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Spacecraft models that mimic the Planck satellite’s behaviour have produced information on cosmic microwave background radiation, assisting physicists in their understanding of the composition and expansion of the universe. For achieving the intended formation, a framework for a discrete fractional difference spacecraft model is constructed by the use of a discrete nabla operator of variable order containing the Mittag–Leffler kernel. The efficacy of the suggested framework is evaluated employing a numerical simulation of the concerning dynamic systems of motion while taking into account multiple considerations such as exterior disruptions, parameterized variations, time-varying feedback delays, and actuator defects. The implementation of the Banach fixed-point approach provides sufficient requirements for the presence of the solution as well as a distinctive feature for such mechanisms Furthermore, the consistent stability is examined. With the aid of discrete nabla operators, we monitor the qualitative behavioural patterns of spacecraft systems to provide justification for structure’s chaos. We acquire the fixed points of the proposed trajectory. At each fixed point, we calculate the eigenvalue of the spacecraft system’s Jacobian matrix and check for zones of instability. The outcomes exhibit a wide range of multifaceted behaviours resulting from the interaction with various fractional orders in the offered system. To maintain stability and synchronize the system, nonlinear controllers are additionally provided. The study highlights the technique’s vulnerability to fractional-order factors, resulting in exclusive, changing trends and equilibrium frameworks. Because of its diverse and convoluted behaviour, the spacecraft chaotic model is an intriguing and crucial subject for research.

We consider a coupling of the Stommel box model and the Lorenz model, with the goal of investigating the so-called crises that are known to occur given sufficient forcing. In this context, a crisis is characterized as the destruction of a chaotic attractor under a critical forcing strength. We document the variety of chaotic attractors and crises possible in our model, focusing on the parameter region where the Lorenz model is always chaotic and where bistability exists in the Stommel box model. The chaotic saddle collisions that occur in a boundary crisis are visualized, with the chaotic saddle computed using the Saddle-Straddle Algorithm. We identify a novel sub-type of boundary crisis, namely a vanishing basin crisis. For forcing strength beyond the crisis, we demonstrate the possibility of a merging between the persisting chaotic attractor and either a chaotic transient or a ghost attractor depending on the type of boundary crisis. An investigation of the finite-time Lyapunov exponents around crisis levels of forcing reveals a convergence between two near-neutral exponents, particularly at points of a trajectory most sensitive to divergence. This points to loss of hyperbolicity associated with crisis occurrence. Finally, we generalize our findings by coupling the Stommel box model to other strange attractors and thereby show that the behaviors are quite generic and robust. [ABSTRACT FROM AUTHOR]

Abstract Accurate development of satellite maneuvers necessitates a broad orbital dynamical system and efficient nonlinear control techniques. For achieving the intended formation, a framework of a discrete fractional difference satellite model is constructed by the use of commensurate and non-commensurate orders for the control and synchronization of fractional-order chaotic satellite system. The efficacy of the suggested framework is evaluated employing a numerical simulation of the concerning dynamic systems of motion while taking into account multiple considerations such as Lyapunov exponent research, phase images and bifurcation schematics. With the aid of discrete nabla operators, we monitor the qualitative behavioural patterns of satellite systems in order to provide justification for the structure’s chaos. We acquire the fixed points of the proposed trajectory. At each fixed point, we calculate the eigenvalue of the satellite system’s Jacobian matrix and check for zones of instability. The outcomes exhibit a wide range of multifaceted behaviours resulting from the interaction with various fractional-orders in the offered system. Additionally, the sample entropy evaluation is employed in the research to determine complexities and endorse the existence of chaos. To maintain stability and synchronize the system, nonlinear controllers are additionally provided. The study highlights the technique’s vulnerability to fractional-order factors, resulting in exclusive, changing trends and equilibrium frameworks. Because of its diverse and convoluted behaviour, the satellite chaotic model is an intriguing and crucial subject for research.

Introduction: For the Navier-Stokes equation, one of the most essential tasks should be to study its completeness of the complex nonlinear systems. Also, its nature and physical practical applications would be depth explored. Moreover, as one of the routes to chaos, this equation with an external force has been investigated numerically in 1989. Recently, some information is worth noting that when the high symmetry was imposed on the velocity field, the complex nonlinear motions should occur even lead to the chaos phenomenon. However, most of the published papers are based on theoretical studies and rarely deal with the above results, which lost of the match between them and the integrity of the scientific system. Methods: This study analyzed the molecular distillation process in detail based on the basic theory of nonlinear chaotic systems. Then, the mathematical model for the process of molecular distillation with one brushless DC motor (BLDCM) is built and named the Molecular-Distillation-Navier-Stokes (MDNS) equation. Also, its complex and potentially chaotic behaviors and chaotic processes are first discovered and demonstrated, such as chaotic attractors, chaotic co-attractors, phase portraits, time-domain waveforms, Lyapunov exponent spectrums, Poincare maps, the bifurcation diagrams, and so on. Results: The good agreement among theoretical analysis, simulation and experimental results verifies the practicability and flexibility of the configured model. Discussion: The related conclusions have supplemented and improved the theoretical system for the Navier Stokes equations. Also, it reflects the significance in molecular distillation processes. Meanwhile, the novel research direction for the fields of the chaotic nonlinear and complex industrial systems have been explored and discovered. [ABSTRACT FROM AUTHOR]

