems, [PERSON] et al. [32] extended the work in [27] to study multi-switching combination synchronization among three non-identical fractional-order chaotic systems. [PERSON] et al. [33] investigated multi-switching combination-combination synchronization among a class of four non-identical fractional-order chaotic systems.