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A significant attention has concentrated on systems with “hidden attractors” in recent years. The aim of this work is to study an oscillator displaying “hidden attractors”. The oscillator has all nonlinear terms but no equilibrium. The oscillator exhibits attractive dynamics such as chaos, bubble bifurcation, symmetrical attractors, and boosting attractors. Oscillator's feasibility is verified by a microcontroller. Utilizing the oscillator for biomedical image encryption illustrates its application. In addition, the adaptive control is designed for the oscillator.

Referring to fractional memristor-based discrete systems, this paper contributes to the field by presenting a new fourth-dimensional (4D) hyperchaotic memristor-based fractional map. The conceived system, obtained by combining a non-integer order discrete memristor with the Grassi-Miller map, is characterized by some special features, which include the absence of equilibrium point and the coexistence of various chaotic and hypechaotic attractors. Numerical techniques including phase plots, Lyapunov exponents and bifurcation diagrams are used to highlight the complex dynamic behavior of the suggested 4D fractional memristor-based Grassi-Miller map.

The volatilome profile of some biofluids (blood, urine, and human semen) identified by Solid-Phase Microextraction–Gas Chromatography/Mass Spectrometry (SPME-GC/MS) and collected from young men living in two high-pollution areas in Italy, i.e., Land of Fires and Valley of Sacco River, have been coupled to sperm parameters obtained by spermiogram analysis to build general multiple regression models. Panels of volatile organic compounds (VOCs) have been selected to optimize the models and used as predictive variables to estimate the different sperm quality parameters (sperm cell concentration, total and progressive motility/immotile cells, total/head/neck/tail morphology anomalies, semen round cell concentration). The results of the multiple linear regression models based on the different subgroups of data joining VOCs from one/two or three biofluids have been compared. Surprisingly, the models based on blood and urine VOCs have allowed an excellent estimate of spermiogram values, paving the way towards a new method of indirect evaluation of semen quality and preventive screening. The significance of VOCs in terms of toxicity and dangerousness was discussed with the support of chemical databases available online.

Memristors offer a crucial element for constructing discrete maps that have garnered significant attention in complex dynamics and various potential applications. In this study, we have integrated memristive and sigmoidal function to propose innovative mapping techniques. Our research confirms that the amalgamation of memristor and sigmoidal functions represents a promising approach for creating both 2D and 3D maps. Particularly noteworthy are the chaotic maps featuring multiple sigmoidal functions and multiple memristors, as highlighted in our findings. Specifically focusing on the novel STMM1 map, we delve into its dynamics and assess its feasibility. Intriguingly, the introduction of sigmoidal functions leads to alterations in the quantity of fixed points and the symmetry of the map.

This paper describes a new four-dimensional fractional discrete neural network with electromagnetic radiation model. In addition, the non-linear dynamics of the suggested model are examined, within the framework of commensurate, incommensurate and variable orders, through different numerical techniques such as Lyapunov exponent, phase portraits, bifurcation diagrams, and the 0–1 test method. The results imply that The behaviours of the proposed fractional discrete neural network model have rich and complex dynamical properties that are influenced by the variation of the system parameters, the commensurate order, the incommensurate order and the variable order. Moreover, the approximate entropy test and Co algorithm are carried out to measure complexity and validate the presence of chaos. Finally, nonlinear controllers are illustrated to stabilize and synchronize the proposed model.

A novel two-dimensional fractional discrete Hopfield neural network is presented in this study, which is based on discrete fractional calculus. This network incorporates both constant and variable orders, and its behavior is examined using phase plots, time evolution, bifurcation, Lyapunov exponents, and complexity analysis. Compared to integer and constant fractional orders, the numerical simulations demonstrate that the proposed variable-order fractional HNN exhibits more complex characteristics, and by selecting different fractional variable orders, novel attractors with chaotic behavior can be obtained. Additionally, a control scheme is proposed to stabilize the suggested neural network by utilizing the stability theorem for fractional discrete time systems. This control scheme is applied to both states in the study.

We explore an oscillator with nonlinear functions and equilibrium lines that displays chaos. The equilibrium stability and complexity of the oscillator have been analysed and investigated. The presence of multiple equilibrium lines sets it apart from previously reported oscillators. The synchronization of the oscillator is considered as an application for secure communications. An observer is designed by considering a transmitted signal as a state, in other words, by injecting a linear function satisfying Lipschitz’s condition to the proposed oscillator. Moreover, the adaptive control of the new oscillator is obtained.

