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Currently, despite advances in the analysis of dynamical systems, there are still doubts about the transition between both stable and chaotic behaviors. In this research, we will explain the transition of a system that develops between two dynamic systems that have already been studied: the classical logistic model and a new chaotic system. This research addresses the study of the transition of both the system and its behaviors using computational techniques, where cobweb diagrams, time series, bifurcation diagrams, and even a graphical visualization for the maximum Lyapunov exponent will be visualized. Using a graphical and numerical methodology, bifurcation points were identified that revealed the transition of behaviors at different points. This resulted in a deep understanding of the dynamics of the system, thus highlighting the importance of incorporating computational analysis in dynamic systems, which greatly contributes to the efficient modeling of natural phenomena. [ABSTRACT FROM AUTHOR]

Analysis of a dynamic system helps scientists understand its properties and utilize it properly in different applications. This study analyzes the effects of various external excitements on a recently proposed mathematical neuron model derived from the original Fitzhugh–Nagumo model. Different bifurcation analyses on this system are conducted to detect chaotic behaviors that are common and of great importance in biological systems, considering the effects of different types of external excitements. Lyapunov exponents (LEs) confirm the existence of chaotic patterns. Furthermore, a bifurcation diagram that looks into the changes in the system dynamics caused by the simultaneous application of the external stimulants is represented. Neurons are bound to play a role in a network in which synchrony is an analytical quality. Therefore, the potential of a network of this model in showing synchronization is examined using the master stability function (MSF) technique. Ultimately, it is concluded that this neural model can produce chaotic behaviors and synchronous networks. [ABSTRACT FROM AUTHOR]

This article investigates a class of Duffing nonlinear dynamic systems with fractional‐order dry friction and conducts in‐depth research on the stability, chaotic characteristics, and erosion of the safety basin of this system; the results are verified through numerical simulation. First, the average method is used to approximate the amplitude–frequency relationship of the system, and the accuracy of the analytical results is verified through numerical experiments. Second, the Melnikov method is used to obtain the conditions for the system to enter chaos in the Smale horseshoe sense, and the Melnikov curve is drawn for further verification. Then, bifurcation diagrams are drawn for the changes in various parameters in the system, with a focus on analyzing the influence of friction factors on chaotic bifurcation. By applying the definition and calculation principle of the maximum Lyapunov exponent, and drawing and utilizing the maximum Lyapunov exponent graph, the chaotic state that the system enters under different parameters is more clearly defined. Finally, the evolution law of the safety basin under various parameter changes, especially dry friction changes, is analyzed, and the erosion and bifurcation mechanism of the safety basin is studied. Comparing with the bifurcation diagram, it reveals that chaos primarily contributes to the erosion of the safety basin. [ABSTRACT FROM AUTHOR]

In this paper, we develop a novel n-dimensional discrete chaotic map. First, the chaotic behaviors of the proposed map are studied. Next, to illustrate the effectiveness of our map, the 4D hyperchaotic map as an example is analyzed using the phase diagram, Lyapunov exponents, bifurcation diagrams and different types of entropy, etc. Moreover, the chaotic signals generated by the proposed map passed the NIST test and were implemented on the hardware DSP. Finally, we apply our chaotic map in image encryption using true numbers and extended XOR. The experimental results express that the presented encryption algorithm has a higher level of security than same other sophisticated methods. [ABSTRACT FROM AUTHOR]

The isolated boost full-bridge converter (IBFBC) is a DC–DC conversion topology that achieves a high boost ratio and provides electrical isolation, making it suitable for applications requiring both. Its operational dynamics are often intricate due to its inherent characteristics and the prevalent usage of nonlinear switching elements, leading to bifurcation and chaos. Chaos theory was employed to investigate the impact of changes in the voltage feedback coefficient K and input voltage E on the dynamic behavior of the IBFBC when operating in discontinuous conduction mode (DCM). Based on an analysis of its operating principles, a discrete iterative mapping model of the system in DCM is constructed using the stroboscopic mapping method. The effects of two control parameters, namely the proportional coefficient and input voltage, on system performance are studied using bifurcation diagrams, fold diagrams, Poincaré sections, and Lyapunov exponents. Simulation experiments are conducted using time-domain and waveform diagrams to validate the discrete mapping model and confirm the correctness of the theoretical analysis. The results indicate that when the IBFBC operates in DCM, its operating state is influenced by the voltage feedback coefficient K and input voltage E . Under varying values of K and E , the system may operate in a single-period stable state, multi-period bifurcation, or chaotic state, exhibiting typical nonlinear behavior. [ABSTRACT FROM AUTHOR]

Host–parasitoid systems are an essential area of research in ecology and evolutionary biology due to their widespread occurrence in nature and significant impact on species evolution, population dynamics, and ecosystem stability. In such systems, the host is the organism being attacked by the parasitoid, while the parasitoid depends on the host to complete its life cycle. This paper investigates the effect of parasitoid aggregation attacks on a host in a host–parasitoid model with self-diffusion on two-dimensional coupled map lattices. We assume that in the simulation of biological populations on a plane, the interactions between individuals follow periodic boundary conditions. The primary objective is to analyze the complex dynamics of the host–parasitoid interaction model induced by an aggregation effect and diffusion in a discrete setting. Using the aggregation coefficient k as the bifurcating parameter and applying central manifold and normal form analysis, it has been shown that the system is capable of Neimark–Sacker and flip bifurcations even without diffusion. Furthermore, with the influence of diffusion, the system exhibits pure Turing instability, Neimark–Sacker–Turing instability, and Flip–Turing instability. The numerical simulation section explores the path from bifurcation to chaos through calculations of the maximum Lyapunov exponent and the construction of a bifurcation diagram. The interconversion between different Turing instabilities is simulated by adjusting the timestep and self-diffusion coefficient values, which is based on pattern dynamics in ecological modeling. This contributes to a deeper understanding of the dynamic behaviors driven by aggregation effects in the host–parasitoid model. [ABSTRACT FROM AUTHOR]

