FPGA realization of fractional-order chaotic oscillators using ABM and EFORK methods

Abstract

Despite the growing theoretical interest in fractional-order chaotic systems, there is a lack of systematic, hardware-oriented methodology for their real-time digital realization that preserves chaotic dynamical properties and optimize computational resource utilization, particularly in the context of FPGA implementations using different fractional-order numerical integration schemes. This work presents the development of two reconfigurable VHDL modules for the digital emulation of fractional-order chaotic oscillators based on the Adams–Bashforth–Moulton (ABM) and explicit fractional-order Runge–Kutta (EFORK) methods. Fractional-order implementations of the Lorenz, Rabinovich–Fabrikant, Li, Chen, and Yu–Wang oscillators are realized on an Artix-7 FPGA platform. To validate the proposed architectures, Matlab simulations of each oscillator using both numerical methods are first conducted, followed by a bifurcation analysis of the Lyapunov exponents (LEs) as a function of the fractional-order. It is shown that, in general, ABM yields positive LEs with larger magnitudes and over wider intervals than EFORK, except for the Chen oscillator. Finally, the logic resources required to emulate each oscillator on an FPGA device are analyzed, and the results show that Runge–Kutta systematically reduces LUT usage due to a more hardware-efficient computational structure, particularly by avoiding division operations, which are more resource-intensive in FPGA implementations. As a reference, Adams–Bashforth–Moulton implementation requires the fewest logic resources for the Lorenz system, with 47,780 LUTs, whereas for Runge–Kutta, Li system requires 38,860 LUTs. Overall, the results highlight a clear trade-off between dynamical complexity and resource efficiency in the digital emulations of fractional-order chaotic oscillators on FPGA devices.

Introduction

Fractional-order dynamical systems have emerged as one of the active research frontiers in nonlinear science and engineering over the past two decades. Unlike their integer-order counterparts, fractional-order systems possess memory and hereditary properties inherent in many real-world processes, which allows them to model complex phenomena with greater accuracy. Moreover, the study of fractional-order nonlinear dynamical systems has grown drastically due to their wide range applications in areas such as chemistry ¹, control ², electronics ³, medicine ⁴, and telecommunications ⁵, among others. Within this landscape, chaotic oscillators are particularly applicable nonlinear systems because they exhibit high sensitivity to perturbations and initial conditions, and although they are deterministic, they produce aperiodic trajectories over time  ^6,7 which makes them useful in several domains. The hardware realization of the fractional-order chaotic oscillators on Field Programmable Gate Arrays (FPGAs) has gained considerable momentum, driven by the need for fast, reconfigurable, and energy-efficient implementations that are suitable for real-time applications⁸. FPGA offer the distinct advantages of parallel processing and low-latency computation that can optimize several iterative discretization schemes required by most fractional differential equations. This makes them an hardware resource that many researchers consider in the acceleration of computationally intensive numerical methods, particularly for real-time or high-performance applications involving fractional differential equations⁹.

However, a key challenge in this domain lies in the selection of appropriate numerical methods to discretize the continous fractional-order differential equations into hardware-amenable difference equations. This has allowed researchers to develop several methods applied to this problem such as Adams–Bashforth–Moulton (ABM)¹⁰, Grünwald–Letnikov (GL)¹¹, and, more recently, fractional-order Runge–Kutta (FORK) schemes¹². The ABM method is a well-established multi-step predictor-corrector approach that leverages Caputo fractional derivative formulation and it is known for its accuracy and stability in capturing long-memory dynamics. Although, ABM avoids the long-term memory accumulation problem by expanding the nonlinear terms of the system in Adomian polynomials which is parallelizable and well suited for FPGA pipelines, it also comes at the cost of significant memory overhead¹³. The comparative study in¹⁴, examined six fractional-order chaotic oscillators including the Lorenz and Chua systems, and demonstrated that GL and ABM algorithms provide feasible solutions for generating chaotic attractors with the ABM method generally exhibiting superior accuracy in capturing the fine structure of attractor topology. Furthermore, the authors show that qualitative differences in trajectory accuracy becomes more pronounced at lower fractional orders and step sizes, where the memory kernel exerts a stronger influence on the systems dynamics. These findings reinforce the status of ABM method as a reliable benchmark for fractional-order chaotic simulation, even if its direct hardware implementation present challenges.

In contrast, FORK methods are a more recently formalized class of methods derived from the classical Runge-Kutta techniques but extended to fractional order settings which offers a computationally lighter alternative with reduced memory requirements while preserving acceptable numerical fidelity. While the ABM and GL methods require accumulation of weighted sums over the entire solution history¹⁵, FORK method evaluates the system equations only at a small number of intermediate stages within each time step without requiring explicit storage of past states beyond the current steps. Nevertheless, there are open challenges in terms of numerical accuracy and hardware benchmarking protocols for fractional-order chaotic oscillators FPGA implementations. Also, most current FPGA implementations focus on commensurate systems leaving incommensurate implementations unexplored where different states of the oscillators have different derivative orders. Researchers have extensively documented the chaotic dynamics of well-known oscillator models extended to the fractional domain. Systems such as the Lorenz  ¹⁶, Rössler¹⁷, Chua¹⁸, Chen¹⁹, Lü²⁰, and Hyperjerk²¹ oscillators have been simulated as fractional-order systems using various numerical methods. In terms of stability, the fractional-order (FO) parameter significantly contributes to the dynamic richness of a chaotic oscillator. Its influence is typically quantified using Lyapunov exponent analysis, which serves as an objective measure of stability by evaluating the average rate at which small perturbations either grow or decay over time  ²². Complementing this, bifurcation diagrams provide a visual representation of the system’s dynamics, illustrating qualitative changes in behavior as system parameters vary. These diagrams reveal transitions from periodic orbits to chaotic regimes, offering a comprehensive view of the structural evolution of the system’s dynamics  ²³.

At present, chaotic systems have applications in different fields, such as thermodynamics ²⁴, neural networks ²⁵, Earth system dynamics ²⁶, among others. In particular, they have significant applications in telecommunications, where their complex dynamics are exploited to generate pseudo-random sequences aimed at protecting information ²⁷. This in turn, motivates the need for the hardware implementations that emulate their behaviors. Of particular interest, chaotic oscillators have been exploited in chaos-based communication systems because of their inherent unpredictability and sensitivity to initial conditions that are of great importance in developing secure transmission schemes. These properties essentially makes them even more attractive for applications such as spread-spectrum systems, and random sequence generation. In this context, efficient hardware implementation becomes a critical requirement, especially for real-time and embedded applications. FPGA-based realizations enables the high-speed and parallel processing of chaos-based systems in practical communication scenarios, including deployment in edge devices and software-defined radio platforms.