The purpose of this work is a study of the dynamics of the systems of ordinary differential equations of the second order, which is obtained using the Lagrange formalism. These systems describe the parametric interaction of oscillators (modes) in the presence of a general quadratic or cubic nonlinearity. Also, we compare the dynamics of the systems of ordinary differential equations of the second order and dynamics of the Vyshkind–Rabinovich and Rabinovich–Fabrikant models in order to determine the possibilities of the latter models when modeling coupled oscillators of the above type. Methods. The study is based on the numerical solution using the methods of the theory of the obtained analytically differential equations. Results. For both systems of second-order differential equations, is was presented a chart of in the parameter plane, a graphs of Lyapunov exponents at the value of the parameter that specifies the dissipation of oscillators, a time dependences of the generalized coordinates of oscillators and its amplitudes, portraits of attractors, a projection of the attractors on a phase planes of oscillators. A comparison with the dynamics of the Vyshkind–Rabinovich and Rabinovich–Fabrikant models is carried out. These models are three-dimensional real approximations of the above systems obtained by the method of slowly varying amplitudes. Conclusion. The study of the constructed systems showed that in the parameter space there are regions corresponding to both various regular regimes, such as the equilibrium position, limit cycle, two-frequency tori, and chaotic regimes. For both systems, it was shown that the transition to chaos occurs as a result of a sequence of period doubling bifurcations of the tori. In addition, a comparison of the dynamics of the constructed systems with the dynamics of the Vyshkind–Rabinovich and Rabinovich–Fabrikant models allows us to assert that if the Vyshkind–Rabinovich model predicts the dynamics of the corresponding initial system well enough, then the Rabinovich–Fabrikant model does not have such a property.

One of the fastest-growing industries in recent years has been e-Healthcare. Many cyberattacks and threats against patient confidentiality exist in electronic health records (EHRs). To shield EHRs from data breaches and to secure the data with integrity, DNA subsequences, SHA-256, and Hyper Chaotic Multi Attractors Chen System (HCMACS) are proposed for effective medical image encryption. A combined HCMACS produces a pseudorandom key sequence to strengthen its resiliency. The two significant advantages of the proposed technique are integrity and robustness, where the secret keys are susceptible to initial states determined by the original image's hash value. Furthermore, the encrypted image is uploaded to a cloud-based service where an authorised user can retrieve the original data. Finally the digital medical images have confidentiality, integrity and availability (CIA). The outcomes of the DNA-based cryptosystem for medical images are validated with several analyses and are efficiently defended against statistical, differential, and chosen-plaintext attacks. [ABSTRACT FROM AUTHOR]

Introduction: For the Navier-Stokes equation, one of the most essential tasks should be to study its completeness of the complex nonlinear systems. Also, its nature and physical practical applications would be depth explored. Moreover, as one of the routes to chaos, this equation with an external force has been investigated numerically in 1989. Recently, some information is worth noting that when the high symmetry was imposed on the velocity field, the complex nonlinear motions should occur even lead to the chaos phenomenon. However, most of the published papers are based on theoretical studies and rarely deal with the above results, which lost of the match between them and the integrity of the scientific system.Methods: This study analyzed the molecular distillation process in detail based on the basic theory of nonlinear chaotic systems. Then, the mathematical model for the process of molecular distillation with one brushless DC motor (BLDCM) is built and named the Molecular-Distillation-Navier-Stokes (MDNS) equation. Also, its complex and potentially chaotic behaviors and chaotic processes are first discovered and demonstrated, such as chaotic attractors, chaotic co-attractors, phase portraits, time-domain waveforms, Lyapunov exponent spectrums, Poincare maps, the bifurcation diagrams, and so on.Results: The good agreement among theoretical analysis, simulation and experimental results verifies the practicability and flexibility of the configured model.Discussion: The related conclusions have supplemented and improved the theoretical system for the Navier Stokes equations. Also, it reflects the significance in molecular distillation processes. Meanwhile, the novel research direction for the fields of the chaotic nonlinear and complex industrial systems have been explored and discovered.