In nonlinear dynamics, there is a continuous exploration of introducing systems with evidence of chaotic behavior. The presence of nonlinearity within system equations is crucial, as it allows for the emergence of chaotic dynamics. Given that quadratic terms represent the simplest form of nonlinearity, our study focuses on introducing a novel chaotic system characterized by only quadratic nonlinearities. We conducted an extensive analysis of this system’s dynamical properties, encompassing the examination of equilibrium stability, bifurcation phenomena, Lyapunov analysis, and the system’s basin of attraction. Our investigations revealed the presence of eight unstable equilibria, the coexistence of symmetrical strange repeller(s), and the potential for multistability in the system.

Fractional-order discrete-time neural networks represent a class of discrete systems described by non-integer order difference operators. Even though the stability of these networks is a prerequisite for their successful applications, very few papers have been published on this topic. This paper aims to make a contribution to these stability issues by presenting a network model based on the nabla Caputo h-discrete operator and by proving its Mittag–Leffler stability. Additionally, a class of variable fractional-order discrete-time neural network is introduced and a novel theorem is proved to assure its asymptotic stability. Finally, simulation results are carried out to highlight the effectiveness of the stability approach illustrated herein.

Referring tothe study of epidemic mathematical models, this manuscript presents a noveldiscrete-time COVID-19 model that includes the number of vaccinated individuals as an additional state variable in the system equations. The paper shows that the proposed compartment model, described by difference equations, has two fixed points, i.e., a disease-free fixed point and an epidemic fixed point. By considering both the forward difference system and the backward difference system, some stability analyses of the disease-free fixed point are carried out.In particular, for the backward difference system a novel theorem is proved, which gives a condition for the disappearance of the pandemic when an inequality involving some epidemic parameters is satisfied. Finally, simulation results of the conceived discrete model are carried out, along with comparisons regarding the performances of both the forward difference system and the backward difference system.

Social media occupies an important place in people’s daily lives where users share various contents and topics such as thoughts, experiences, events and feelings. The massive use of social media has led to the generation of huge volumes of data. These data constitute a treasure trove, allowing the extraction of high volumes of relevant information particularly by involving deep learning techniques. Based on this context, various research studies have been carried out with the aim of studying the detection of mental disorders, notably depression and anxiety, through the analysis of data extracted from the Twitter platform. However, although these studies were able to achieve very satisfactory results, they nevertheless relied mainly on binary classification models by treating each mental disorder separately. Indeed, it would be better if we managed to develop systems capable of dealing with several mental disorders at the same time. To address this point, we propose a well-defined methodology involving the use of deep learning to develop effective multi-class models for detecting both depression and anxiety disorders through the analysis of tweets. The idea consists in testing a large number of deep learning models ranging from simple to hybrid variants to examine their strengths and weaknesses. Moreover, we involve the grid search technique to help find suitable values for the learning rate hyper-parameter due to its importance in training models. Our work is validated through several experiments and comparisons by considering various datasets and other binary classification models. The aim is to show the effectiveness of both the assumptions used to collect the data and the use of multi-class models rather than binary class models. Overall, the results obtained are satisfactory and very competitive compared to related works.

The absence of satisfactory results that address the stability analysis of nonlinear incommensurate Fractional-order Difference Systems (FoDSs) as well as the impossibility of establishing a proper Lyapunov function that helps us to reveal any stability outcome for those systems, are considered some significant nonlinear problems that should be taken into account. To overcome such dilemmas, the present work provides some novel convenient conditions that can be utilized to ensure the asymptotic stability of those systems. In particular, through utilizing some features of the Z-transform scheme, we intend to propose certain results that aim to reveal the stability of the nonlinear incommensurate fractional-order difference system formulated in the sense of the Caputo difference operator. These novel results are completely confirmed by demonstrating the stability of the numerical solutions for two nonlinear difference systems with incommensurate fractional-orders.

The use of the advancements in memristor technology to construct chaotic maps has garnered significant research attention in recent years. The combination of memristors and nonlinear terms provides an effective approach to proposing novel maps. In this study, we have leveraged memristors and sine terms to develop three-dimensional maps, capable of processing special fixed points. Additionally, we have conducted an in depth study of a specific example (TDMM1 map) to demonstrate its dynamics, feasibility, and application for lightweight encryption. Notably, our general approach could be extended to develop higher-dimensional maps, including four- and five-dimensional ones, thereby opening up the possibility to create numerous higher-dimensional maps.