The main objective of this study is to investigate the chaotic dynamics and analog circuit implementation of a nonautonomous single inertial neuron. First, the dimensionless mathematical model of such neuron is established. The equilibria with three kinds of stabilities are then depicted, and system symmetry and Hamilton energy function are analyzed, respectively. In numerical simulations, by using the two-dimensional/one-dimensional bifurcation diagrams, Lyapunov exponent spectra, phase plots, and basins of attraction, bifurcation of coexisting bubbles as well as bi-stable patterns are revealed, where the generated coexisting attractors are symmetric about the origin. In addition, we confirm that the complexity of the basins of attraction deteriorates with the increase of the frequency parameter. Finally, analog circuit experiments with few analog circuit elements are deployed, via which the bi-stable patterns of the model are well verified. [ABSTRACT FROM AUTHOR]

The Basener and Ross mathematical model is widely recognized for its ability to characterize the interaction between the population dynamics and resource utilization of Easter Island. In this study, we develop and investigate a discrete-time version of the Basener and Ross model. First, the existence and the stability of fixed points for the present model are investigated. Next, we investigated various bifurcation scenarios by establishing criteria for the occurrence of different types of codimension-one bifurcations, including flip and Neimark–Sacker bifurcations. These criteria are derived using the center manifold theorem and bifurcation theory. Furthermore, we demonstrated the existence of codimension-two bifurcations characterized by 1:2, 1:3, and 1:4 resonances, emphasizing the model's complex dynamical structure. Numerical simulations are employed to validate and illustrate the theoretical predictions. Finally, through bifurcation diagrams, maximal Lyapunov exponents, and phase portraits, we uncover a diversity of dynamical characteristics, including limit cycles, periodic solutions, and chaotic attractors. [ABSTRACT FROM AUTHOR]

In this paper, we investigated a three-dimensional chaotic system that models key aspects of a monetary system, including interest rates, investment demand, and price levels. The proposed system is described by a set of autonomous quadratic ordinary differential equations. We analyze the dynamic behavior of this system through equilibrium points and their stability, Lyapunov exponents (LEs), and bifurcation diagrams. The system demonstrates a variety of behaviors, including chaotic, periodic, and equilibrium states depending on parameter values. Additionally, we explore the multistability of the system and present a radial basis function neural network (RBFNN) controller design to stabilize the chaotic behavior. The effectiveness of the controller is validated through numerical simulations, highlighting its potential applications in economic and financial modeling. [ABSTRACT FROM AUTHOR]

The study of chaotic systems with special properties addresses one of the ongoing research topics. In this paper, we introduce a new class of third-order chaotic systems possessing a cyclic symmetry obeying unidirectional or mutual coupling. These new models are built by considering nonlinearities with three, four, or five segments and are distinguished from each other by the topological structure of the associated chaotic attractor. The projection of the novel chaotic attractors on certain planes reveals their multiscroll or multiwing character. For illustration, one of these models is studied in detail using analytical and numerical methods. The mechanism giving rise to the establishment of the chaotic regime starting from the state of equilibrium under the variation of a parameter is described using bifurcation diagrams, phase portraits, basins of attraction as well as the maximum Lyapunov exponent. Several zones of parameters are identified where the model experiences two or more attractors (i.e. multistability). In order to demonstrate the practical feasibility of the proposed circulant models, a series of experimental measurements are carried out using a physical implementation with an Arduino microcontroller. [ABSTRACT FROM AUTHOR]

The effects of a rigid barrier on the stability of the structure while it is undergoing phase-locked motions due to the surrounding fluid–structure interactions, are studied. The primary structure and the near wake dynamics are modeled as a harmonic oscillator and a Van der Pol oscillator, respectively, and are weakly coupled via acceleration coupling. Qualitative changes in the dynamical behavior of this system are investigated in the context of discontinuity-induced bifurcations that result from the interaction of fluid flow and nonsmoothness from the impact of the primary structure. Phenomenological behaviors like the co-existence of attractors and period-adding cascades of limit cycles separated by chaotic orbits are observed. The behavior of orbits in the local neighborhood of the discontinuity boundary is defined using a higher-order transverse discontinuity map, and the corresponding stability analysis is carried out using Floquet theory. The derived mapping is implemented to obtain the respective Lyapunov exponents. Results obtained using the mapping are demonstrated to accurately predict both the stable and chaotic regimes, as observed from the corresponding numerical bifurcation diagrams. [ABSTRACT FROM AUTHOR]

This paper proposes an image encryption scheme based on an improved four-dimensional chaotic system. First, a 4D chaotic system is constructed by introducing new state variables based on the Chen chaotic system, and its chaotic behavior is verified by phase diagrams, bifurcation diagrams, Lyapunov exponents, NIST tests, etc. Second, the initial chaotic key is generated using the hash function SHA-512 and plain image information. Parity scrambling is performed on the plain image using the chaotic sequence generated by the chaotic system. The image is then converted into a hexadecimal character matrix, divided into two planes according to the high and low bits of the characters and scrambled by generating two position index matrices using chaotic sequences. The two planes are then restored to a hexadecimal character matrix, which is further converted into the form of an image matrix. Finally, different combined operation diffusion formulas are selected for diffusion according to the chaotic sequence to obtain the encrypted image. Based on simulation experiments and security evaluations, the scheme effectively encrypts gray images and shows strong security against various types of attacks. [ABSTRACT FROM AUTHOR]

This study examines the dynamical properties of the Ikeda map, with a focus on bifurcations and chaotic behavior. We investigate how variations in dissipation parameters influence the system, uncovering shrimp-shaped structures that represent intricate transitions between regular and chaotic dynamics. Key findings include the analysis of period-doubling bifurcations and the onset of chaos. We utilize Lyapunov exponents to distinguish between stable and chaotic regions. These insights contribute to a deeper understanding of nonlinear and chaotic dynamics in optical systems. [ABSTRACT FROM AUTHOR]