Since the seminal work of Henri Poincaré on the three-body problem, and more recent research dating back to the second half of the 20th century on chaotic dynamical systems, many applications have emerged in different domains (economics, electronic, cryptography, physics, etc). We try to describe the evolution of the last 50 years on the subject and to find out whether applications have compromised the purity and beauty of theoretical research. [ABSTRACT FROM AUTHOR]

In this paper, we suggest and assess a stochastic model for pneumonia-typhoid co-infection to investigate their distinctive correlation under the impact of preventative techniques considering environmental noise and piecewise fractional derivative operators. Initially, we conducted a descriptive investigation of the model, and the basic reproductive number is defined in terms of the existence and stability of dynamic equilibrium. Then, we obtain the sufficient requirements for the existence of an ergodic stationary distribution by utilizing a novel methodology for constructing stochastic Lyapunov candidates. Besides that, the basic stochastic reproductive R0s as a threshold that will examine the extinction and persistence of the disease. Through a rigorous analysis, this study presents the concept of piecewise derivative with the goal of modelling the co-dynamics of pneumonia and typhoid fever with varying kernels. We viewed various possibilities and described numerical strategies for addressing difficulties. Visual observations, such as chaotic and dynamical behaviour patterns, are provided to demonstrate the efficacy of the proposed notion. Thus, the innovative considerations of fractional calculus include more versatile configurations, allowing us to more effectively acclimate to the dynamic system behaviours of real-world manifestations. Finally, we discovered that treating pneumonia with typhoid fever preventative measures is the least expensive. As a result, for advantageous and cost-effective regulation of both pathogens, legislators must prioritize preventative measures while not overlooking treatment of affected patients.

Real mechanisms that are advancing in a complex network exhibit chaotic behaviour. This behaviour is crucial in physical and complex systems involving numerical modelling frameworks because it essentially determines the framework’s evolutionary process. In this context, notwithstanding its difficulty, the potential of intentional oversight of the phenomenon has feasible effects; this is why theoretical approaches are advantageous in such scenarios. This study investigates the functioning of a Willamowski–Rössler (W–R) mechanism, including the synchronization of two minimal W–R structures depending on the responsive suggestion technique for regulation, with the purpose of achieving chaos influence in chemical interactions. We investigate the reliability of the steady state at various fractional order (FO) factors. Employing maximum Lyapunov exponents (MLEs), phase depictions, bifurcation schematics, the 0–1 evaluation and approximated entropy, it is demonstrated that adjusting the FOs causes a system’s behavioural pattern to undergo a transition from steady to chaotic. In addition to demonstrating that the proposed scheme fits chaotically under certain circumstances, simulation outcomes demonstrate that mathematical modelling is used to illustrate theoretical debates. To verify that the community detects chaos, the MLE and bifurcation illustrations, whose hallmark factors are plotted, display erratic behaviour while effectively attempting to control the chaos.

Recently, we reported that a chaotic attractor in a discontinuous one-dimensional map may undergo a so-called exterior border collision bifurcation, which causes additional bands of the attractor to appear. In the present paper, we suppose that the chaotic attractor's basin boundary contains a chaotic repeller, and discuss a bifurcation pattern consisting of an exterior border collision bifurcation and an expansion bifurcation (interior crisis). In the generic case, where neither the border point the chaotic attractor collides with, nor any of its images belong to the chaotic repeller, the exterior border collision bifurcation is followed by the expansion bifurcation, and the distance between both bifurcations may be arbitrarily small but positive. In the non-generic (codimension-2) case, these bifurcations occur simultaneously, so that a border collision bifurcation of a chaotic attractor leads directly to its expansion. [ABSTRACT FROM AUTHOR]

Although Lyapunov exponents have been widely used to characterize the dynamics of nonlinear systems, few methods are available so far to obtain a priori bounds on their magnitudes. Recently, sufficient conditions to rule out the existence of attractors with positive Lyapunov exponents have been derived via a Lyapunov approach based on the second additive compound matrices of the system Jacobian. This paper first provides some insights into this approach by showing how the several available techniques for computing Lyapunov functions can be fruitfully applied to Lorenz and Thomas systems to derive explicit conditions on their system parameters, which ensure that there are no attractors with positive Lyapunov exponents. Then, the approach is extended to the case of nonlinear systems with a first integral of motion and its application to the memristor Chua's circuit is discussed. [ABSTRACT FROM AUTHOR]

Oncolytic virotherapy is a viable chemotherapeutic agent that identifies and kills tumor cells using replication-competent pathogens. Oncolytic alphavirus M1 is a naturally existing disease that has been shown to have rising specificity and potency in cancer progression. The objective of this research is to introduce and analyze an oncolytic M1 virotherapy framework with spatial variability and anti-tumor immune function via piecewise fractional differential operator techniques. To begin, we potentially demonstrate that the stochastic system's solution is non-negative and global by formulating innovative stochastic Lyapunov candidates. Then, we derive the existence-uniqueness of an ergodic stationary distribution of the stochastic framework and we establish a sufficient assumption $ \mathbb{R}_{0}^{p} < 1 $ extermination of tumor cells and oncolytic M1 virus. Using meticulous interpretation, this model allows us to analyze and anticipate the procedure from the start to the end of the tumor because it allows us to examine a variety of behaviours ranging from crossover to random mechanisms. Furthermore, the piecewise differential operators, which can be assembled with operators including classical, Caputo, Caputo-Fabrizio, Atangana-Baleanu, and stochastic derivative, have decided to open up innovative avenues for readers in various domains, allowing them to encapsulate distinct characteristics in multiple time intervals. Consequently, by applying these operators to serious challenges, scientists can accomplish better outcomes in documenting facts.