Dynamics and control of discrete chaotic systems of fractional-order have received considerable attention over the last few years. So far, nonlinear control laws have been mainly used for stabilizing at zero the chaotic dynamics of fractional maps. This article provides a further contribution to such research field by presenting simple linear control laws for stabilizing three fractional chaotic maps in regard to their dynamics. Specifically, a one-dimensional linear control law and a scalar control law are proposed for stabilizing at the origin the chaotic dynamics of the Zeraoulia-Sprott rational map and the Ikeda map, respectively. Additionally, a two-dimensional linear control law is developed to stabilize the chaotic fractional flow map. All the results have been achieved by exploiting new theorems based on the Lyapunov method as well as on the properties of the Caputo h-difference operator. The relevant simulation findings are implemented to confirm the validity of the established linear control scheme.

The Fitzhugh-Nagumo model (FN model), which is successfully employed in modeling the function of the so-called membrane potential, exhibits various formations in neuronal networks and rich complex dynamics. This work deals with the problem of control and synchronization of the FN reaction-diffusion model. The proposed control law in this study is designed to be uni-dimensional and linear law for the purpose of reducing the cost of implementation. In order to analytically prove this assertion, Lyapunov’s second method is utilized and illustrated numerically in one- and/or two-spatial dimensions.

This study aims to generalize the discrete integer-order SEIR model to obtain the novel discrete fractional-order SEIR model of COVID-19 and study its dynamic characteristics. Here, we determine the equilibrium points of the model and discuss the stability analysis of these points in detail. Then, the non-linear dynamic behaviors of the suggested discrete fractional model for commensurate and incommensurate fractional orders are investigated through several numerical techniques, including maximum Lyapunov exponents, phase attractors, bifurcation diagrams and C0 algorithm. Finally, we fitted the model with actual data to verify the accuracy of our mathematical study of the stability of the fractional discrete COVID-19 model.

Reaction–diffusion systems have a broad variety of applications, particularly in biology, and it is well known that fractional calculus has been successfully used with this type of system. However, analyzing these systems using discrete fractional calculus is novel and requires significant research in a diversity of disciplines. Thus, in this paper, we investigate the discrete-time fractional-order Lengyel–Epstein system as a model of the chlorite iodide malonic acid (CIMA) chemical reaction. With the help of the second order difference operator, we describe the fractional discrete model. Furthermore, using the linearization approach, we established acceptable requirements for the local asymptotic stability of the system’s unique equilibrium. Moreover, we employ a Lyapunov functional to show that when the iodide feeding rate is moderate, the constant equilibrium solution is globally asymptotically stable. Finally, numerical models are presented to validate the theoretical conclusions and demonstrate the impact of discretization and fractional-order on system dynamics. The continuous version of the fractional-order Lengyel–Epstein reaction–diffusion system is compared to the discrete-time system under consideration.

The aim of this paper is to propose a new secure communications approach, obtained by combining into one scheme chaotic modulation, recursive encryption and chaotic masking. The objective is achieved by designing control laws to synchronize two hyperchaotic Lorenz systems with encrypted states via a backstepping technique. The encryption process relies on a Unified chaotic system, driven by the hyperchaotic Lorenz system, as well as on a recursive encryption algorithm that exploits an n-fold composition of an invertible function. A novel theorem is proved, which assures that the transmitted message is recovered at the receiver side by virtue of the synchronization of the two Lorenz systems via backstepping design. Numerical simulations are carried out using image and text as transmitted signals, with the aim to show the effectiveness of the conceived chaos-based encryption scheme.

This paper describes a new fractional predator–prey discrete system of the Leslie type. In addition, the non-linear dynamics of the suggested model are examined within the framework of commensurate and non-commensurate orders, using different numerical techniques such as Lyapunov exponent, phase portraits, and bifurcation diagrams. These behaviours imply that the fractional predator–prey discrete system of Leslie type has rich and complex dynamical properties that are influenced by commensurate and incommensurate orders. Moreover, the sample entropy test is carried out to measure the complexity and validate the presence of chaos. Finally, nonlinear controllers are illustrated to stabilize and synchronize the proposed model.

Nowadays, a lot of research papers are concentrating on the diffusion dynamics of infectious diseases, especially the most recent one: COVID-19. The primary goal of this work is to explore the stability analysis of a new version of the SEIR model formulated with incommensurate fractional-order derivatives. In particular, several existence and uniqueness results of the solution of the proposed model are derived by means of the Picard–Lindelöf method. Several stability analysis results related to the disease-free equilibrium of the model are reported in light of computing the so-called basic reproduction number, as well as in view of utilising a certain Lyapunov function. In conclusion, various numerical simulations are performed to confirm the theoretical findings.