Moreover, FO increases the complexity of chaotic waveforms and yields encryption sequences that are more difficult to break.

In particular, Adeyemi et. al.²⁸ implemented a parameter switching scheme in VHDL to stabilize spherical FO systems, synchronized master–slave oscillators using a Hamiltonian approach, and demonstrated an application of secure image (RGB and grayscale) encryption/transmission on Artix-7 AC701 and Intel Cyclone V boards, also they reported correlation metrics and logic-resource usage, while all the discretizations of the FO system was carried out with GL method. For their part, Soriano–Sánchez et. al.²⁹ synchronized complex networks of nine FO Liu oscillators coupled via the second state variable and showed an FPGA implementation on a Zynq-7000 board using fixed-point arithmetic and a LabVIEW–FPGA software/hardware chain. Meanwhile, Pano-Azucena described an end-to-end architecture to solve different families of FO chaotic oscillators via GL on a Cyclone IV, applying the short-memory principle implemented in rom/ram blocks, a control unit for hardware reuse, and the experimental observation of attractors through a digital-to-analog converter (DAC) DAS1612, reporting operating frequencies between 77.59 and 84.9 MHz ³⁰.

More recently, several complementary research directions further motivate accurate and efficient hardware emulation of chaotic dynamics beyond purely numerical-method considerations. On the control side, Liu et. al.³¹ propose a noise-tolerant fuzzy-type zeroing neural network (NTFTZNN) with a fuzzy time-varying convergent parameter for robust synchronization of chaotic systems, where an FLS-generated parameter adaptively tunes the convergence rate according to the synchronization error; they provide predefined-time and noise-tolerance analysis and validate the approach through comparative synchronization experiments on chaotic systems of different dimensions. In a closely related work, Jin et. al.³² introduce a time-varying fuzzy-parameter ZNN (TVFP-ZNN) for chaotic synchronization under external noise, benchmarking it against multiple ZNN variants and additionally demonstrating practical feasibility by displaying the synchronization process on an FPGA-based oscilloscope setup. From the hardware-platform viewpoint, Huang et. al.³³ present EdgeLLM, a highly efficient CPU–FPGA heterogeneous edge accelerator that combines mix-precision systolic processing elements with an end-to-end compilation flow, illustrating best practices for mapping compute-intensive workloads onto heterogeneous CPU–FPGA systems. Finally, recent advances also reveal that deploying synchronized chaotic cryptosystems in hardware introduces new security concerns: Almada-Prieto et. al.³⁴ demonstrate a chaos-based hardware-Trojan covert-channel concept that exploits the synchronization phase of chaotic cryptosystems to leak secret information, underscoring that faithful FPGA realizations of chaotic oscillators are critical not only for functionality and performance, but also for evaluating security-relevant behaviors in edge-device deployments.

In this work, the FPGA realization of the ABM and explicit fractional-order Runge-Kutta (EFORK) numerical methods is presented through a modular and reconfigurable architecture that enables straightforward replacement of the governing equations and flexible tuning of oscillator parameters, facilitating reproducible and didactic experimentation. As a case study, the proposed approach is evaluated using five widely cited chaotic oscillators: Lorenz ³⁵, Rabinovich–Fabrikant ³⁶, Li ³⁷, Chen ³⁸, and Yu–Wang ³⁹.

Additionally, all five oscillators are simulated in Matlab, where bifurcation diagrams and LE diagrams are obtained as the FO is varied. This analysis aims to identify fractional–order values that yield pronounced dynamical divergence while preserving stable chaotic behavior. The main contributions of this work are:

• FPGA implementation of the ABM and EFORK methods to emulate FO chaotic systems. The architecture was designed in VHDL with 32-bit words on the Artix 7 AC701 board;

• FPGA emulation of the Lorenz, Rabinovich–Fabrikant, Li, Chen, and Yu–Wang oscillators using the ABM and EFORK methods;

• Matlab-based bifurcation analysis using the LEs for the five FO oscillators solved with ABM and EFORK.

This article is organized as follows: Section 2 presents the mathematical models of the oscillators, along with the numerical methods used. In Section 3, the simulations of the five FO chaotic oscillators are shown, as well as the maximum bifurcation analysis based on the LEs. In Section 4, the VHDL implementation of the ABM and EFORK methods is presented. In Section 5, the results are discussed. Finally, in Section 6, the main conclusions are presented.

Theoretical framework

This section presents a brief description of fractional calculus, together with the equations of the Lorenz, Rabinovich–Fabrikant, Li, Chen, and Yu–Wang oscillators. Afterwards, the description of ABM and EFORK methods used to solve FO chaotic systems are presented. Finally, LEs and Benettin–Wolf algorithm for computing LEs in FO systems are introduced.

Fractional calculus

Although fractional calculus was initially regarded as a mathematical curiosity, it has gradually evolved into a fundamental tool for modeling complex phenomena. By extending integral and differential operators to real, and complex domains, fractional calculus overcomes the traditional restriction to integer-order operations⁴⁰. Although the computation of FO derivatives and integrals may initially appear nontrivial, a common starting point is the definition of a generalized integral operator of order . This formulation is presented in equation (1).

1

where is the order of the integral (positive integer), is the independent variable, is the function dependent on , is the integration variable, and denotes the derivative operator with negative exponent, which represents an integration. As observed in (1), the factorial depends on the order of integration; therefore, in this form it is not possible to obtain a FO integral. Cauchy noticed this and proposed replacing the factorial with the Gamma function () ⁴¹, which generalizes the factorial to real and complex numbers. Mathematically, is defined as shown in equation (2),

2

where is the integration variable, is a real or complex number, and is the Gamma function evaluated at . In this way, Cauchy succeeded in expressing in a general form an integral of real or FO order. However, it was not until 1832 that Liouville, building on Cauchy’s concepts ⁴², formalized the FO integral as shown in equation (3),

3

where represents the integration order, which can be any real number. On the other hand, to obtain an FO derivative, there exist several definitions; among the most popular is the Caputo fractional derivative, whose definition adapts the fractional formula so that the initial conditions are expressed in terms of integer-order derivatives of the function. This makes it more intuitive and provides the opportunity to model real dynamical systems using systems of FO differential equations ⁴². Formally, the Caputo fractional derivative is expressed as shown in equation (4),

4

where is the FO differential operator, is the smallest integer such that , is the integer-order derivative of order with respect to .

Table 1 lists equations (5)–(9), which correspond to the Lorenz, Rabinovich–Fabrikant, Li, Chen, and Yu–Wang oscillators in their FO form, together with the parameter values for which a chaotic behavior is obtained.

Table 1.

Parameters of each FO chaotic oscillator.