In this paper, we suggest and assess a stochastic model for pneumonia-typhoid co-infection to investigate their distinctive correlation under the impact of preventative techniques considering environmental noise and piecewise fractional derivative operators. Initially, we conducted a descriptive investigation of the model, and the basic reproductive number is defined in terms of the existence and stability of dynamic equilibrium. Then, we obtain the sufficient requirements for the existence of an ergodic stationary distribution by utilizing a novel methodology for constructing stochastic Lyapunov candidates. Besides that, the basic stochastic reproductive R 0 s as a threshold that will examine the extinction and persistence of the disease. Through a rigorous analysis, this study presents the concept of piecewise derivative with the goal of modelling the co-dynamics of pneumonia and typhoid fever with varying kernels. We viewed various possibilities and described numerical strategies for addressing difficulties. Visual observations, such as chaotic and dynamical behaviour patterns, are provided to demonstrate the efficacy of the proposed notion. Thus, the innovative considerations of fractional calculus include more versatile configurations, allowing us to more effectively acclimate to the dynamic system behaviours of real-world manifestations. Finally, we discovered that treating pneumonia with typhoid fever preventative measures is the least expensive. As a result, for advantageous and cost-effective regulation of both pathogens, legislators must prioritize preventative measures while not overlooking treatment of affected patients. [ABSTRACT FROM AUTHOR]

Purpose of this work is a numerical study of the Rabinovich–Fabrikant system and its generalized model, which describe the occurrence of chaos during the parametric interaction of three modes in a nonequilibrium medium with cubic nonlinearity, in the case when the parameters that have the meaning of dissipation coefficients take negative values. These models demonstrate a rich dynamics that differs in many respects from what was observed for them, but in the case of positive values of the parameters. Methods. The study is based on the numerical solution of the differential equations, and their numerical bifurcation analysis using the MаtCont program. Results. For investigated models we present a charts of dynamic regimes in the control parameters plane, Lyapunov exponents depending on the parameters, attractors and their basins. On the parameters plane, which have the meaning of dissipation coefficients, bifurcation lines and points are numerically found. They are plotted for equilibrium point and period one limit cycle. For both models we compared dynamics observed in the case when the parameters that have the meaning of dissipation coefficients take negative values, with the one observed in the case when these parameters take positive values. And it is shown that in the first case parameter space has a simpler structure. Conclusion. The Rabinovich– Fabrikant system and its generalized model were studied in detail in the case when the parameters which have the meaning of dissipation coefficients take negative values. It is shown that there are a number of differences in comparison with the case of positive values of these parameters. For example, a new type of chaotic attractor appears, multistability that is not related to the symmetry of the system disappears, etc. The obtained results are new, since the Rabinovich–Fabrikant system and its generalized model were studied in detail for the first time in the region of negative values of parameters which have the meaning of dissipation coefficients.

Oncolytic virotherapy is a viable chemotherapeutic agent that identifies and kills tumor cells using replication-competent pathogens. Oncolytic alphavirus M1 is a naturally existing disease that has been shown to have rising specificity and potency in cancer progression. The objective of this research is to introduce and analyze an oncolytic M1 virotherapy framework with spatial variability and anti-tumor immune function via piecewise fractional differential operator techniques. To begin, we potentially demonstrate that the stochastic system's solution is non-negative and global by formulating innovative stochastic Lyapunov candidates. Then, we derive the existence-uniqueness of an ergodic stationary distribution of the stochastic framework and we establish a sufficient assumption ... extermination of tumor cells and oncolytic M1 virus. Using meticulous interpretation, this model allows us to analyze and anticipate the procedure from the start to the end of the tumor because it allows us to examine a variety of behaviours ranging from crossover to random mechanisms. Furthermore, the piecewise differential operators, which can be assembled with operators including classical, Caputo, Caputo-Fabrizio, Atangana-Baleanu, and stochastic derivative, have decided to open up innovative avenues for readers in various domains, allowing them to encapsulate distinct characteristics in multiple time intervals. Consequently, by applying these operators to serious challenges, scientists can accomplish better outcomes in documenting facts. [ABSTRACT FROM AUTHOR]

We analyze the classical and quantum dynamics of the driven dissipative Bose–Hubbard dimer. Under variation of the driving frequency, the classical system is shown to exhibit a bifurcation to the limit cycle, where its steady-state solution corresponds to periodic oscillation with the frequency unrelated to the driving frequency. This bifurcation is shown to lead to a peculiarity in the stationary single-particle density matrix of the quantum system. The case of the Bose–Hubbard trimer, where the discussed limit cycle bifurcates into a chaotic attractor, is briefly discussed. [ABSTRACT FROM AUTHOR]

Compared to voltage mode circuits, current mode circuits have advantages such as large dynamic range, fast speed, wide frequency band, and good linearity. In recent years, the development of call flow modeling technology has been rapid and has become an important foundation for analog integrated circuits. In this paper, a current mode chaotic oscillation circuit based on current differential transconductance amplifier (CDTA) is proposed. This proposed circuit fully utilizes the advantages of current differential transconductance amplifier: a current input and output device with a large dynamic range, virtual ground at the input, extremely low input impedance, and high output impedance. The linear and non-linear parts of the proposed circuit operate in current mode, enabling a true current mode multi scroll chaotic circuit. Pspice simulation results show that the current mode chaotic circuit proposed can generate multi scroll chaotic attractors.