This study investigates inventory management systems using a sample of listed manufacturing companies in China from 2019 to 2023. By constructing a static mathematical model, the impact of inventory management on corporate performance was empirically tested. Additionally, based on a classical inventory management dynamical model and considering inventory delay characteristics, a new class of two-dimensional inventory management systems was reconstructed. The system's periodic and chaotic nonlinear characteristics were verified using 0-1 tests, bifurcation diagrams, Lyapunov exponents, and system eigenvalue plots. Furthermore, MATLAB simulations were employed to examine the effect of resource transfer rates on the nonlinear dynamic behavior of the inventory management system. The results from both static mathematical models and dynamical models provide a theoretical basis for inventory management and safety stock level predictions in the manufacturing industry. [ABSTRACT FROM AUTHOR]

This paper introduces a new one-dimensional chaotic system and a new image encryption algorithm. Firstly, the new chaotic system is analyzed. The bifurcation diagram and Lyapunov exponent show that the system has strong chaotic characteristics and is suitable for the field of image encryption. The chaotic sequence generated by the system is used in image encryption, scrambled according to its sequence value, and several sequences required for DNA coding operation are generated. The encrypted image is obtained by encoding, decoding, and diffusion operation according to the sequence. The initial values and parameters of system are generated by Hash algorithm based on plain image, so it has strong sensitivity and can effectively resist attacks such as selective plaintext attack. Experimental results show that the algorithm has high security and robustness, and can resist common attacks. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of a new discrete form of the classical continuous Rosenzweig–MacArthur predator–prey model and the biological implications of the dynamics. The discretization method used here is to modify the continuous model to another with piecewise-constant arguments and then to integrate the modified model, which is quite different from the traditional Euler discretization method. First, the existence and local stability of fixed points have been thoroughly discussed. Then, all codimension-1 bifurcations have been studied, including the transcritical bifurcations at the trivial fixed point and the boundary fixed point, and the Neimark–Sacker bifurcation at the unique positive fixed point. Furthermore, it is proved that there are no codimension-2 bifurcations. Finally, the control of the bifurcations is studied. Regarding the biological significance, we have considered the following four aspects: When no harvesting effort is applied to both predators and prey, it is observed that providing more resources to the prey species leads to predator extinction, causing the ecosystem to lose stability. This implies that the paradox of enrichment occurs. When predators are harvested, the system exhibits the hydra effect, i.e. increasing the harvesting effort on predators will lead to an increase in the mean population density of predators. When the prey is harvested, numerical examples indicate that increasing the harvesting effort initially reduces prey density. Upon reaching the bifurcation point, prey density stabilizes while the predator density continues to decline. When both prey and predators are harvested, the biological significance is similar to the previous case. The outcomes of this paper have significant theoretical meaning in the study of population ecology. By MATLAB, the bifurcation diagrams, maximal Lyapunov exponent diagrams, phase portraits and mean population density curves are plotted. Numerical simulations not only show the correctness of the theoretical findings but also reveal many new and interesting dynamics. [ABSTRACT FROM AUTHOR]

To solve the problem that the previously constructed S-boxes based on chaotic systems were difficult to achieve good cryptographic performance, the design scheme for S-boxes was proposed based on hyperchaotic system and genetic particle swarm optimization algorithm. Introducing sine and cosine functions and exponential factors, the two-dimensional hyperchaotic system was constructed based on the one-dimensional chaotic mapping. The performance analysis was conducted by system bifurcation diagram, phase diagram and Lyapunov exponent diagram to reveal that the chaotic system exhibited continuous hyper-chaotic intervals in the parameter range with complex chaotic behavior. By varying initial values, parameters and iteration times of the chaotic system, S-boxes were dynamically generated. Combining particle swarm optimization algorithm and genetic algorithm, the genetic particle swarm optimization algorithm for S-boxes was proposed, and the S-boxes generated by chaotic system were used as initial population. The particle swarm optimization algorithm was leveraged to enhance the crossover operation in the genetic algorithm, and a new mutation strategy was introduced in conjunction with hill-climbing algorithm. To verify the performance of the generated S-box, the simulation tests were conducted on bijective property, nonlinearity, strict avalanche criterion, differential probability and bit independence criterion. The simulation results show that the proposed optimization algorithm can generate S-boxes with good performance in terms of nonlinearity, differential uniformity and bit independence criterion. [ABSTRACT FROM AUTHOR]

In recent years, the introduction of memristors in discrete chaotic map has attracted much attention due to its enhancement of the complexity and controllability of chaotic maps, especially in the fields of secure communication and random number generation, which have shown promising applications. In this work, a three-dimensional discrete memristive hyperchaotic map (3D-DMCHM) based on cosine memristor is constructed. First, we analyze the fixed points of the map and their stability, showing that the map can either have a linear fixed point or none at all, and the stability depends on the parameters and initial state of the map. Then, phase diagrams, bifurcation diagrams, Lyapunov exponents, timing diagrams, and attractor basins are used to analyze the complex dynamical behaviors of the 3D-DMCHM, revealing that the 3D-DMCHM enters into a chaotic state through a period-doubling bifurcation path, and some special dynamical phenomena such as multi-layer differentiation, multi-amplitude control, and offset boosting behaviors are also observed. In particular, with the change of memristor initial conditions, there exists an offset that only homogeneous hidden chaotic attractors or a mixed state offset with coexistence of point attractors and chaotic attractors. Finally, we confirmed the high complexity of 3D-DMCHM through complexity tests and successfully implemented it using a digital signal processing circuit, demonstrating its hardware feasibility. [ABSTRACT FROM AUTHOR]