Chaotic Oscillator	Equations	Parameters
Lorenz35
Rabinovich–Fabrikant36
Li37
Chen38
Yu–Wang39

Despite the availability of analytical methods for solving FO systems, many physical systems are so complex that it is very difficult or even impossible to obtain an analytical solution that describes their behavior. To overcome these limitations, numerical methods have emerged as an effective alternative for obtaining numerical approximations, since they rely on discrete computational algorithms that achieve results with a reasonable degree of accuracy for systems that do not admit an analytical solution. In what follows, the ABM and EFORK methods for numerically solving FO chaotic oscillators are presented.

Adams-Bashforth-Moulton method

The ABM predictor–corrector method employs successive predictor and corrector iterations to improve numerical accuracy while avoiding complex mathematical formulations, which facilitates its efficient implementation in software and hardware.

Equations (10) and (11) correspond to the ABM predictor–corrector method,

10

11

where is time, is the FO, is the step size between iterations, is the dependent variable, is the sample index, is the initial value, is the system function, is the predictor value, and is the corrected value. The coefficients and are obtained from (12) and (13), respectively,

12

13

In equations (12)–(13), the summation over runs through all previous states, preserving the dependence on past values at each evaluation. For the ABM predictor–corrector scheme ⁴³, the typical error is of order ; therefore, if a maximum error of is desired, it is required that .

Explicit fractional-order Runge-Kutta method

Although standard Runge–Kutta methods are not directly applicable to FO systems, their popularity, ease of implementation, and reliability have motivated their adaptation to FO formulations, commonly referred to as FORK inclusive of both implicit and explicit methods. Similar to classical Runge–Kutta schemes, explicit fractional-order Runge-Kutta (EFORK) method is characterized by an order index, where higher-order methods provide increased accuracy at the expense of greater computational complexity. The general EFORK scheme of order is shown in equations (14)–(15).

14

15

In the above expressions, denotes the state variables of the system, the current state, and the next state; is the system vector field; is the step size between iterations; is the FO; is the order of the method; are prescribed constants; and , depend on .

For this study, a second-order EFORK scheme was selected because it offers an adequate balance between numerical performance and implementation complexity. Higher-order EFORK schemes require additional intermediate stages, which increase the number of operations, and temporary storage elements in the FPGA design. The second-order EFORK is given by equations (16) and (17),

16

17

where and represent estimates of the slope of the solution of the differential equation, is the value of the current time instant, are the current values of the differential equations, and the next state is given by (18),

18

with the parameters , , and defined in equations (19)–(21),

19

20

21

If we define as in (22),

22

and this yields equations (23)–(25),

23

24

25

For the second-order EFORK scheme ⁴⁴, the typical error is ; thus, for a maximum error of , it is required that .

Lyapunov exponents

LE were developed to analyze the stability of deterministic systems of differential equations, focusing on the effect of small perturbations on the evolution of a system and determining whether, under such perturbations, the system remains stable. As mentioned above, chaotic systems are deterministic and exhibit high sensitivity to arbitrarily small changes in their initial conditions; this sensitivity can be characterized using LEs. LEs are widely used in dynamical-systems theory to distinguish among stable, periodic, and chaotic behaviors ⁴⁵. A continuous dynamical system can be expressed as in equation (26),

26

where x represents the state of the system and f(x) defines its evolution. To compute the LE in a one-dimensional system, two nearby trajectories in phase space are selected and the distance between them is evaluated, as in equation (27),

27

where and denote two nearby trajectories and d(t) their separation. In a chaotic system, d(t) is expected to grow exponentially. The average rate of exponential divergence, that is, the Lyapunov exponent, is defined as in equation (28),

28

where is the LE of the system. For a single dimension, if the dynamics are periodic or quasi–periodic; if , the system converges; and if , the system exhibits divergent behavior^45,46. In k-dimensional systems, there are k LEs, so one speaks of an LE spectrum, and the same procedure used to compute a single LE is applied; however, the computation becomes more complex, and it is necessary to use the Jacobian matrix to determine the divergence rate among the different trajectories in phase space. The Jacobian matrix is defined as in equation (29),

29

where k is the dimension of the system, x is the state vector, f(x) is the vector field, and is the Jacobian matrix. Likewise, it is established that for a three-dimensional system ⁴⁷:

• If all three LEs are negative, the system converges to a point.

• If two LEs are negative and one is 0, the behavior is cyclic.

• If one LE is negative and the other two are 0, the system exhibits a stable two-dimensional torus.

• If at least one of the three LEs is positive, the system is chaotic.

Bennettin-Wolf algorithm to calculate LEs of FO systems

The Benettin–Wolf algorithm makes it possible to compute LE for k-dimensional systems; it was proposed by Benettin in 1980 ⁴⁸ and later implemented in assembly language by Wolf in 1985 ⁴⁹. In broad terms, it consists of the following steps:

1. Initialization. Select a set of k initial vectors or nearby trajectories around the initial state of the reference trajectory in phase space.

2. Solve the system. Integrate the oscillator system of differential equations. In the FO case, use an appropriate integrator such as ABM or EFORK to evolve the main trajectory and the tangent vectors.

3. Orthogonalization and rescale. After a fixed number of integration steps, the tangent vectors tend to align due to exponential divergence. Gram–Schmidt orthogonalization is then applied to maintain their independence. The Gram–Schmidt procedure generates an orthonormal family from an arbitrary family that spans the same subspace ⁵⁰. The list of orthogonal vectors is based on the generalization of equation (30),

   30

   where denotes the dot product and i represents the dimension, taking integer values from 1 to k.

4. Calculate LEs. Once the Gram–Schmidt procedure has been applied, the logarithmic expansion rate of the orthogonalized vectors is computed to determine the LEs using equation (28).

An important advantage of the Benettin–Wolf algorithm is that the integration of dynamic systems is modular with respect to the orthogonalization and rate-computation stages; therefore, the scheme is directly applicable to FO systems by using, in Step 2, the appropriate integrator (ABM or EFORK).

Matlab simulation

In this section, we present Matlab simulations of five FO chaotic oscillators solved using the ABM and EFORK methods. We also report how the LE diverge as the FO of each oscillator is varied.