Owing to the COVID-19 pandemic, which broke out in December 2019 and is still disrupting human life across the world, attention has been recently focused on the study of epidemic mathematical models able to describe the spread of the disease. The number of people who have received vaccinations is a new state variable in the COVID-19 model that this paper introduces to further the discussion of the subject. The study demonstrates that the proposed compartment model, which is described by differential equations of integer order, has two fixed points, a disease-free fixed point and an endemic fixed point. The global stability of the disease-free fixed point is guaranteed by a new theorem that is proven. This implies the disappearance of the pandemic, provided that an inequality involving the vaccination rate is satisfied. Finally, simulation results are carried out, with the aim of highlighting the usefulness of the conceived COVID-19 compartment model.

Abstract The paper investigates control and synchronization of fractional-order maps described by the Caputo h-difference operator. At first, two new fractional maps are introduced, i.e., the Two-Dimensional Fractional-order Lorenz Discrete System (2D-FoLDS) and Three-Dimensional Fractional-order Wang Discrete System (3D-FoWDS). Then, some novel theorems based on the Lyapunov approach are proved, with the aim of controlling and synchronizing the map dynamics. In particular, a new hybrid scheme is proposed, which enables synchronization to be achieved between a master system based on a 2D-FoLDS and a slave system based on a 3D-FoWDS. Simulation results are reported to highlight the effectiveness of the conceived approach.

The study of the chaotic dynamics in fractional-order discrete-time systems has received great attention over the last years. Some efforts have been also devoted to analyze fractional maps with special features. This paper makes a contribution to the topic by introducing a new fractional map that is characterized by both particular dynamic behaviors and specific properties related to the system equilibria. In particular, the conceived one dimensional map is algebraically simpler than all the proposed fractional maps in the literature. Using numerical simulation, we investigate the dynamic and complexity of the fractional map. The results indicate that the new one-dimensional fractional map displays various types of coexisting attractors. The approximate entropy is used to observe the changes in the sequence sequence complexity when the fractional order and system parameter. Finally, the fractional map is applied to the problem of encrypting electrophysiological signals. For the encryption process, random numbers were generated using the values of the fractional map. Some statistical tests are given to show the performance of the encryption.

The paper concerns the problem of synchronization-control in nonlinear bacterial cultures reaction-diffusion model, linear and nonlinear controllers have been proposed to study the complete synchronization of couples of the Degn-Harrison system with identical and non-identical coefficients. Throughout the paper, we use numerical simulation to show the effectiveness of the proposed results.

In this work, we recall some definitions on fractional calculus with discrete-time. Then, we introduce a discrete-time Hopfield neural network (D.T.H.N.N) with non-commensurate fractional variable-order (V.O) for three neurons. After that, phase-plot portraits, bifurcation and Lyapunov exponents diagrams are employed to verify that the proposed discrete time Hopfield neural network with non-commensurate fractional variable order has chaotic behavior. Furthermore, we use the 0-1 test and C0 complexity algorithm to confirm and prove the results obtained about the presence of chaos. Finally, simulations are carried out in Matlab to illustrate the results.

Background: During the COVID-19 pandemic, the risk of SARS-CoV-2 infection, the public health measures of social distancing, the freedom limitations, quarantine, and the enforced homeworking under the lockdown period, as well as medical causes including COVID-19 infection per se, may have caused major emotional distress, especially in the most vulnerable patients. We aimed to evaluate the variations in the number of admissions due to Takotsubo syndrome (TTS) during the COVID-19 pandemic in the Veneto region. Methods: We retrospectively reviewed and analyzed the number of admissions because of TTS in 13 Divisions of Cardiology located in the Veneto region, the northeastern area of Italy, covering a population of more than 2.5 million inhabitants, during the two major pandemic waves of COVID-19 (the first between 15 March and 30 April 2020 and the second between 15 November and 30 December 2020) that occurred in 2020. Results: In total, 807 acute coronary syndromes were admitted in the 13 enrolling hospitals. Among these, 3.9% had TTS. Compared to the corresponding 2018 and 2019 time periods, we observed a significant increase in the number of TTS cases (+15.6%, p = 0.03 and +12.5%, p = 0.04, comparing 2018 to 2020 and 2019 to 2020, respectively). Geographical distribution of the TTS cases reflected the broad spread of the SARS-CoV-2 infection with a significant direct relationship between TTS incidence and the number of COVID-19 infections according to Pearson’s correlation (r = 0.798, p < 0.001). Conclusions: The higher incidence of TTS during the 2020 COVID-19 pandemic waves, especially in the areas that were hit hardest in terms of morbidity and mortality by the SARS-CoV-2 infection, suggest a strong direct and/or indirect role of COVID-19 in the pathogenesis of TTS.