Different functional areas of the brain are composed of varying numbers of heterogeneous neurons, which transmit signals via synapses. As an electronic element, memristor can store resistance values and simulate the memory properties of synapses. At present, research on coupling heterogeneous neurons through memristive synapses is relatively scarce. A heterogeneous neural network model is proposed in this paper, where the memristor acts as a coupled synapse between a Two-Dimensional Hindmarsh–Rose (2D HR) neuron and a Three-Dimensional Hopfield Neural Network (3D HNN) neuron. The activation function of the neural network can simulate the firing behavior and nonlinearly transform the input signals. Therefore, an activation gradient is added to the 3D HNN neuron. In this paper, heterogeneous neurons and activation gradients are simultaneously introduced into the neural network. The model provides a more realistic artificial neural network, which can simulate the diversity of neurons in the brain. The firing behaviors varying with the coupling strength between the memristor and neurons are analyzed, and the influence of the activation gradient on the neural network model is studied. Various firing patterns are discovered, including chaotic/periodic spiking, chaotic/periodic bursting, stochastic bursting and transient chaotic spiking. By observing the Lyapunov Exponents (LEs) and bifurcation diagram under the initial values of two memristors, it is found that there are infinitely many attractors, indicating the extreme multistability firing in this heterogeneous neural network model. Finally, circuit design is implemented by Multisim simulation, and hardware implementation is completed on the Field-Programmable Gate Array (FPGA) platform. [ABSTRACT FROM AUTHOR]

This article investigates the chaotic analysis and attractive domain of a fractional‐order Φ6$$ {\Phi}^6 $$‐van der Pol with time delay velocity under harmonic excitation. Firstly, eight different types of bifurcation states of the system under different parameters are calculated by using the undisturbed system. Secondly, the Melnikov method is used to explore the effect of time delay velocity on the threshold of chaos in the Smale horseshoe sense under the double‐well potential and three‐well potential of the system. Finally, through numerical analysis of the phase diagram, bifurcation diagram, and maximum Lyapunov exponent, the influence of time delay velocity on system chaos is studied. The results indicate that an increase in the delay velocity coefficient will lead to the system transitioning from a chaotic state to a periodic state, while an increase in the delay velocity term will lead to the system transitioning from a periodic state to a chaotic state. In the study of system bifurcation, it is found that the position of the equilibrium points of the system changes during periodic motion. Therefore, cell mapping is used to draw the attractive domain of the system is studying the influence of initial conditions on the equilibrium point of the system and the results show that there is a close relationship between the attraction domain and the process of chaos occurrence. [ABSTRACT FROM AUTHOR]

This work analyzes the effect of parametric dissipation on the dynamics of the classical oscillator counterpart of a quantum nonlinear oscillator driven by a position-dependent mass. The multiple scale method is used to search the different possible states of resonances and two types of resonances are analyzed. The influence of system parameters and in particular that of parametric dissipation on the resonance amplitude is studied. The complete dynamics and transition to chaos of the oscillator are analyzed numerically using the Runge-Kutta algorithm of order 4. Bifurcation diagrams, Lyapunov exponents and phase spaces are used and the frequency doubling phenomenon is obtained and the limits of the system oscillations are widened. [ABSTRACT FROM AUTHOR]

In the present article, we investigate the charged micro-particle dynamics in the surface radio-frequency trap (SRFT). We have developed a new configuration of the SRFT that consists of three curved electrodes on a glass substrate for massive micro-particles trapping. We provide the results of numerical simulations for the dynamical regimes of charged silica micro-particles in the SRFT. Here, we introduce a term of a "main route" to chaos, i.e., the sequence of dynamical regimes for the given particles with the increase of the strength of an electric field. Using the Lyapunov exponent formalism, typical Reynolds number map, Poincaré sections, bifurcation diagrams, and attractor basin boundaries, we have classified three typical main routes to chaos depending on the particle size. Interestingly, in the system described here, all main scenarios of a transition to chaos are implemented, including the Feigenbaum scenario, the Landau–Ruelle–Takens–Newhouse scenario as well as intermittency. [ABSTRACT FROM AUTHOR]

This paper presents a novel numerical method for analvwing chaotic systems, focusing on applications to real-world problems. The Caputo-Fabrizio operator, a fractional derivative without a singular kernel, is used to investigate chaotic behavior. A fractional-order chaotic model is analvwed using numerical solutions derived from this operator, which captures the complexity of chaotic dynamics. In this paper, the uniqueness and boundedness of the solution are established using fixed-point theory. Due to the non-linearity of the system, an appropriate numerical scheme is developed. We further explore the model's dynamical properties through phase portraits, Lyapunov exponents, and bifurcation diagrams. These tools allow us to observe the system's sensitivity to varying parameters and derivative orders. Ultimately, this work extends the application of fractional calculus to chaotic systems and provides a robust methodology for obtaining insights into complex behaviors. [ABSTRACT FROM AUTHOR]

This paper introduces a modified Morris–Lecar neuron model that incorporates a memristor with a ReLU-based activation function. The impact of the memristor on the dynamics of the ML neuron model is analyzed using bifurcation diagrams and Lyapunov exponents. The findings reveal chaotic behavior within specific parameter ranges, while increased magnetic strength tends to maintain periodic dynamics. The emergence of various firing patterns, including periodic and chaotic spiking as well as square-wave and triangle-wave bursting is also evident. The modified model also demonstrates multistability across certain parameter ranges. Additionally, the dynamics of a network of these modified models are explored. This study shows that synchronization depends on the strength of the magnetic flux, with synchronization occurring at lower coupling strengths as the magnetic flux increases. The network patterns also reveal the formation of different chimera states, such as traveling and non-stationary chimera states. [ABSTRACT FROM AUTHOR]

In dynamical systems, events that deviate significantly from usual or expected behavior are referred to as extreme events. This paper investigates the mechanism of extreme event generation in a 3D jerk system based on a generalized memristive device. In addition, regions of coexisting parallel bifurcation branches are explored as a way of investigating the multistability of the memristive system. The system is examined using bifurcation diagrams, Lyapunov exponents, time series, probability density functions of events, and inter-event intervals. It is found that extreme events occur via a period-doubling route and are due to an interior crisis that manifests itself as a sudden shift from low-amplitude to high-amplitude oscillations. Multistability is also identified when both control parameters and initial values are modified. Finally, an analog circuit based on the memristive jerk system is designed and experimentally realized. To our knowledge, this is the first time that extreme events have been reported in a memristive jerk system in particular and in jerk systems in general. [ABSTRACT FROM AUTHOR]