The purpose of this work is to numerically study of the generalized Rabinovich–Fabrikant model. This model is obtained using the Lagrange formalism and describing the three-mode interaction in the presence of a general cubic nonlinearity. The model demonstrates very rich dynamics due to the presence of third-order nonlinearity in the equations. Methods. The study is based on the numerical solution of the obtained analytically differential equations, and their numerical bifurcation analysis using the MаtCont program. Results. For the generalized model we present a charts of dynamic regimes in the control parameter plane, Lyapunov exponents depending on parameters, portraits of attractors and their basins. On the plane of control parameters, bifurcation lines and points are numerically found. They are plotted for equilibrium point and period one limit cycle. It is shown that the dynamics of the generalized model depends on the signature of the characteristic expressions presented in the equations. A comparison with the dynamics of the Rabinovich– Fabrikant model is carried out. We indicated a region in the parameter plane in which there is a complete or partial coincidence of dynamics. Conclusion. The generalized model is new and describes the interaction of three modes, in the case when the cubic nonlinearity that determines their interaction is given in a general form. In addition, since the considered model is a certain natural extension of the well-known Rabinovich–Fabrikant model, then it is universal. And it can simulate systems of various physical nature (including radio engineering), in which there is a three-mode interaction and there is a general cubic nonlinearity

Early detection of diseases is vital for patient recovery. This article explains the design and technical matters of a computer-supported diagnostic system for eye melanoma detection implementing a security approach using chaotic-based encryption to guarantee communication security. The system is intended to provide a diagnosis; it can be applied in a cooperative environment for hospitals or telemedicine and can be extended to detect other types of eye diseases. The introduced method has been tested to assess the secret key, sensitivity, histogram, correlation, Number of Pixel Change Rate (NPCR), Unified Averaged Changed Intensity (UACI), and information entropy analysis. The main contribution is to offer a proposal for a diagnostic aid system for uveal melanoma. Considering the average values for 145 processed images, the results show that near-maximum NPCR values of 0.996 are obtained along with near-safe UACI values of 0.296 and high entropy of 7.954 for the ciphered images. The presented design demonstrates an encryption technique based on chaotic attractors for image transfer through the network. In this article, important theoretical considerations for implementing this system are provided, the requirements and architecture of the system are explained, and the stages in which the diagnosis is carries out are described. Finally, the encryption process is explained and the results and conclusions are presented. [ABSTRACT FROM AUTHOR]

Differential operators based on convolution have been recognized as powerful mathematical operators able to depict and capture chaotic behaviors, especially those that are not able to be depicted using classical differential and integral operators. While these differential operators have being applied with great success in many fields of science, especially in the case of dynamical system, we have to confess that they were not able depict some chaotic behaviors, especially those with additionally similar patterns. To solve this issue new class of differential and integral operators were proposed and applied in few problems. In this paper, we aim to depict chaotic behavior using the newly defined differential and integral operators with fractional order and fractal dimension. Additionally we introduced a new chaotic operators with strange attractors. Several simulations have been conducted and illustrations of the results are provided to show the efficiency of the models.

The paper features the simulation of wave forces and moments acting on the ship for synthesizing the motion control algorithms. The available approaches to describing the wave forces and moments are shown so have some weak points. An alternative approach to wave simulation is proposed, based on chaotic oscillations generated in the dynamical systems of chaotic attractor type. The wave spectral characteristics are analyzed, and criteria to define the major frequency range of the signal spectrum are proposed. It is proposed to compare the major frequency range, signal variance, and distribution characteristics of the simulated process and the wave of the preset frequency and average height under comparison. The known Arneodo and Chen attractors are considered, their spectral properties are studied, and phase patterns are constructed. It is proposed to use an attractor phase coordinate as a magnitude modeling the wave. The vehicle mathematical model is supplemented with the attractor equations, and the process becomes quasistochastic due to the attractor features. The adequacy of the proposed wave model is estimated using the estimates of spectra and distributions. The ship rolling has been simulated using the described approach. [ABSTRACT FROM AUTHOR]

Generally, there is no analytical method to calculate the chaotic solution of a dynamical system. In this paper, a simple proof of the existence of analytical solution in a discontinuous 2D discrete time piecewise chaotic map is reported, i.e., we show that a map that has a simple analytical solution can also have a chaotic attractor by the use of the classical definition of the Jordan canonical form defined for matrices available in most kinds of literature on linear algebra. [ABSTRACT FROM AUTHOR]