The dynamics of the Caputo-fractional variable-order difference form of the Tinkerbell map are studied. The phase portraits, bifurcation, and largest Lyapunov exponent (LLE) were employed to demonstrate the presence of chaos over a different fractional variable-order and establish the nature of the dynamics. In addition, the 0–1 test tool was used to detect chaos. Finally, the numerical results were confirmed using the approximate entropy.

Background: The present study aimed to examine longitudinal trends in hospitalizations for acute coronary syndrome (ACS) before and during the COVID-19 pandemic, by reviewing the data from 13 hospitals of the Veneto Region, in the north-east of Italy. Methods: We performed a multicenter, retrospective analysis including all the consecutive patients presenting with ACS and other acute cardiovascular (CV) conditions (defined as heart failure, arrhythmias, cardiac arrest and venous thromboembolism) hospitalized in 13 different hospitals of the Veneto Region covering a population of 2,554,818 inhabitants, during the first (between 15 March 2020 and 30 April 2020) and second (between 15 November 2020 and 30 December 2020) COVID-19 pandemic waves (the 2020 cohort). Data were compared with those obtained at the same time-windows of years 2018 and 2019 (the historical cohorts). Results: Compared to the historical cohorts, a significant decrease in the number of ACS cases was observed in 2020 (−27.3%, p = 0.01 and −32%, p < 0.001, comparing 2018 versus 2020 and 2019 and 2020, respectively). The proportion of patients hospitalized for acute CV conditions decreased during the first and second wave COVID-19 pandemic when compared to the historical cohorts (−36.5%, p < 0.001 and −40.6%, p < 0.001, comparing 2018 versus 2020 and 2019 and 2020, respectively). Pearson’s correlation evidenced a significant inverse relationship between the number of COVID-19 cases and both ACS hospital admissions (r = −0.881, p = 0.005) and hospitalizations for acute CV conditions (r = −0.738, p = 0.01), respectively. Conclusions: The decrease in hospitalizations for ACS and other acute CV conditions will strongly affect future patients’ management since undiagnosed nonfatal CV events represent a source of increased (and unknown) CV morbidity and mortality.

Fractional-order systems have proved to be accurate in describing the spread of the COVID-19 pandemic by virtue of their capability to include the memory effects into the system dynamics. This manuscript presents a novel fractional discrete-time COVID-19 model that includes the number of vaccinated individuals as an additional state variable in the system equations. The paper shows that the proposed compartment model, described by difference equations, has two fixed points, i.e., a disease-free fixed point and an epidemic fixed point. A new theorem is proven which highlights that the pandemic disappears when an inequality involving the percentage of the population in quarantine is satisfied. Finally, numerical simulations are carried out to show that the proposed incommensurate fractional-order model is effective in describing the spread of the COVID-19 pandemic.