With the development and progress of Internet and data technology, the consensus control of multi-agent systems has been an important topic in nonlinear science. How to effectively achieve the consensus of leader–follower multi-agent systems at a low cost is a difficult problem. This paper analyzes the consensus control of complex financial systems. Firstly, the dynamic characteristics of the financial system are analyzed by the equilibrium points, bifurcation diagrams, and Lyapunov exponent spectra. The behavior of the financial system is discussed by different parameter values. Secondly, according to the Lyapunov stability theorem, the consensus of master–slave systems is proposed by linear feedback control, wherein the controllers are simple and low cost. And an adaptive control method for the consensus of master–slave systems is investigated based on financial systems with unknown parameters. In theory, the consensus of the leader–follower multi-agent systems is proved by the parameter identification laws and linear feedback control method. Finally, the effectiveness and reliability of the consensus of leader–follower multi-agent systems are verified through the experimental simulation results and circuit implementation. [ABSTRACT FROM AUTHOR]

Medical images make up for more than 25% of global attacks on privacy. Securing them is therefore of utmost importance. Chaos based image encryption is one of the most method suggested in the literature for image security due to their intrinsic characteristic, including ergodicity, aperiodicity, high sensitivity to initials conditions and system parameters. Dynamic systems such as bridge circuit, jerk circuit, Van der Pol circuit, Colpitts oscillator and many other pseudo-random numbers generators have been used in the process of encrypting images. Among them, are the jerk oscillators that have been used with different nonlinearities. In this paper, a new, simple, off-shell component of jerk oscillator (jerk quintic) with an interesting nonlinear function is proposed. Its dynamical behaviors are investigated using classical tools like bifurcation diagrams, Maximum Lyapunov exponent plot, basin of attraction, phase portraits. We showed that the nonlinear function is responsible of complex nonlinear behaviors displayed by the novel circuit, including symmetric/asymmetric bifurcation and coexisting bubbles, multistability just to name a few. The real implementation of the interesting circuit is embedded in a microcontroller verifies these dynamics. As an application of this contribution in multimedia, an encryption algorithm built on a new confusion-diffusion architecture using pseudo random number generated in high chaoticity regime of the new circuit is proposed. The cryptosystem underwent thorough security tests and proved to be fast thanks to the 3D map used, given its complex dynamical behaviors and large chaotic area. This approach yields a robust cipher that underwent thorough security tests better than the one in the literature like average NPCR=99.61, UACI=33.48, key space-sensitivity, entropy=7.9994, average correlation=0.0040. Furthermore, it proved to be robust in terms of noise and data loss in the transmission channel, offering a large key space of 10180 and an entropy close to the standard value, thus rendering the cryptosystem robust against various attacks, especially brute force and exhaustive attacks. [ABSTRACT FROM AUTHOR]

This study investigates the behavior of the Izhikevich discrete neuron model across various parameter configurations. Bifurcation diagrams and Lyapunov exponents are utilized to examine the impact of these parameters on the behavior of the system. The study specifically identifies important parameter ranges in which the attractor undergoes a sudden expansion, displaying characteristics of extreme events. Within the system, two distinct categories of extreme events can be identified: rare occurrences of small probability events located in the tail of the probability distribution and Dragon-King (DK) events, which possess a high probability amplitude. DK events are verified through the use of the DK test. The research concludes by examining the practical ramifications of these findings. The significance of forecasting and controlling extreme events in intricate systems is underscored, along with the cruciality of identifying their happening. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of a four-dimensional plankton-fish system that contains three communities of plankton: non-toxic phytoplankton, toxic-producing phytoplankton, and zooplankton. The zooplankton consumes both non-toxic and toxic-producing phytoplankton. First, the positivity and boundedness of the solutions in the plankton-fish model are evaluated. Then, we derive the conditions for the existence of ecologically possible equilibria and their local and global asymptotic nature. Furthermore, we evaluate the criteria for the existence of Hopf-bifurcation and transcritical bifurcation. By applying Pontryagin’s maximum principle, we investigate the optimal control of fish harvesting, where fishing effort is taken as a control parameter. By using phase-space diagrams, time responses, bifurcation diagrams, and numerical analyses validate the theoretical results. The numerical results including maximum Lyapunov exponent verify chaotic dynamics of the system. We notice that a suitable choice of harvesting, phytoplankton growth rate, and additional food parameters can control the chaotic behavior of the presented model. The proposed novel mathematical model gives deeper insights to theoretical ecologists on how toxic-producing phytoplankton affects zooplankton and fish populations in aquatic systems. Furthermore, the proposed model is more appropriate for fish harvesting to maximize profit as well as species conversion in the presence of toxic phytoplankton. [ABSTRACT FROM AUTHOR]

This paper presents a novel 4D hyperchaotic system derived from a modified 3D Lorenz chaotic system. A key aspect of this system is the presence of a single equilibrium point, and its stability is carefully analyzed. The dynamic properties, including the Lyapunov exponent spectrum, bifurcation diagram, and chaotic attractors, are investigated using MATLAB simulations. The results reveal that the system displays hyperchaotic behavior across a wide range of the parameter d , with its chaotic attractor transitioning through four states: hyperchaotic, chaotic, periodic, and quasi-periodic, showcasing the system's complex dynamics. Experimental validation using STM32 embedded hardware successfully reproduces these four types of chaotic attractors, confirming the theoretical predictions. The proposed hyperchaotic system is then applied to image encryption, introducing a novel encryption method. The hyperchaotic key sequence generated by this system meets 15 tests of the NIST SP800-22 standard, and further experimental validation with STM32 hardware demonstrates the algorithm's effectiveness, simplicity, non-linearity, and high security. The encrypted images and sequences are rigorously tested key space analysis, histogram similarity analysis, information entropy analysis, statistical attack analysis, differential attack analysis, key sensitivity analysis, and correlation analysis, highlighting the robustness and reliability of the proposed system. This method is versatile and can be extended to various fields, including audio and video encryption, text encryption, IoT security, financial transaction security, and medical data protection. [ABSTRACT FROM AUTHOR]