Simulation of FO chaotic oscillators

In this section, we present Matlab simulations of the FO chaotic Lorenz, Rabinovich–Fabrikant, Li, Chen, and Yu–Wang oscillators. We solve each oscillator using both the ABM and EFORK methods. Figure 1 compares the simulations obtained with the two numerical methods. The selected fractional orders were initially guided by values reported in previous studies. Subsequently, preliminary simulations were performed to refine these choices, seeking FO sufficiently separated from the corresponding IO case while preserving the characteristic trajectories of each chaotic system. Furthermore, for each oscillator, the same parameters were used in both the ABM and EFORK simulations to ensure a fair comparison between the two numerical methods. In terms of dynamic richness, the selection of the fractional order for each oscillator was guided by a systematic criterion aimed at balancing irregularity and numerical stability. Specifically, the chosen values of ensure a clear deviation from the corresponding integer-order behavior, while preserving bounded chaotic attractors. This selection was supported by bifurcation analysis and Lyapunov exponent evaluation, which allowed identifying regions where the systems exhibit sustained chaotic dynamics. In this way, the selected fractional orders are not arbitrary, but correspond to operating points that maintain the essential characteristics of fractional-order chaotic behavior.

Fig. 1.

Simulation of phase plane of FO chaotic oscillators, Lorenz commensurate: (a) ABM, (b) EFORK. Lorenz non–commensurate: (c) ABM, (d) EFORK. Rabinovich–Fabrikant: (e) ABM, (f) EFORK. Li: (g) ABM, (h) EFORK. Chen: (i) ABM, (j) EFORK. Yu–Wang: (k) ABM, (l) EFORK.

When simulating the Lorenz chaotic system with the same commensurate FO using both numerical methods, the following values were used: , , , , , and . Figure 1a shows the commensurate Lorenz oscillator in the phase plane – solved with ABM, while Fig. 1b shows the solution obtained with EFORK. Likewise, to simulate the Lorenz chaotic oscillator with different, non-commensurate FO, only was modified to and . Figure 1c shows the non-commensurate Lorenz oscillator in the phase plane – solved with ABM, while Fig. 1d shows the solution obtained with EFORK.

On the other hand, to simulate the Rabinovich–Fabrikant chaotic system with FO using both numerical methods, we use the following parameters: , , , , , and . Figure 1e shows the Rabinovich–Fabrikant oscillator in the phase plane – solved with ABM, while Fig. 1f shows the solution obtained with EFORK.

To simulate the Li chaotic system with FO using both numerical methods, we use the following parameters: , , , , , and . Figure 1g shows the Li oscillator in the phase plane – solved with ABM, while Fig. 1h shows the solution obtained with EFORK.

Subsequently, when simulating the Chen chaotic system with FO using both numerical methods, we use the following parameters: , , , , , and . Figure 1i shows the Chen oscillator in the phase plane – solved with ABM, while Fig. 1j shows the solution obtained with EFORK.

Finally, to simulate the Yu–Wang chaotic system with FO using both numerical methods, we use the following parameters: , , , , , and . Figure 1k shows the Yu–Wang oscillator in the phase plane – solved with ABM, whereas Fig. 1l shows the solution obtained with EFORK.

Bifurcation of LEs

In this section, the bifurcation diagrams and the LE spectra obtained by varying the FO of the chaotic Lorenz, Rabinovich–Fabrikant, Li, Chen, and Yu–Wang oscillators are presented. Each oscillator was solved using the ABM and EFORK methods, respectively. In all cases, the Lyapunov exponents were computed in MATLAB. Gram–Schmidt orthogonalization was performed every 1 samples, and the transient interval removed prior to computation was 1 samples, equivalent to 1 s. Convergence was assessed using a final-window stability criterion, whereby the difference among the last 100 estimated values was required to be less than 1. This practical condition is consistent with standard LE estimation procedures in which the computation is stopped once the obtained exponent stabilizes⁵¹.

In this work, Lyapunov exponents are used to characterize the dynamical behavior of the fractional-order chaotic systems under different numerical methods. It is important to note that the role of Lyapunov exponents in this study is to analyze the dynamical regimes of the systems, rather than to directly validate the FPGA implementation.

To obtain the bifurcation diagrams and the LE spectra of the Lorenz oscillator with both numerical methods, we sweep the FO over the range using samples. Figure 2 shows the bifurcation diagrams and the LE spectra for the Lorenz system, where Fig. 2a corresponds to the system solved with ABM and Fig. 2b to the system solved with EFORK.

Fig. 2.

Bifurcation diagrams and LE spectra: Lorenz oscillator solved by (a) ABM, (b) EFORK; Rabinovich–Fabrikant oscillator solved by (c) ABM, (d) EFORK; Li oscillator solved by (e) ABM, (f) EFORK; Chen oscillator solved by (g) ABM, (h) EFORK; Wang oscillator solved by (i) ABM, (j) EFORK.

To compute the bifurcation diagrams and the LE spectra of the Rabinovich–Fabrikant oscillator with both numerical methods, we sweep the FO over the range using samples. The bifurcation diagrams and the LE spectra for the Rabinovich–Fabrikant system are shown in Fig. 2c for the system solved with ABM and in Fig. 2d for the system solved with EFORK.

To evaluate the system’s dynamics, the bifurcation diagrams and the LE spectra of the Li oscillator with both numerical methods, we sweep the FO over the range using samples. The bifurcation diagrams and the LE spectra for the Li system are shown in Fig. 2e for the system solved with ABM and in Fig. 2f for the system solved with EFORK.

To construct the bifurcation diagrams and the LE spectra of the Chen oscillator with both numerical methods, we sweep the FO over the range using samples. The bifurcation diagrams and the LE spectra for the Chen system are shown in Fig. 2g for the system solved with ABM and in Fig. 2h for the system solved with EFORK.

To generate the bifurcation diagrams and LE spectra of the Yu–Wang oscillator with both numerical methods, we sweep the FO over the range using samples. The bifurcation diagrams and the LE spectra for the Yu–Wang system are shown in Fig. 2i for the system solved with ABM and in Fig. 2j for the system solved with EFORK.

VHDL implementation of numerical methods

The VHDL language operates on a bit-by-bit basis, which complicates the handling of large data volumes. Many libraries convert bit vectors to base-10 representations, but they are often intended only for simulation and are not synthesized on an FPGA; therefore, only IEEE.STD_LOGIC_1164 and IEEE.NUMERIC_STD were used, since they support basic arithmetic on bit arrays and generate synthesized code for reconfigurable FPGA modules.

FO chaotic oscillators require high numerical precision, but the word length on an FPGA is limited: too few bits lead to information loss and instability, whereas too many bits increase resource utilization. A 32-bit signed fixed-point format was adopted for the state variables, parameters, initial conditions, FO, and sampling time, achieving six decimal digits of precision; thus, 32-bit variables represent numbers from -214.7483648 to 214.7483647. This choice balances precision and resource usage and enables the FO chaotic systems to be solved using ABM and EFORK.