Background & aims: Obeticholic acid (OCA) is the second-line treatment approved for patients with primary biliary cholangitis (PBC) and an inadequate response or intolerance to ursodeoxycholic acid. We aimed to evaluate the effectiveness and safety of OCA under real-world conditions. Methods: Patients were recruited into the Italian PBC Registry, a multicentre, observational cohort study that monitors patients with PBC at national level. The primary endpoint was the biochemical response according to Poise criteria; the secondary endpoint was the biochemical response according to normal range criteria, defined as normal levels of bilirubin, alkaline phosphatase (ALP), and alanine aminotransferase (ALT) at 12 months. Safety and tolerability were also assessed. Results: We analysed 191 patients until at least 12 months of follow-up. Median age was 57 years, 94% female, 61 (32%) had cirrhosis, 28 (15%) had histologically proven overlap with autoimmune hepatitis (PBC-AIH). At 12 months, significant median reductions of ALP (-32.3%), ALT (-31.4%), and bilirubin (-11.2%) were observed. Response rates were 42.9% according to Poise criteria, and 11% by normal range criteria. Patients with cirrhosis had lower response than patients without cirrhosis (29.5% vs. 49.2%, p = 0.01), owing to a higher rate of OCA discontinuation (30% vs. 12%, p = 0.004), although with similar ALP reduction (29.4% vs. 34%, p = 0.53). Overlap PBC-AIH had a similar response to pure PBC (46.4% vs. 42.3%, p = 0.68), with higher ALT reduction at 6 months (-38% vs. -29%, p = 0.04). Thirty-three patients (17%) prematurely discontinued OCA because of adverse events, of whom 11 experienced serious adverse events. Treatment-induced pruritus was the leading cause of OCA discontinuation (67%). Conclusions: Effectiveness and safety of OCA under real-world conditions mirror those in the Poise trial. Patients with cirrhosis had lower tolerability. Overlap PBC-AIH showed higher ALT reduction at 6 months compared with patients with pure PBC. Lay summary: Obeticholic acid (OCA) was shown to be effective in more than one-third of patients not responding to ursodeoxycholic acid in a real-world context in Italy. Patients with cirrhosis had more side effects with OCA, and this led to suspension of the drug in one-third of patients. OCA was also effective in patients who had overlap between autoimmune hepatitis and primary biliary cholangitis.

Abstract In this paper, we propose three fractional chaotic maps based on the well known 3D Stefanski, Rössler, and Wang maps. The dynamics of the proposed fractional maps are investigated experimentally by means of phase portraits, bifurcation diagrams, and Lyapunov exponents. In addition, three control laws are introduced for these fractional maps and the convergence of the controlled states towards zero is guaranteed by means of the stability theory of linear fractional discrete systems. Furthermore, a combined synchronization scheme is introduced whereby the fractional Rössler map is considered as a drive system with the response system being a combination of the remaining two maps. Numerical results are presented throughout the paper to illustrate the findings.

Abstract In this paper, the problem of synchronization of a class of spatiotemporal fractional-order partial differential systems is studied. Subject to homogeneous Neumann boundary conditions and using fractional Lyapunov approach, nonlinear and linear control schemes have been proposed to synchronize coupled general fractional reaction–diffusion systems. As a numerical application, we investigate complete synchronization behaviors of coupled fractional Lengyel–Epstein systems.

Abstract In this paper, we propose a fractional form of a new three-dimensional generalized Hénon map and study the existence of chaos and its control. Using bifurcation diagrams, phase portraits and Lyapunov exponents, we show that the general behavior of the proposed fractional map depends on the fractional order. We also present two control schemes for the proposed map, one that adaptively stabilizes the fractional map, and another to achieve the synchronization of the proposed fractional map.

During the broadcast of Coronavirus across the globe, many mathematicians made several mathematical models. This was, of course, in order to understand the forecast and behavior of this epidemic’s spread precisely. Nevertheless, due to the lack of much information about it, the application of many models has become difficult in reality and sometimes impossible, unlike the simple SIR model. In this work, a simple, novel fractional-order discrete model is proposed in order to study the behavior of the COVID-19 epidemic. Such a model has shown its ability to adapt to the periodic change in the number of infections. The existence and uniqueness of the solution for the proposed model are examined with the help of the Picard Lindelöf method. Some theoretical results are established in view of the connection between the stability of the fixed points of this model and the basic reproduction number. Several numerical simulations are performed to verify the gained results.

Referring to chaotic systems, it is well-known that they are nonlinear dynamical systems, which are distinguished by sensitive dependence on initial conditions and by having evolution through phase space that appears to be quite random [...]

This study intends to examine different dynamics of the chaotic incommensurate fractional-order Hopfield neural network model. The stability of the proposed incommensurate-order model is analyzed numerically by continuously varying the values of the fractional-order derivative and the values of the system parameters. It turned out that the formulated system using the Caputo differential operator exhibits many rich complex dynamics, including symmetry, bistability, and coexisting chaotic attractors. On the other hand, it has been detected that by adapting the corresponding controlled constants, such systems possess the so-called offset boosting of three variables. Besides, the resultant periodic and chaotic attractors can be scattered in several forms, including 1D line, 2D lattice, and 3D grid, and even in an arbitrary location of the phase space. Several numerical simulations are implemented, and the obtained findings are illustrated through constructing bifurcation diagrams, computing Lyapunov exponents, calculating Lyapunov dimensions, and sketching the phase portraits in 2D and 3D projections.