The subject of the model research contained in this paper is a new design solution of the energy harvesting system with a star-shaped structure of elastic elements and variable configuration. Numerical experiments focused mainly on the assessment of the configuration of elastic elements in the context of energy harvesting efficiency. The results of computer simulations were limited to zero initial conditions as it is the natural position of the static equilibrium. The article compares the energy efficiency for the selected range of the dimensionless excitation frequency. For this purpose, four cases of elastic element configurations were compared. The results are visualized based on the diagram of RMS voltage induced on piezoelectric electrodes, bifurcation diagrams, Lyapunov exponents, and Poincaré maps, showing the impact of individual solutions on the efficiency of energy harvesting. The results of the simulations show that the harvester's efficiency ranges from 4 V to 20 V depending on the configuration and the frequency range of the excitation, but the design allows for a smooth adjustment to the given conditions. [ABSTRACT FROM AUTHOR]

In this article, a new method for generating grid multi-wing chaotic attractors from fractional-order linear differential systems is proposed. In order to generate grid multi-wing attractors, we extend the method of constructing heteroclinic loops from classical differential equations to fractional-order differential equations. Firstly, two basic fractional-order linear systems are obtained by linearization at two symmetric equilibrium points of the fractional-order Rucklidge system. Then a heteroclinic loop is constructed and all equilibrium points of the two basic fractional-order linear systems are connected by saturation function switching control. Secondly, the theoretical methods of switching control and construction of heteromorphic rings of fractal-order two-wing and multi-wing chaotic attractors are studied. Finally, the feasibility of the proposed method is verified by numerical simulation.

Abstract This research aims to discuss and control the chaotic behaviour of an autonomous fractional biological oscillator. Indeed, the concept of fractional calculus is used to include memory in the modelling formulation. In addition, we take into account a new auxiliary parameter in order to keep away from dimensional mismatching. Further, we explore the chaotic attractors of the considered model through its corresponding phase-portraits. Additionally, the stability and equilibrium point of the system are studied and investigated. Next, we design a feedback control scheme for the purpose of chaos control and stabilization. Afterwards, we introduce an efficient active control method to achieve synchronization between two chaotic fractional biological oscillators. The efficiency of the proposed stabilizing and synchronizing controllers is verified via theoretical analysis as well as simulations and numerical experiments.

We analyze the classical and quantum dynamics of the driven dissipative Bose–Hubbard dimer. Under variation of the driving frequency, the classical system is shown to exhibit a bifurcation to the limit cycle, where its steady-state solution corresponds to periodic oscillation with the frequency unrelated to the driving frequency. This bifurcation is shown to lead to a peculiarity in the stationary single-particle density matrix of the quantum system. The case of the Bose–Hubbard trimer, where the discussed limit cycle bifurcates into a chaotic attractor, is briefly discussed.

In this paper, we propose and analyze a nonsmoothly two-dimensional map arising in a seasonal influenza model. Such map consists of both linear and nonlinear dynamics depending on where the map acts on its domain. The map exhibits a complicated and unpredictable dynamics such as fixed points, period points, chaotic attractors, or multistability depending on the ranges of a certain parameters. Surprisingly, bistable states include not only the coexistence of a stable fixed point and stable period three points but also that of stable period three points and a chaotic attractor. Among other things, we are able to prove rigorously the coexistence of the stable equilibrium and stable period three points for a certain range of the parameters. Our results also indicate that heterogeneity of the population drives the complication and unpredictability of the dynamics. Specifically, the most complex dynamics occur when the underlying basic reproduction number with respect to our model is an intermediate value and the large portion of the population in the same compartment changes in states the following season. [ABSTRACT FROM AUTHOR]

Accurate development of satellite maneuvers necessitates a broad orbital dynamical system and efficient nonlinear control techniques. For achieving the intended formation, a framework of a discrete fractional difference satellite model is constructed by the use of commensurate and non-commensurate orders for the control and synchronization of fractional-order chaotic satellite system. The efficacy of the suggested framework is evaluated employing a numerical simulation of the concerning dynamic systems of motion while taking into account multiple considerations such as Lyapunov exponent research, phase images and bifurcation schematics. With the aid of discrete nabla operators, we monitor the qualitative behavioural patterns of satellite systems in order to provide justification for the structure's chaos. We acquire the fixed points of the proposed trajectory. At each fixed point, we calculate the eigenvalue of the satellite system's Jacobian matrix and check for zones of instability. The outcomes exhibit a wide range of multifaceted behaviours resulting from the interaction with various fractional-orders in the offered system. Additionally, the sample entropy evaluation is employed in the research to determine complexities and endorse the existence of chaos. To maintain stability and synchronize the system, nonlinear controllers are additionally provided. The study highlights the technique's vulnerability to fractional-order factors, resulting in exclusive, changing trends and equilibrium frameworks. Because of its diverse and convoluted behaviour, the satellite chaotic model is an intriguing and crucial subject for research., (© 2024. The Author(s).)