We discuss 1D, 2D and 3D bifurcation diagrams of two nonlinear dynamical systems: an electric arc system having both chaotic and periodic steady-state responses and a cytosolic calcium system with both periodic/chaotic and constant steady-state outputs. The diagrams are mostly obtained by using the 0–1 test for chaos, but other types of diagrams are also mentioned; for example, typical 1D diagrams with local maxiumum values of oscillatory responses (periodic and chaotic), the entropy method and the largest Lyapunov exponent approach. Important features and properties of each of the three classes of diagrams with one, two and three varying parameters in the 1D, 2D and 3D cases, respectively, are presented and illustrated via certain diagrams of the K values, − 1 ≤ K ≤ 1 , from the 0–1 test and the sample entropy values S a E n > 0 . The K values close to 0 indicate periodic and quasi-periodic responses, while those close to 1 are for chaotic ones. The sample entropy 3D diagrams for an electric arc system are also provided to illustrate the variety of possible bifurcation diagrams available. We also provide a comparative study of the diagrams obtained using different methods with the goal of obtaining diagrams that appear similar (or close to each other) for the same dynamical system. Three examples of such comparisons are provided, each in the 1D, 2D and 3D cases. Additionally, this paper serves as a brief review of the many possible types of diagrams one can employ to identify and classify time-series obtained either as numerical solutions of models of nonlinear dynamical systems or recorded in a laboratory environment when a mathematical model is unknown. In the concluding section, we present a brief overview of the advantages and disadvantages of using the 1D, 2D and 3D diagrams. Several illustrative examples are included. [ABSTRACT FROM AUTHOR]

In this paper, we study a three-dimensional discrete-time model to describe the behavior of cancer cells in the presence of healthy cells and HIV-infected cells. Based on the Caputo-like difference operator, we construct the fractional-order biological system. This study's significance lies in developing a new approach to presenting a biological dynamical system. Since the qualitative analysis related to existence, uniqueness, and stability is almost the same as can be found in numerous existing papers, and comparing this study to other research, constructing a biological discrete system using the Caputo difference operator can be particularly important. Using powerful tools of nonlinear theory such as phase plots, bifurcation diagrams, Lyapunov exponent spectrum, and the 0-1 test, we establish that the proposed system can exhibit different biological states, including stable, periodic, and chaotic behaviors. Here, the route leading to chaos is period-doubling bifurcation. Furthermore, the level of chaos in the system is quantified using C0 complexity and approximate entropy algorithms. The stabilization or suppression of chaotic motions in the fractional-order system is presented, where an efficient controller is designed based on the stability theory of the discrete-time fractional-order systems. Numerical simulations are provided to validate the theoretical results derived in this research paper. [ABSTRACT FROM AUTHOR]

The growing demand for maintaining secure communication channels with the advancements in digital technologies has led to an intensified interest in designing reliable image encryption schemes. Despite various encryption schemes that have been used, some are considered to have insecurity regarding data transmission and multimedia. Motivated by this concern, this paper proposes a new color image encryption algorithm of a multi-key generation-based nD-Hyperchaotic (HC) system. The new algorithm achieves Shannon’s confusion and diffusion principles using a two S-box generation approach, where the first S-box is generated from the proposed nD-HC system and the resulting sequence is used to increase encryption complexity, while the second S-box is generated from (n + i)D-HC system. Afterward, mathematical analysis is carried out to showcase the robustness and efficiency of the proposed algorithm, as well as its resistance to visual, statistical, differential, and brute-force attacks. The proposed scheme successfully passes all NIST SP 800 suite tests. The cryptographic system demonstrated by the proposed scheme has proven to have outstanding performance through simulation tests, which indicates promising potential applicability aspects in secure and real-time image communication settings. [ABSTRACT FROM AUTHOR]

To ensure the secure and efficient transmission of images, we propose a new image encryption algorithm based on a chaotic system with free control and zigzag scrambling. The performance of the chaotic system is analyzed using the phase trajectory, Lyapunov exponents (LEs), and bifurcation diagram, which demonstrate its good ergodicity, complex chaotic behavior, large and continuous chaotic range, and stable Lyapunov exponent spectrum. In addition, based on this chaotic system, this algorithm also provided the security for the protection of image data. The algorithm involves scrambling pixel positions using an improved zigzag algorithm and substituting pixel values. The simulation and analysis results show that the proposed algorithm has high security and low time complexity. It can resist various attacks, including statistical analysis, differential attacks, brute-force attacks, known plaintext attacks, and chosen plaintext attacks. The experimental simulations and its performance tests prove that the proposed scheme is effective and secure in image data. [ABSTRACT FROM AUTHOR]

Most previous works in the field of nonlinear memristive chaotic systems mainly focus on non‐material memristor while seldom on material memristor. In this letter, the model and the corresponding parameter identification of the Knowm memristor is presented. The chaotic circuit based on the Knowm memristor is designed. Equilibrium points and stability of this Knowm‐memristor‐based chaotic system is analyzed. Nonlinear dynamic behavior is presented by the bifurcation diagram and Lyapunov exponents. Finally, the hardware circuit of this Knowm‐memristor‐based chaotic system is designed and the experimental results are presented for confirmation. [ABSTRACT FROM AUTHOR]

This paper presents a new third-order symmetric difference equation transformed into a 3D discrete symmetric map. The nonlinear dynamics and symmetry of the proposed map are analyzed with two initial conditions for exploring the sensitivity of the map and highlighting the influence of the map parameters on its behaviors, thus comparing the findings. Moreover, the stability of the zero fixed point and symmetry are examined by theoretical analysis, and it is proved that the map generates diverse nonlinear traits comprising multistability, chaos, and hyperchaos, which is confirmed by phase attractors in 2D and 3D space, Lyapunov exponents ( L E s ) analysis and bifurcation diagrams; also, 0-1 test and sample entropy (SampEn) are used to confirm the existence and measure the complexity of chaos. In addition, a nonlinear controller is introduced to stabilize the symmetry map and synchronize a duo of unified symmetry maps. Finally, numerical results are provided to illustrate the findings. [ABSTRACT FROM AUTHOR]