ABM implementation

The implementation of the FO chaotic oscillators solved with ABM was carried out in VHDL using Vivado 2020.2. Since VHDL does not directly provide functions such as or real-valued powers, the constants that involve these terms are supplied as constant inputs, in order to facilitate tests with different values, and, in a final FPGA version, to replace such inputs with constants stored in memory. To reduce resource usage, in the ABM predictor–corrector step, only the previous state is stored, accepting a slight loss of accuracy; moreover, and are fixed as constants per state variable.

Table 2 shows the input and output ports corresponding to a general entity that emulates FO chaotic systems solved with ABM. Here, corresponds to the clock input, resets the emulation when set to 1; correspond to the initial values of the state variables of the system, with denoting the number of variables; corresponds to the FO of each equation of the chaotic system; is the bifurcation parameter of the system; correspond to the constant values of the Gamma function with respect to the FO; correspond to the constant values of raised to the FO of each equation; correspond to the constant values of raised to the FO of each equation; is the sampling time of the emulation; corresponds to the sampling time raised to the FO of each equation; and are the outputs that display the behavior of each state variable of the oscillator.

Table 2.

I/O for ABM VHDL implementation.

Signal’s name	Type	Length	Signal’s name	Type	Length	Signal’s name	Type	Length
	IN	std_logic		IN	std_logic (31 downto 0)	...	IN	std_logic (31 downto 0)
	IN	std_logic		IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)
	IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)
	IN	std_logic (31 downto 0)	...	IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)
...	IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)
	IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)	...	IN	std_logic (31 downto 0)
	IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)
	IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)		OUT	std_logic (31 downto 0)
...	IN	std_logic (31 downto 0)	...	IN	std_logic (31 downto 0)		OUT	std_logic (31 downto 0)
	IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)	...	OUT	std_logic (31 downto 0)
	IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)		OUT	std_logic (31 downto 0)
	IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)

The operation of the ABM module in VHDL is divided into functional blocks that perform each of the operations in equations (10)–(13). The first block corresponds to each equation of the system; this block is denoted , and there are as many such blocks as variables in the system. This block receives as inputs the values of the system variables and the parameter , and it outputs the result of the corresponding equation. The second function, denoted , performs the operations required to compute the summation in equation (10). This block takes as inputs the output of , the corresponding value of the constant , and the Gamma-related constant. Subsequently, once the predictor for each variable has been calculated, the function is used to obtain the summation in equation (11). This block takes as inputs the output of , the corresponding value of the constant , the Gamma-related constant, the value of , and the output of evaluated using the predictor values as inputs. Figure 3 shows the RTL diagram of the module used to emulate n-dimensional FO chaotic oscillators using the ABM method.

Fig. 3.

RTL Diagram of a module to emulate FO chaotic oscillators solved by ABM.

Once implemented, the emulation was carried out in Vivado 2020.2. Figure 4 shows the three state variables of the Lorenz system solved with ABM; Fig. 4b corresponds to the emulation in Vivado 2020.2, whereas Fig. 4a corresponds to the Matlab simulation. For both figures, the following parameters were used: , , , and , and the Vivado 2020.2 emulation was run at a frequency of .

Fig. 4.

Lorenz system solved by ABM with: (a) Matlab simulation; (b) Vivado 2020.2 emulation.

A drawback of running emulations in Vivado 2020.2 is that the value of each input must be assigned every time the emulation is executed; when testing different values, this becomes impractical. One solution is to use System Generator for DSP, which adds a Vivado 2020.2 library to Matlab and migrates the VHDL designs as blocks in Simulink, thereby facilitating experimentation. In addition, by exporting the FPGA results to Matlab, the analysis becomes more flexible, since Matlab provides specialized tools for generating and analyzing plots.

Finally, Fig. 5a shows the module generated by System Generator for DSP, which encapsulates the complete VHDL code to emulate the Lorenz system using ABM; whereas Fig. 5b shows the block diagram for the emulation of the Lorenz system solved with ABM in System Generator for DSP. It is important to note that this process was applied to the five FO chaotic systems.

Fig. 5.

Lorenz oscillator solved by ABM in System Generator for DSP: (a) VHDL black box; (b) block diagram to emulate the system.

EFORK implementation

To implement reconfigurable modules that emulate FO chaotic oscillators using the EFORK method, VHDL was employed within the Vivado 2020.2 design environment. As in the ABM-based implementation, certain constant values associated with operations that are computationally demanding on FPGA hardware were exposed as input ports to facilitate testing and parameter variation. Once suitable parameter values are identified, most of these inputs can be replaced by constants stored in system memory, enabling a fully self-contained FPGA implementation.

Table 3 shows the input and output ports corresponding to a general entity that emulates FO chaotic systems solved with EFORK. Here, corresponds to the clock input; resets the emulation when set to 1; correspond to the initial values of the state variables of the system, with denoting the number of variables; correspond to the FO of each equation of the chaotic system; is the bifurcation parameter of the system; correspond to the constant values for each equation; correspond to the constant values for each equation; correspond to the constant values for each equation; is the sampling time of the emulation; correspond to the sampling time raised to the FO of each equation; and are the outputs that display the behavior of each state variable of the oscillator.

Table 3.

I/O for EFORK VHDL implementation.

Signal’s name	Type	Lenght	Signal’s name	Type	Lenght	Signal’s name	Type	Lenght
	IN	std_logic		IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)
	IN	std_logic		IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)
	IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)
	IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)
...	IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)	...	IN	std_logic (31 downto 0)
	IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)
	IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)		OUT	std_logic (31 downto 0)
	IN	std_logic (31 downto 0)	...	IN	std_logic (31 downto 0)		OUT	std_logic (31 downto 0)
...	IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)	...	OUT	std_logic (31 downto 0)
	IN	std_logic (31 downto 0)		IN	std_logic (31 downto 0)		OUT	std_logic (31 downto 0)

The operation of the EFORK module in VHDL is divided into functional blocks that perform each of the operations in equations (16)–(25). The first block is , as in subsection ABM implementation. The second function is denoted and performs the operations required to obtain the values of and ; this block takes as inputs the output of and . The third function is denoted and multiplies the constant by , and then adds the corresponding state variable value , , , or . Subsequently, the value of each equation is computed using these intermediate values, as indicated in equation (17). This block takes as inputs the corresponding constant , , and the current value of the state variable being processed. The fourth function is denoted and performs the operations in equation (18) to compute the next state of , , , or . This block takes as inputs the corresponding constants and , , , and the current value of the state variable being processed. Figure 6 shows the RTL diagram of the module used to emulate n-dimensional FO chaotic oscillators using the EFORK method.

Fig. 6.

RTL Diagram of a module to emulate FO chaotic oscillators solved by EFORK.