In one-dimensional piecewise smooth maps with multiple borders, chaotic attractors may undergo border collision bifurcations, leading to a sudden change in their structure. We describe two types of such border collision bifurcations and explain the mechanisms causing the changes in the geometrical structure of the attractors, in particular, in the number of their bands (connected components). [ABSTRACT FROM AUTHOR]

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Copyright of Frontiers of Information Technology & Electronic Engineering is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Early detection of diseases is vital for patient recovery. This article explains the design and technical matters of a computer-supported diagnostic system for eye melanoma detection implementing a security approach using chaotic-based encryption to guarantee communication security. The system is intended to provide a diagnosis; it can be applied in a cooperative environment for hospitals or telemedicine and can be extended to detect other types of eye diseases. The introduced method has been tested to assess the secret key, sensitivity, histogram, correlation, Number of Pixel Change Rate (NPCR), Unified Averaged Changed Intensity (UACI), and information entropy analysis. The main contribution is to offer a proposal for a diagnostic aid system for uveal melanoma. Considering the average values for 145 processed images, the results show that near-maximum NPCR values of 0.996 are obtained along with near-safe UACI values of 0.296 and high entropy of 7.954 for the ciphered images. The presented design demonstrates an encryption technique based on chaotic attractors for image transfer through the network. In this article, important theoretical considerations for implementing this system are provided, the requirements and architecture of the system are explained, and the stages in which the diagnosis is carries out are described. Finally, the encryption process is explained and the results and conclusions are presented.

The relationship between currencies in foreign exchange markets has been a topic of significance in economics. Previous studies have focused more on correlations between currencies. However, the detection of causality can reveal their inherent laws. Although the traditional Granger causality test can identify causality, it cannot take into account the nature and intensity of the causality. Thus, the objective of this paper is to identify the causalities of currencies from the perspective of dynamics. In this paper, we select 25 currencies (with the US dollar (USD) as the numeraire) from foreign exchange markets, as they occupy large shares in their regions. To detect the causalities of the foreign exchange markets, we combine PC (pattern causality) theory and complex networks to construct directed and weighted causality networks, in which the nodes represent the currencies and the directed edges represent the causal intensities. Furthermore, we study the symmetry of each causality and quantify the symmetry degree. The results demonstrate that causalities exist between currencies that differ in terms of nature and intensity. The positive causality network exhibits substantial robustness, which can be regarded as the dominant causal relationship in the foreign exchange markets, although a few exceptions are encountered, such as the dominant negative and disordered causalities between currency pairs. In addition, the dominant causalities between most currencies are symmetric in terms of nature, and they also exhibit symmetry in terms of intensity. Furthermore, by gradually deleting the network by thresholding according to the edge weights, we identify the important driving currencies of the markets. This paper may provide valuable information for investors and supervisory departments. [ABSTRACT FROM AUTHOR]

In this paper, we propose a methodology to design chaos-based communication systems, which exploits the topological structure of 3-D chaotic attractors. The first step consists in defining a proper partition of a Poincaré section of the attractor and the subsequent encoding of the chaotic trajectories. Then, the evolution mechanism of the chaotic attractor, according to the dynamical restrictions imposed by the chaotic flow, is represented by a state diagram, where each state represents a region of the Poincaré section or a branch in the template of the chaotic attractor. The state transitions are associated with segments of chaotic trajectories that connect the corresponding regions of the Poincaré section. The chaotic signals are transmitted over both additive white Gaussian noise and Rayleigh flat fading channels, and a trellis structure derived from the state diagram is used at the decoder to estimate the transmitted information sequence. Finally, the bit error rate performance of the system is analyzed.

The methods of recurrence plots (RPs) and recurrence quantification analysis (RQA) have been used to investigate the tribosystem. The morphology of RPs and RQA measures are strongly dependent on the embedding parameters of the recursive matrix and the segment sizes of the time-series. To improve the calculation accuracy of recursive characteristics analysis, the influences of the embedding parameters and segment sizes on the morphology of RPs and RQA measures have been studied in this letter. Three kinds of theoretical chaotic time-series and measured coefficient of friction (COF) signals during the running-in process were chosen as research objects, and the morphology of RPs and RQA measures were obtained using CRP toolbox afterward. The results indicate that no embedding was actually needed if the data sets are to be qualitatively analyzed using RPs and RQA. Additionally, the morphology of RPs and RQA measures are sensitive to the segment sizes for theoretical chaotic time-series, while the RQA measures of COF signal in the steady-state period are rather stable due to its self-similarity. Finally, according to the guidelines of the parameter settings, the dynamical evolution of measured COF signals during the running-in process have been investigated. It is indicated that recursive characteristics of COF signals could reveal the tribological behaviors' evolution and conduct the running-in status identification. The results in this paper are significant for improving the calculation accuracy and saving computational time when using the method of recursive characteristics analysis on the tribological behaviors. [ABSTRACT FROM AUTHOR]