In this paper, Mixed Reality (MR) has been exploited in the operating rooms to perform laparoscopic and open surgery with the aim of providing remote mentoring to the medical doctors under training during the Covid-19 pandemic. The employed architecture, which has put together MR smartglasses, a Digital Imaging Player, and a Mixed Reality Toolkit, has been used for cancer surgery at the IRCCS Hospital ‘Giovanni Paolo II’ in southern Italy. The feasibility of using the conceived platform for real-time remote mentoring has been assessed on the basis of surveys distributed to the trainees after each surgery.

This work aims to present a study on the stability analysis of linear and nonlinear incommensurate h-nabla fractional-order difference systems. Several theoretical results are inferred with the help of using some theoretical schemes, such as the Z-transform method, Cauchy–Hadamard theorem, Taylor development approach, final-value theorem and Banach fixed point theorem. These results are verified numerically via two illustrative numerical examples that show the stabilities of the solutions of systems at hand.

Few papers have been published to date regarding the stability of neural networks described by fractional difference operators. This paper makes a contribution to the topic by presenting a variable-order fractional discrete neural network model and by proving its Ulam–Hyers stability. In particular, two novel theorems are illustrated, one regarding the existence of the solution for the proposed variable-order network and the other regarding its Ulam–Hyers stability. Finally, numerical simulations of three-dimensional and two-dimensional variable-order fractional neural networks were carried out to highlight the effectiveness of the conceived theoretical approach.

Chaotic systems with no equilibrium are a very important topic in nonlinear dynamics. In this paper, a new fractional order discrete-time system with no equilibrium is proposed, and the complex dynamical behaviors of such a system are discussed numerically by means of a bifurcation diagram, the largest Lyapunov exponents, a phase portrait, and a 0–1 test. In addition, a one-dimensional controller is proposed. The asymptotic convergence of the proposed controller is established by means of the stability theory of linear fractional order discrete-time systems. Next, a synchronization control scheme for two different fractional order discrete-time systems with hidden attractors is reported, where the master system is a two-dimensional system that has been reported in the literature. Numerical results are presented to confirm the results.

Investigating dynamic properties of discrete chaotic systems with fractional order has been receiving much attention recently. This paper provides a contribution to the topic by presenting a novel version of the fractional Grassi–Miller map, along with improved schemes for controlling and synchronizing its dynamics. By exploiting the Caputo h-difference operator, at first, the chaotic dynamics of the map are analyzed via bifurcation diagrams and phase plots. Then, a novel theorem is proved in order to stabilize the dynamics of the map at the origin by linear control laws. Additionally, two chaotic fractional Grassi–Miller maps are synchronized via linear controllers by utilizing a novel theorem based on a suitable Lyapunov function. Finally, simulation results are reported to show the effectiveness of the approach developed herein.

Abstract This paper presents new results related to the coexistence of function-based hybrid synchronization types between non-identical incommensurate fractional-order systems characterized by different dimensions and orders. Specifically, a new theorem is illustrated, which ensures the coexistence of the full-state hybrid function projective synchronization (FSHFPS) and the inverse full-state hybrid function projective synchronization (IFSHFPS) between wide classes of three-dimensional master systems and four-dimensional slave systems. In order to show the capability of the approach, a numerical example is reported, which illustrates the coexistence of FSHFPS and IFSHFPS between the incommensurate chaotic fractional-order unified system and the incommensurate hyperchaotic fractional-order Lorenz system.

Abstract In this paper, we introduce two approaches to the generalized synchronized synchronization and the inverse generalized synchronization of fractional discrete-time chaotic systems with non-identical dimensions. The convergence of the proposed approaches is established by means of recently developed stability theory. Numerical results are presented based on well-known maps in the literature. Two examples are considered: a 3D generalized synchronization and a 2D inverse generalized synchronization.

Abstract The topic related to the coexistence of different synchronization types between fractional-order chaotic systems is almost unexplored in the literature. Referring to commensurate and incommensurate fractional systems, this paper presents a new approach to rigorously study the coexistence of some synchronization types between nonidentical systems characterized by different dimensions and different orders. In particular, the paper shows that identical synchronization (IS), antiphase synchronization (AS), and inverse full state hybrid projective synchronization (IFSHPS) coexist when synchronizing a three-dimensional master system with a fourth-dimensional slave system. The approach, which can be applied to a wide class of chaotic/hyperchaotic fractional-order systems in the master-slave configuration, is based on two new theorems involving the fractional Lyapunov method and stability theory of linear fractional systems. Two examples are provided to highlight the capability of the conceived method. In particular, referring to commensurate systems, the coexistence of IS, AS, and IFSHPS is successfully achieved between the chaotic three-dimensional Rössler system of order 2.7 and the hyperchaotic four-dimensional Chen system of order 3.84. Finally, referring to incommensurate systems, the coexistence of IS, AS, and IFSHPS is successfully achieved between the chaotic three-dimensional Lü system of order 2.955 and the hyperchaotic four-dimensional Lorenz system of order 3.86.