Once implemented, the emulation was carried out in Vivado 2020.2. Figure 7 shows the three state variables of the Lorenz system solved with EFORK; Figure 7bcorresponds to the emulation in Vivado 2020.2, whereas Fig. 7a corresponds to the Matlab simulation. This system was emulated using the following parameters: , , , and , and the Vivado 2020.2 emulation was run at a frequency of .

Fig. 7.

Lorenz system solved by EFORK: (a) Matlab simulation; (b) Vivado 2020.2 emulation.

Furthermore, Fig. 8a shows the module generated by System Generator for DSP, which encapsulates the complete VHDL code to emulate the Lorenz system using EFORK; whereas Fig. 8b shows the block diagram for the emulation of the Lorenz system solved with EFORK in System Generator for DSP. It is important to note that this process was applied to the five FO chaotic systems.

Fig. 8.

Lorenz oscillator solved by EFORK in System Generator for DSP: (a) VHDL black box; (b) block diagram to emulate the system.

Finally, both the ABM and EFORK modules compute the next value of all oscillator state variables in every clock cycle. Table 4 presents a theoretical complexity comparison between the ABM and EFORK methods, indicating the number of multiplications and divisions required by each numerical method for each state variable of the chaotic oscillator. This comparison excludes the additional operations associated with the specific system of equations of each chaotic oscillator and does not include the normalization division required after each multiplication between two variables.

Table 4.

Theoretical arithmetic complexity comparison between the ABM and EFORK methods.

Method	Multiplications per variable	Divisions per variable	3D total multiplications	3D total divisions	4D total multiplications	4D total divisions	Clock cycles per update
ABM	3	2	9	6	12	8	1
FORK	4	0	12	0	16	0	1

Simulation of FO chaotic oscillators for FPGA implementation

In this section, we present the FPGA simulation of the FO chaotic Lorenz, Rabinovich–Fabrikant, Li, Chen, and Yu–Wang oscillators solved using the ABM and EFORK methods. First, we develop the VHDL implementation for the Artix-7 board in Vivado 2020.2. Next, we use System Generator for DSP to integrate the Xilinx library into Matlab/Simulink and carry out the simulation of the system. Figure 9 shows the emulations of the five FO chaotic oscillators; in each case, we adopt the same parameters used for the simulations in Section Simulation of FO chaotic oscillators.

Fig. 9.

Emulation of phase plane of FO chaotic oscillators, Lorenz commensurate: (a) ABM, (b) EFORK. Lorenz non–commensurate: (c) ABM, (d) EFORK. Rabinovich–Fabrikant: (e) ABM, (f) EFORK. Li: (g) ABM, (h) EFORK. Chen: (i) ABM, (j) EFORK. Yu–Wang: (k) ABM, (l) EFORK.

Figure 9a shows the emulation of the commensurate Lorenz oscillator in the phase plane – solved with ABM, whereas Fig. 9b shows the solution obtained with EFORK. Likewise, Fig. 9c shows the emulation of the non-commensurate Lorenz oscillator in the phase plane – solved with ABM, whereas Fig. 9d shows the solution obtained with EFORK. Figure 9e shows the emulation of the Rabinovich–Fabrikant oscillator in the phase plane – solved with ABM, whereas Fig. 9f shows the solution obtained with EFORK. Figure 9g shows the emulation of the Li oscillator in the phase plane – solved with ABM, whereas Fig. 9h shows the solution obtained with EFORK. Figure 9i shows the emulation of the Chen oscillator in the phase plane – solved with ABM, whereas Fig. 9k shows the solution obtained with EFORK. Finally, Fig. 9j shows the emulation of the Yu–Wang oscillator in the phase plane – solved with ABM, whereas Fig. 9l shows the solution obtained with EFORK.

In Table 5, we present a quantitative comparative analysis between the MATLAB simulations and the FPGA emulations for the state variables of the chaotic oscillators solved with the ABM and EFORK methods. We compare both implementations through the NRMSE, mean, standard deviation, minimum, and maximum values of each state variable.

Table 5.

Quantitative comparison between MATLAB simulation and FPGA emulation for the state variables of the chaotic oscillators solved with ABM and EFORK.

Chaotic oscillator	Variable	ABM
		NRMSE	Mean MATLAB	Mean FPGA	Std MATLAB	Std FPGA	Min MATLAB	Min FPGA	Max MATLAB	Max FPGA
Lorenz commensurate		0.252419	-0.226215	0.215418	12.0889	12.0894	−29.7673	−28.1132	33.9166	33.9166
		0.191794	−0.229492	0.236078	15.2762	15.3335	−48.7110	−44.6572	59.4755	59.4743
		0.180440	54.7322	54.5687	14.2570	14.6312	0.0865224	0.0865540	109.794	109.799
Lorenz non-commensurate		0.228654	0.619618	0.0110464	12.2684	12.2701	−34.6118	−34.6421	40.4875	40.4891
		0.178975	0.615984	−0.00209646	13.9272	14.0131	−49.0488	−49.1312	60.8504	60.8501
		0.170171	56.5441	56.2810	11.1118	11.8032	10.0000	10.0000	103.898	103.900
Rabinovich–Fabrikant		0.0762663	0.00239191	0.0442279	0.398561	0.379014	−3.60117	−2.52440	3.68703	2.84012
		0.128187	−0.0263857	0.00305677	0.494729	0.511742	−2.75075	−2.31324	2.79297	2.42159
		0.244687	1.04655	1.02215	0.908153	0.788624	0.0209756	0.0212210	5.06229	3.66907
Li		0.127112	0.000353342	0.00104602	1.43848	1.40068	−7.86628	−6.87914	7.95595	6.73051
		0.134973	0.000150266	−0.00121977	2.02100	1.95829	−10.4118	−9.08712	10.5535	8.88692
		0.210368	0.882624	0.789686	2.53260	2.20256	−4.61269	−3.75602	11.5277	9.64102
		0.183786	0.00342111	−0.00635026	1.49774	1.39497	−5.32787	−4.77934	5.90474	5.03252
Chen		0.217607	0.627851	0.340969	8.46408	8.50046	−23.7791	−23.1203	30.2677	30.2675
		0.189859	0.624579	0.336533	9.21841	9.22126	−31.2175	−31.2200	36.0499	36.0497
		0.165695	23.9795	23.9613	7.24603	6.96673	0.0835902	0.0836250	60.1355	60.1383
Wang		0.271572	0.00616691	0.00687852	0.820361	0.823823	−2.19865	−2.19624	2.21255	2.21197
		0.219028	0.00608428	0.00685304	0.927848	0.931434	−3.00440	−2.97552	3.15113	3.14942
		0.268489	6.62601	6.65206	3.29235	3.28267	0.100000	0.100000	17.2160	17.2002
EFORK
Lorenz commensurate		0.282712	0.470232	0.587233	12.0823	12.0654	−28.5496	−29.4091	33.8728	33.8729
		0.209273	0.475639	0.565311	15.2465	15.3091	−45.7467	−47.8554	59.4220	59.4211
		0.186569	54.7465	54.5389	14.1965	14.6247	0.0865466	0.0865980	109.603	109.600
Lorenz non-commensurate		0.264169	0.762413	−0.541506	12.2532	12.3049	−29.7060	−31.7794	36.3269	37.8982
		0.202499	0.757954	−0.556071	13.9760	14.1008	−42.3419	−47.0275	56.1622	60.3540
		0.161279	56.5486	56.8750	10.9407	10.2883	10.0000	10.0000	102.811	103.529
Rabinovich–Fabrikant		0.0792441	0.0211098	0.0115680	0.393924	0.376828	−3.54089	−3.32582	3.62144	2.66748
		0.134396	−0.00665802	−0.00240883	0.492621	0.514841	−2.72428	−2.62449	2.76586	2.35871
		0.241423	1.02986	1.03867	0.906966	0.802761	0.0206904	0.0382620	5.00386	4.43297
Li		0.127773	−0.000778634	0.0000733599	1.39481	1.38433	−8.75741	−6.61350	6.89139	6.47015
		0.127287	−0.000508546	0.000640684	1.94280	1.92513	−11.6603	−8.77597	10.2366	8.57679
		0.167561	0.726442	0.702583	2.22249	2.08766	−5.16926	−3.65628	12.8669	9.12658
		0.167894	0.00122704	0.00118166	1.41705	1.38163	−5.27036	−4.67571	6.30160	4.37764
Chen		0.218985	−0.124843	−0.165254	8.46731	8.48881	−24.0606	−22.3153	30.2149	30.2149
		0.193529	−0.123006	−0.167203	9.18799	9.19321	−30.9794	−30.9808	36.0011	36.0011
		0.162856	23.8764	23.9139	7.01680	6.82753	0.0836218	0.0836680	59.7867	59.7876
Wang		0.262206	0.0143434	−0.00109813	0.806929	0.813535	−2.18870	−2.18319	2.19396	2.19303
		0.210395	0.0147190	−0.00116113	0.910877	0.917544	−3.04093	−2.98113	3.12910	3.12788
		0.287603	6.40163	6.45877	3.33252	3.30545	0.100000	0.100000	16.5769	16.5692