The memristor, a novel device, has been widely utilized due to its small size, low power consumption, and memory characteristics. In this paper, we propose a new three-dimensional discrete memristor map based on coupling a one-dimensional chaotic map amplifier with a memristor. Firstly, we analyzed the memristor model to understand its characteristics. Then, a Simulink model for this three-dimensional discrete memristor map was developed. Lastly, the complex dynamical characteristics of the system were analyzed via equilibrium points, bifurcation diagrams, Lyapunov exponent spectra, complexity, and multistability. This study revealed the phenomena of coexisting attractors and hyperchaotic attractors. Simulink modeling confirmed that the discrete memristors effectively enhanced the chaos complexity in the three-dimensional discrete memristor map. This approach addresses the shortcomings of randomness, the lack of ergodicity, and the small key space in a one-dimensional chaotic map, thereby enriching the theoretical analysis and circuit implementation of chaos. [ABSTRACT FROM AUTHOR]

The objective of this work is to investigate the complex dynamics of a discrete predator–prey system using the method of piecewise constant argument for discretization. An analysis is conducted to examine the presence and stability of fixed points. Furthermore, the system is shown to undergo period-doubling (PD) and Neimark–Sacker (NS) bifurcations by the use of center manifold and bifurcation theories. The feedback and hybrid control strategies are used to regulate the system's bifurcating and chaotic behaviors. Both strategies seem to be effective in managing bifurcation and chaos inside the system. Finally, the main results are validated by numerical evidence. Parameters of the system are varied to produce time graphs, phase portraits, bifurcation diagrams, and maximum Lyapunov exponent (MLE) graphs. The discrete model displays rich dynamics, as seen in the numerical simulations and graphs, indicating a complex and chaotic system. [ABSTRACT FROM AUTHOR]

In continuous neural modeling, memristor coupling has been investigated widely. Yet, there is little research on discrete neural networks in the field. Discrete models with synaptic crosstalk are even less common. In this paper, two locally active discrete memristors are used to couple two discrete Aihara neurons to form a map called DMCAN. Then, the synapse is modeled using a discrete memristor and the DMCAN map with crosstalk is constructed. The DMCAN map is investigated using phase diagram, chaotic sequence, Lyapunov exponent spectrum (LEs) and bifurcation diagrams (BD). Its rich and complex dynamical behavior, which includes attractor coexistence, state transfer, Feigenbaum trees, and complexity, is systematically analyzed. In addition, the DMCAN map is implemented in hardware on a DSP platform. Numerical simulations are further validated for correctness. Numerical and experimental findings show that the synaptic connections of neurons can be modeled by discrete memristor coupling which leads to the construction of more complicated discrete neural networks. [ABSTRACT FROM AUTHOR]

The security performance of image encryption schemes based on chaotic systems has been greatly improved. Specifically, those chaotic systems with high-dimension or special attractors have shown more benefits for enhancing performance. By introducing the tangent function, a 4-dimensional chaotic system with infinite coexisting attractors is constructed. And the dynamical behaviors are analyzed through phase diagrams, bifurcation diagrams, Lyapunov exponent spectrum and spectral entropy complexity diagram. The results demonstrate that the system exhibits self-replication with respect to the initial value y0 and possesses rich dynamical properties, such as infinite coexisting attractors, sensitivity to initial values and period-doubling bifurcation. These characteristics make it suitable for chaotic cryptography applications. Subsequently, a lossless double color image encryption scheme is designed based on the constructed system. The scheme adopts a diffusion-scrambling-diffusion processing method, and cleverly utilizes the information of the original plaintext image in the scrambling process, which significantly enhances the ability to resist known plaintext attacks or selected plaintext attacks. The experimental results verify that the designed algorithm not only effectively encrypts color and grayscale images, but also allows for encryption images of any size. Moreover, the algorithm implementation process is efficient and ensures high security performance, effectively resisting differential attacks, rotation attacks and cropping attacks. This research exploration on the chaotic characteristics of the nonlinear high-dimensional system and its application in image encryption is expected to provide theoretical guidance in the field of secure communication. [ABSTRACT FROM AUTHOR]

This paper introduces a new symmetric fractional-order discrete system. The dynamics and symmetry of the suggested model are studied under two initial conditions, mainly a comparison of the commensurate order and incommensurate order maps, which highlights their effect on symmetry-breaking bifurcations. In addition, a theoretical analysis examines the stability of the zero equilibrium point. It proves that the map generates typical nonlinear features, including chaos, which is confirmed numerically: phase attractors are plotted in a two-dimensional (2D) and three-dimensional (3D) space, bifurcation diagrams are drawn with variations in the derivative fractional values and in the system parameters, and we calculate the Maximum Lyapunov Exponents (MLEs) associated with the bifurcation diagram. Additionally, we use the C 0 algorithm and entropy approach to measure the complexity of the chaotic symmetric fractional map. Finally, nonlinear 3D controllers are revealed to stabilize the symmetric fractional order map's states in commensurate and incommensurate cases. [ABSTRACT FROM AUTHOR]

In recent years, memristors have received widespread attention due to their inherent nonlinear characteristics. To this end, a three-dimensional memristor chaotic system based on secondary nonlinear magnetron memristors has been constructed. Through the analysis of phase portraits, 0–1 tests, Poincare sections, bifurcation diagrams, and Lyapunov exponent spectra, it was found that the system exhibits rich dynamical behaviors, including immovable points, periodic orbits, and chaos. The coexistence attractors of the system were studied, and their basins were also plotted. Comparative complexity analysis indicates that the chaotic system possesses high complexity, thus providing higher security for image encryption. The practicality of the chaotic system was verified through circuit simulation and FPGA. Finally, the chaotic system was applied to DNA image encryption. The focus was on analyzing the histogram, adjacent pixel correlation, information entropy, security, and runtime. The analysis results demonstrate that the algorithm has high security and practicality. [ABSTRACT FROM AUTHOR]