Oscillations and oscillators appear in various fields and find applications in numerous areas. We present an oscillator with infinite equilibria in this work. The oscillator includes only nonlinear elements (quadratic, absolute, and cubic ones). It is different from common oscillators, in which there are linear elements. Special features of the oscillator are suitable for secure applications. The oscillator’s dynamics have been discovered via simulations and an electronic circuit. Chaotic attractors, bifurcation diagrams, Lyapunov exponents, and the boosting feature are presented while measurements of the implemented oscillator are reported by using an oscilloscope. We introduce a random number generator using such an oscillator, which is applied in biomedical image encryption. Moreover, the security and performance analysis are considered to confirm the correctness of encryption and decryption processes.

Researchers have recently paid significant attention to special chaotic systems. In this work, we introduce an oscillator with different special features. In addition, the oscillator is symmetrical. The features and oscillator dynamics are discovered through different tools of nonlinear dynamics. An electronic circuit is designed to mimic the oscillator’s dynamics. Moreover, the combined synchronization of two drives and one response oscillator is reported. Numerical examples illustrate the correction of our approach.

Most of the papers published so far in literature have focused on the theoretical phenomena underlying the formation of chaos, rather than on the investigation of potential applications of chaos to the real world. This paper aims to bridge the gap between chaos theory and chaos applications by presenting a survey of very recent applications of chaos. In particular, the manuscript covers the last three years by describing different applications of chaos as reported in the literature published during the years 2018 to 2020, including the matter related to the symmetry properties of chaotic systems. The topics covered herein include applications of chaos to communications, to distributed sensing, to robotic motion, to bio-impedance modelling, to hardware implementation of encryption systems, to computing and to random number generation.

This paper proceeds from the perspective that most strongly nonlinear oscillators of fractional-order do not enjoy exact analytical solutions. Undoubtedly, this is a good enough reason to employ one of the major recent approximate methods, namely an Optimal Homotopy Asymptotic Method (OHAM), to offer approximate analytic solutions for two strongly fractional-order nonlinear benchmark oscillatory problems, namely: the fractional-order Duffing-relativistic oscillator and the fractional-order stretched elastic wire oscillator (with a mass attached to its midpoint). In this work, a further modification has been proposed for such method and then carried out through establishing an optimal auxiliary linear operator, an auxiliary function, and an auxiliary control parameter. In view of the two aforesaid applications, it has been demonstrated that the OHAM is a reliable approach for controlling the convergence of approximate solutions and, hence, it is an effective tool for dealing with such problems. This assertion is completely confirmed by performing several graphical comparisons between the OHAM and the Homotopy Analysis Method (HAM).

Synchronization and control in high dimensional spatial-temporal systems have received increasing interest in recent years. In this paper, the problem of complete synchronization for reaction-diffusion systems is investigated. Linear and nonlinear synchronization control schemes have been proposed to exhibit synchronization between coupled reaction-diffusion systems. Synchronization behaviors of coupled Lengyel-Epstein systems are obtained to demonstrate the effectiveness and feasibility of the proposed control techniques.

This article investigates a non-equilibrium chaotic system in view of commensurate and incommensurate fractional orders and with only one signum function. By varying some values of the fractional-order derivative together with some parameter values of the proposed system, different dynamical behaviors of the system are explored and discussed via several numerical simulations. This system displays complex hidden dynamics such as inversion property, chaotic bursting oscillation, multistabilty, and coexisting attractors. Besides, by means of adapting certain controlled constants, it is shown that this system possesses a three-variable offset boosting system. In conformity with the performed simulations, it also turns out that the resultant hidden attractors can be distributively ordered in a grid of three dimensions, a lattice of two dimensions, a line of one dimension, and even arbitrariness in the phase space. Through considering the Caputo fractional-order operator in all performed simulations, phase portraits in two- and three-dimensional projections, Lyapunov exponents, and the bifurcation diagrams are numerically reported in this work as beneficial exit results.