Due to the inherent sensitivity to initial conditions in chaotic systems, a point-by-point comparison between MATLAB simulations and FPGA-generated trajectories is not expected to yield identical results. Small numerical discrepancies arising from differences in floating-point precision, fixed-point arithmetic, and rounding schemes lead to exponential divergence over time, even when both implementations correctly reproduce the underlying system dynamics. Consequently, the validation methodology employed in this study relies on statistical consistency and the preservation of global dynamical behavior, rather than direct trajectory matching. Metrics, including normalized root mean square error (NRMSE), mean, standard deviation, and extreme values, are utilized to demonstrate that FPGA emulations retain the essential characteristics of the corresponding numerical simulations. Finally, Table 6 reports the logic resource utilization required by each FO chaotic oscillator solved with ABM or EFORK to be emulated on the Xilinx Artix-7 AC701.

Table 6.

Logic resources utilization of Xilinx Artix-7 AC701 for each chaotic oscillator solved by ABM and EFORK.

ENTITY	LUT’s (134,000)	Registers (269,200)	BRAM (365)	DSP (740)	I/O (400)	BUFF (32)
Lorenz Oscillator (ABM)	47,780 (35.50%)	432 (0.16%)	0	94 (12.70%)	0	0
Rabinovich–Fabrikant Oscillator (ABM)	63,450 (47.14%)	483 (0.18%)	0	62 (21.89%)	0	0
Li Oscillator (ABM)	48,685 (36.17%)	694 (0.26%)	0	96 (12.97%)	0	0
Chen Oscillator (ABM	51,054 (37.93%)	483 (0.18%)	0	102 (13.78%)	0	0
Wang Oscillator (ABM)	72,498 (53.86%)	499 (0.19%)	0	232 (31.35%)	0	0
Lorenz Oscillator (EFORK)	40,430 (30.04%)	193 (0.07%)	0	88 (11.89%)	0	0
Rabinovich–Fabrikant Oscillator (EFORK)	52,861 (39.27%)	193 (0.07%)	0	156 (21.08%)	0	0
Li Oscillator (EFORK)	38,860 (28.87%)	289 (0.11%)	0	88 (11.89%)	0	0
Chen Oscillator (EFORK)	41,390 (30.75%)	225 (0.08%)	0	96 (12.97%)	0	0
Wang Oscillator (EFORK)	65,538 (48.69%)	225 (0.08%)	0	226 (30.54%)	0	0

The reduction in LUT utilization observed for the EFORK method can be primarily attributed to the absence of division operations, which are significantly more resource-intensive in FPGA implementations. Division units typically require iterative or pipelined architectures, leading to increased hardware utilization and latency. Although EFORK involves a slightly higher number of multiplications per state variable, these operations can be efficiently mapped onto dedicated DSP blocks available in modern FPGA devices. This efficient mapping reduces the demand on general logic resources, resulting in a lower overall LUT usage. From a design perspective, these results indicate that EFORK is better suited for resource-constrained implementations, where minimizing hardware cost is a priority. In contrast, the ABM method, while more demanding in terms of logic resources due to the inclusion of division operations, tends to preserve richer chaotic dynamics, as evidenced by the broader regions of positive Lyapunov exponents.

Discussion

By comparing the MATLAB simulations in Fig. 1 with the corresponding FPGA simulation in Fig. 9, together with the quantitative results summarized in Table 5, we observe that each emulated oscillator preserves its characteristic phase-space structure and reproduces the main statistical properties of the numerical trajectories. Although the FPGA implementation does not replicate the MATLAB trajectories exactly on a sample-by-sample basis, the statistical descriptors remain close in both cases. In particular, the absolute difference between the MATLAB and FPGA means represents, on average, only 0.33% of the signal range for ABM and 0.31% for EFORK. Likewise, the standard deviation differs on average by 3.28% for ABM and 2.46% for EFORK, whereas the minimum and maximum values differ by about 2.82% and 4.28% of the signal range for ABM, and 3.22% and 4.60% for EFORK. These results indicate that, despite local discrepancies, the FPGA realizations preserve the global dynamical behavior of the FO chaotic oscillators. This difference is expected because MATLAB computes the trajectories using double-precision floating-point arithmetic, whereas the FPGA implementation relies on a fixed-point architecture, which introduces quantization effects that accumulate over time. In addition, the implementations support both commensurate and non-commensurate FO operation, as illustrated by the Lorenz system: the commensurate-order results appear in Figures (a–b) and (a–b), whereas the non-commensurate-order results appear in Figures (c–d) and (c–d).

