**1. Introduction**

Homeostasis is the tendency of organisms to auto-regular and maintain their internal environment in a stable state [1], For instance, an excellent model to describe the homeostatic process in the organism is the glucose-insulin system [1,2]. On one side, when the glucose level is low, arises diverse pathologies (anxiety, tremors, obfuscation, coma, etc.). On the other side, microvascular damages in the retina, kidney, and neuronal injuries, which lead to chronic renal insufficiency and blindness, are originated by high glucose concentrations. The principal pathology of glucose homeostasis is diabetes. Diabetes can be stated as a chronic disorder provoked when the pancreas does not produce insulin in enough quantities, and also if the body cannot successfully process it, resulting in atypical high blood sugar levels. Because of the autoimmune annihilation of *β*-cells, the insulin released by the pancreas to the human organism is not enough to maintain a certain healthy level, and therefore, the Type 1 Diabetes Mellitus (T1DM) may be manifested. On the contrary, when the beta-cells can produce supranormal, or average concentrations of insulin, but cannot be adequately used in reducing glycemia, the Type 2 Diabetes Mellitus (T2DM) appears [3].

Health agencies [4] demonstrate people, mainly adults, living with diabetes has almost quadrupled from 1980 to 422 million. This rise is primarily due to the rise in T2DM and factors driving it, including overweight and obesity. Diverse mathematical models using ODEs and OdEs have provided a common path to understand multiple complex systems [5,6]. In the last years, because of the long-memory of fractional-order operators, fractional-order systems have gained extensive attention for describing and understanding physical and biological systems [5,7–10]. For instance, the analysis of diverse biological systems, such as bio-impedance, drug diffusion, respiratory tissue, and so forth, was reported in [11] using fractional calculus. In [12], the authors characterize the trade-offs between HIV infection and the tumor-immune system employing fractional-order biological systems. Kheiri and Jafari [13] formulated a fractional-order theory that was focused on the multi-patch HIV/AIDS model to analyze whether human migration has effects on the propagation of the HIV/AIDS outbreak. In [14], the authors reported a fractional-order Izhikevich system to obtain insights on distinct neuronal spike responses, including bursting, fast-, regular-spiking, and chattering, as the no-integer order varies. The functioning at electrical level of a fractional-order system of an isolated *β*-cell is presented in [15,16]. In [17], was reported the comparison among diverse scenarios, such as integer, constant, no-constant and fractional-order derivatives, in order to explain the memory index. In [18], the transmission issues of a susceptible-infected-recovered model were analyzed. They found a proper yield of memory of the fractional-order systems to forecast the pandemic spread. Finally, [19] discovered that the neuron's firing rate could be emulated with a fractional derivative and a slowly varying of the parameters. Therefore, the unique rat neocortical pyramidal neurons have several time-scales.

Therefore, the study of fractional-order biological models continues been critical for an accurate analysis of several health conditions, as well as being essential to understanding this significant open-topic.

Additionally, the synchronization and collective dynamics play an essential role both in physical [20–22] and biological systems [15,23–26]. By one side, the synchronization is necessary for systems with stable behavior, for instance, the human heartbeat and respiration [27,28]; nevertheless, on the other hand, the synchronized state could provoke severe pathologies such as the Parkinson's disease where the excessive synchronization correlates with a motor deficit. For the glucose-insulin regulatory system, a synchronization state is mandatory, since this condition is closely related to a disorder-free state where the glucose-insulin concentrations synchronize. In [29], the *β*-cell synchronization is fundamental to effectuate a pulsatile insulin liberation, which carries more substantial hypoglycemic results compared to constant secretion. In [30], the synchronization between gap-junctions and adjacent cells is essential to limit the heterogeneity and biological noise, thence obtaining a robust activation of the *β*-cells population within the islet. Pecora and Carroll in [31] showed that two nonlinear dynamical systems could be synchronized by introducing appropriate coupling, since then, a variety of approaches were proposed to deal with synchronized states of fractional-order chaos generators, and these include backstepping, adaptive, and active controls [32,33]. Regarding the chaos synchronization applying the active control method [34], it has been demonstrated that an active controller can be straightforwardly designed to achieve synchronization globally if the nonlinearity of the system is known. Thereupon, it is considered to be a promising control strategy due to its straightforwardness. [33].

In addition, the experimental verification of fractional-order models is a topic that has been attracting the attention of researchers [35–39]. In this scenario, ARM-based embedded systems have become in a central block integrated with non-embedded technologies, such as FPGAs, DSPs, and microcontrollers. Subsequently, now it is possible to implement complex software and hardware functionality on a single chip [40]. Several digital hardware have been reported and verified contemplating integer-order chaotic systems [41–43]. Notwithstanding, just a short-list of papers have studied the implementation of fractional chaos oscillators on ARM platforms [44,45].

Motivated by the discussion mentioned above, we analyze the effect of a non-local fractional operator in an asymmetrical glucose-insulin regulatory system. More specifically, the system parameters that were related to hypoglycemia, hyperinsulinemia, T1DM, and T2DM, were studied by using both analytical (stability of equilibrium) and numerical (bifurcation diagrams and basins of attraction) techniques. Additionally, we found that the system presents a chaotic attractor, as demonstrated by its phase portraits and Lyapunov exponent spectrum. Besides, the theoretical insights of the glucose-insulin interaction can be validated by using the proposed electronic implementation based on the embedded SoC ARM Broadcom BCM2837B0. Latter, the synchronization between a chaotic behavior (disorder) and periodic behaviors (average condition) were achieved by applying a simple control strategy.

The sections of the manuscript contain the following. The fractional-order glucose-insulin system, in the sense of Caputo, is presented in Section 2. The stability of the equilibrium points is analyzed in Section 3. Several dynamical analysis for the hypoglycemia, hyperinsulinemia, T1D, and T2D are presented in Section 4. Two synchronization schemes using the active control approach are introduced in Section 5. The ARM implementation is carried out in Section 6, which is followed by the conclusions in Section 7.