• Non-linear multi-scale diffusion cancer invasion model is developed. • Interactions of tumor cells, matrix-metalloproteinases/enzymes and oxygen are analyzed. • Multi-parametric evolution of conditions leading to cancer invasion is carried out. • Chaotic versus regular attractors and phase spaces are illustrated and discussed. • Amplitude level contours and oxygen concentrations are obtained. In this study we performed a simulation of carcinogenesis evolution based on varying system parameters in the multi-parametric space. A non-linear multi-scale diffusion cancer invasion model that describes the interactions of the tumor cells, matrix-metalloproteinases (MM), matrix-degradative enzymes (MDE) and oxygen was employed for simulation. To quantify chaotic cancer attractors, the technique based on the wandering trajectories analysis was applied. Multi-parametric evolution of conditions leading to cancer invasion was defined and illustrated in numerous figures. Chaotic attractors and phase spaces of the regular oscillations of basic indicators of a biological system with parameters that correspond to the chaotic and regular regions, respectively, have been presented. Amplitude level contours of MM, MDE and oxygen concentrations have been obtained and juxtaposed with the corresponding parametric planes. In all cases the carcinogenesis is accompanied by significant increase in chemical oscillation amplitudes of MM, MDE and oxygen concentrations. The obtained results derived from all control parameter planes could be used for evaluations of conditions resulting in cancer invasion as well as the modes to inhibit and/or stabilize carcinogenesis. [ABSTRACT FROM AUTHOR]

This research aims to discuss and control the chaotic behaviour of an autonomous fractional biological oscillator. Indeed, the concept of fractional calculus is used to include memory in the modelling formulation. In addition, we take into account a new auxiliary parameter in order to keep away from dimensional mismatching. Further, we explore the chaotic attractors of the considered model through its corresponding phase-portraits. Additionally, the stability and equilibrium point of the system are studied and investigated. Next, we design a feedback control scheme for the purpose of chaos control and stabilization. Afterwards, we introduce an efficient active control method to achieve synchronization between two chaotic fractional biological oscillators. The efficiency of the proposed stabilizing and synchronizing controllers is verified via theoretical analysis as well as simulations and numerical experiments. [ABSTRACT FROM AUTHOR]

The subject of the model research contained in this paper is a new design solution of the energy harvesting system with a star-shaped structure of elastic elements and variable configuration. Numerical experiments focused mainly on the assessment of the configuration of elastic elements in the context of energy harvesting efficiency. The results of computer simulations were limited to zero initial conditions as it is the natural position of the static equilibrium. The article compares the energy efficiency for the selected range of the dimensionless excitation frequency. For this purpose, four cases of elastic element configurations were compared. The results are visualized based on the diagram of RMS voltage induced on piezoelectric electrodes, bifurcation diagrams, Lyapunov exponents, and Poincaré maps, showing the impact of individual solutions on the efficiency of energy harvesting. The results of the simulations show that the harvester’s efficiency ranges from 4 V to 20 V depending on the configuration and the frequency range of the excitation, but the design allows for a smooth adjustment to the given conditions.

We propose a trellis-coded chaotic modulation scheme employing the symbolic dynamics generated by three-dimensional chaotic flows. The continuous chaotic trajectories are discretized using a labeled partition of a Poincaré section. The discrete dynamics is modeled by a graph representing the evolution mechanism of the chaotic trajectories according to the restrictions imposed by the chaotic flow. This graph is employed to design finite-state encoders to transmit binary information sequences using the restricted discrete chaotic dynamics. We show that the proposed system outperforms a recently proposed trellis-coded chaotic modulation scheme. [ABSTRACT FROM AUTHOR]

Chaotic systems are dynamical systems that are highly sensitive to initial conditions. Such systems are used to model many real-world phenomena in science and engineering. The main purpose of this paper is to present several efficient numerical treatments for chaotic systems involving fractal-fractional operators. Several numerical examples test the performance of the proposed methods. Simulations with different values of the fractional and fractal parameters are also conducted. It is demonstrated that the fractal-fractional derivative enables one to capture all the useful information from the history of the phenomena under consideration. The numerical schemes can also be implemented for other chaotic systems with fractal-fractional operators.

In the present paper, we discuss bifurcations of chaotic attractors in piecewise smooth one-dimensional maps with a high number of switching manifolds. As an example, we consider models of DC/AC power electronic converters (inverters). We demonstrate that chaotic attractors in the considered class of models may contain parts of a very low density, which are unlikely to be observed, neither in physical experiments nor in numerical simulations. We explain how the usual bifurcations of chaotic attractors (merging, expansion and final bifurcations) in piecewise smooth maps with a high number of switching manifolds occur in a specific way, involving low-density parts of attractors, and how this leads to an unusual shape of the bifurcation diagrams. [ABSTRACT FROM AUTHOR]