In Section Bifurcation of LEs, we compare the bifurcation diagrams with the corresponding LE spectra for each oscillator and observe that whenever the LE spectrum includes a positive component, the bifurcation diagram displays a chaotic regime; moreover, as the positive LE increases, the bifurcation outcomes become more dispersed, which indicates stronger divergence in the underlying dynamics. From the LE spectra, we also find method-dependent differences: when we solve the Lorenz oscillator with ABM, the positive LE attains larger values and remains positive over a wider range of than when we solve the system with EFORK, and we observe the same trend for the Rabinovich–Fabrikant and Yu–Wang oscillators; in contrast, for the Li oscillator, both methods yield the same positive LE magnitude and the same interval over which it remains positive. For the Chen oscillator, EFORK yields a larger positive LE than ABM, while both methods keep this exponent positive over nearly the same range. Overall, these results indicate that ABM tends to produce larger positive LEs than EFORK for most of the studied FO chaotic oscillators, implying stronger divergence and potentially increasing the complexity of chaos-based encryption keys.

Table 6 indicates that LUTs dominate the hardware cost of the FO chaotic oscillator implementations, and the Yu–Wang system reaches the highest utilization with 72,498 LUTs when we solve it with ABM and implement it on the Artix-7 AC701 board. Among the ABM-based designs, we implement the Lorenz oscillator with the lowest LUT count (47,780 LUTs), whereas among the EFORK-based designs, we implement the Li oscillator with the lowest LUT count (38,860 LUTs). Although the Li system performs fewer operations in its dynamics, it evolves in a four-state-variable space; moreover, our ABM architecture allocates eight variables per state dimension, whereas our EFORK architecture allocates seven variables per state dimension. Consequently, ABM incurs a higher logic cost per dimension, so the Li system can consume more LUTs than the Lorenz system on the FPGA despite its lower arithmetic workload. These results agree with the theoretical analysis presented in Table 4, since, although ABM requires one fewer multiplication per state dimension than EFORK, ABM also requires two divisions, whereas EFORK requires none. Division operations are considerably more demanding in terms of logical resources, which helps explain the higher LUT utilization observed for the ABM-based implementations. Finally, EFORK consistently reduces LUT utilization across all oscillators because it requires fewer computations and fewer state-related variables than ABM.

From an application perspective, these results are particularly relevant for chaos-based communication systems, where both dynamical complexity and hardware efficiency play a critical role. Systems that prioritize stronger chaotic behavior, such as those implemented with ABM, may provide enhanced security characteristics, while EFORK-based implementations offer a more suitable alternative for resource-constrained platforms requiring real-time processing. This trade-off is especially important in practical deployments such as embedded systems, edge computing devices, software-defined radios, and FPGA where hardware limitations directly impact system performance.

Conclusions

This work presents an FPGA implementation of reconfigurable modules that emulate FO oscillators solved using the ABM and EFORK methods. We use these modules to virtually emulate five FO chaotic oscillators: Lorenz, Rabinovich–Fabrikant, Li, Chen, and Yu–Wang. We first report Matlab simulations of each FO oscillator obtained with ABM and EFORK, and we analyze the bifurcation diagrams and LE spectra as the FO varies. Finally, we present the corresponding System Generator for DSP emulations and quantify the logic resources required to implement each oscillator under each numerical method.

From the FPGA simulation results, together with the quantitative comparison with the MATLAB simulations, we observe that each oscillator preserves its characteristic phase-space structure and the main statistical properties of the corresponding numerical trajectories under both commensurate and non-commensurate FO conditions. Although the FPGA implementation does not reproduce the MATLAB trajectories exactly on a sample-by-sample basis, the overall dynamical behavior remains consistent. Therefore, we conclude that the digital implementation of the FO chaotic oscillators is correct and that it provides a reliable hardware realization of their global dynamics.

On the other hand, based on the bifurcation diagrams and the LE spectra, we conclude that solving an FO chaotic oscillator with ABM generally yields a positive LE that matches or exceeds the value obtained with EFORK. This behavior indicates stronger dynamical divergence and makes ABM better suited for chaos-based encryption systems. This trend holds for all the systems considered, except for the Chen oscillator, for which EFORK yields a maximum positive LE that is 16.6% higher than the value obtained with ABM.

Likewise, by comparing the logic resources required to emulate the five chaotic systems on an FPGA device, we conclude that EFORK consistently requires fewer logic resources than ABM for the same FO chaotic oscillator. This result agrees with the theoretical complexity analysis, since, although EFORK requires one additional multiplication per state dimension, it avoids division operations and requires fewer state-related variables than ABM, which leads to a more hardware-efficient implementation.

Finally, ABM provides the most suitable option for generating complex dynamics without destabilizing the oscillator because it generally yields a larger positive LE; in contrast, when logic resource optimization is the primary goal, the appropriate choice is the EFORK-based module.

Author contributions

Conceptualization, J.-C.N.-P., R.-U.A.-P., O.-M.A., and M.-A.E.-V. ; methodology, J.-C.N.-P. R.-U.A.-P., O.-M.A.,and M.-A.E.-V.; software, R.-U.A.-P.,O.-M.A.,M.-A.E.-V., F.- J.P.-P.; validation, J.-C.N.-P., O.-M.A.,M.-A.E.-V.,F.- J.P.-P.; formal analysis, J.-C.N.-P., R.-U.A.-P.,O.-M.A.,M.-A.E.-V., F.- J.P.-P.; investigation, J.-C.N.-P., R.-U.A.-P.,O.-M.A.,and M.-A.E.-V.; resources, J.-C.N.-P.; writing—original draft preparation, J.-C.N.-P., R.-U.A.-P.,O.-M.A.,M.-A.E.-V. and F.- J.P.-P.; writing—review and editing, J.-C.N.-P., R.-U.A.-P.,O.-M.A.,M.-A.E.-V., F.- J.P.-P.; visualization, J.-C.N.-P., R.-U.A.-P.,O.-M.A.,M.-A.E.-V. and F.- J.P.-P.; supervision, J.-C.N.-P.; project administration, J.-C.N.-P.; funding acquisition, J.-C.N.-P. All authors have read and agreed to the published version of the manuscript.

Funding

Funder name: Instituto Politécnico Nacional, Secretaría de Investigación y Posgrado (Project SIP 20260451).

Data availability

Data availability All data generated or analysed during this study are included in this published article.

Declarations

Competing interests

The authors declare no competing interests.