In this paper, we consider the modified single-mode rate equations of semiconductor lasers (SCLs) wherefrom the high dimension of complex dynamics of hyperchaotic islands is studied. Detailed analytical and numerical analysis are investigated under the conjugated impact of three decisive parameters of SCL-systems most specifically, the impact of a new hyperchaotic parameter. Thereby, various quantifying dynamic methods are applied and underscore high-dimensional hyperchaotic states as well as fractal characteristics of new hyperchaotic attractors featuring an entangled central part. The high dimension of the said hyperchaos is also supported by a constructive interference of specific control parameters used which give arise to a complex dynamics of the maximum Lyapunov exponent by means of various two-parameter bifurcation diagrams. Additionally, several cross-sections of Poincaré map bring out odd folded dynamical structures and irregular truncated torus pertaining thereto. In this light, the electronic circuit of the modified model is constructed and simulated in the PSpice environment so as to further support analytical and numerical findings. Finally, controlling hyperchaos is structurally investigated, resulting in the suppression of hyperchaotic states with the generation of continuous wave emission, periodic and quasi-periodic attractors. [ABSTRACT FROM AUTHOR]

Multi-scroll chaotic systems have complex dynamic behaviors, and the multi-scroll chaotic system design and analysis of their dynamic characteristics is an open research issue. This study explores a new multi-scroll chaotic system derived from an asymptotically stable linear system and designed with a uniformly bounded controller. The main contributions of this paper are given as follows: (1) The controlled system can cause chaotic behavior with an appropriate control position and parameters values, and a new multi-scroll chaotic system is proposed using a bounded sine function controller. Meanwhile, the dynamical characteristics of the controlled system are analyzed through the stability of the equilibrium point, a bifurcation diagram, and Lyapunov exponent spectrum. (2) According to the Poincaré section, the existence of a topological horseshoe is proven using the rigorous computer-aided proof in the controlled system. (3) Numerical results of the multi-scroll chaotic system are shown using Matlab R2020b, and the circuit design is also given to verify the multi-scroll chaotic attractors. [ABSTRACT FROM AUTHOR]

Improving neurons in a real physiological environment and studying their electrical behavior is crucial for understanding human cognitive brain functions and neural dynamics. Neuronal cells reside in a complex physiological environment, where the electromagnetic fields generated by ion transmembrane movements affect their discharge activity. Therefore, to better simulate the real conditions of biological neurons, this paper incorporated the characteristics of the memristor and constructed a four-dimensional Morris-Lecar (ML) neuron model by adding a magneto-controlled memristor into the three-dimensional ML neuron model. Through the study of time series diagrams, phase plane diagrams, inter-spike interval (ISI) bifurcation diagrams, we explored the effects of the feedback gain coefficient and the relationship coefficient between membrane potential and magnetic flux on the firing behavior of neurons in the model. It was found that variations in these two parameters can lead to complex firing patterns in neurons. We also utilized the maximum Lyapunov exponent and dissipative theory to investigate the chaotic synchronization phenomenon in the memristor-based ML neuron model. Additionally, we explored the impact of noise on neuronal synchronization behavior within the system, finding that an appropriate noise intensity can effectively accelerate the neurons' attainment of a synchronized state. Finally, applying the chaotic synchronization system to secure Communication, the simulation results and related analysis demonstrate that the system excels in encrypting and decrypting voice signals, offering high levels of security and confidentiality. [ABSTRACT FROM AUTHOR]

Providing robust security within an image cryptosystem during network communication is more essential than ever, highlighting the fundamental aspects of confusion and diffusion. This study presents a novel approach for securely transmitting images through insecure channels, using array index manipulation for scrambling and simple pixel-level confusion and diffusion through XOR operations, effectively involving row and column permutation to achieve robust scrambling. This method operates two layers of confusion by rearranging array indexes using a Tent map and constructing an S-box algorithm derived from the Henon map, while the diffusion process uses a pseudo-random sequence generator based on the Henon map, with their chaotic dynamics analyzed through investigations of Lyapunov exponents and bifurcation diagrams. The expanded chaotic range and improved effects of chaotic maps produce new S-boxes with satisfactory cryptographic performance, including balancedness and non-linearity, with S-box3 achieving the highest non-linearity at 108. Following a performance and security assessment of the proposed technique, it has been found that the entropy value for the cipher image is nearly 7.998; also, the NPCR and UACI values for the cipher images are 99.6 and 33.4, respectively, and it demonstrates closeness to the ideal value. Finally, the proposed method demonstrates high randomness, undergoes evaluation using the NIST test suite, has good operation efficiency, and exhibits resilience against various attack forms such as statistical, differential, data-loss, and noise attacks, affirming its security and relevance for real-time cryptosystems. [ABSTRACT FROM AUTHOR]

This study discusses the dynamic behaviors of the prey–predator model subject to the Allee effect and the harvesting of prey species. The existence of fixed points and the topological categorization of the co-existing fixed point of the model are determined. It is shown that the discrete-time prey–predator model can undergo Flip and Neimark–Sacker bifurcations under some parametric assumptions using bifurcation theory and the center manifold theorem. A chaos control technique called the feedback-control method is utilized to eliminate chaos. Numerical examples are given to support the theoretical findings and investigate chaos strategies' effectiveness and feasibility. Additionally, bifurcation diagrams, phase portraits, maximum Lyapunov exponents, and a graph showing chaos control are demonstrated. [ABSTRACT FROM AUTHOR]

The coupling between neurons can lead to diverse neural network architectures, with the Hopfield neural network (HNN) being particularly noteworthy for its resemblance to human brain function and its potential in modeling chaotic systems. This paper introduces a novel approach: a fractional-order HNN coupled with a hyperbolic tangent-type memristor. Initially, we propose a new model for the hyperbolic tangent-type memristor and fingerprints. Subsequently, we construct a memristor-coupled fractional-order Hopfield neural network (mFOHNN) and explore its dynamic behavior using various analytical tools, including phase diagrams, bifurcation diagrams, Lyapunov exponent diagrams, Poincaré maps, and attractor basins. Our findings reveal rich coexisting bifurcation behavior in the neural network model, influenced by different initial values of coexisting attractors. Finally, we validate the model through analysis and implementation using Multisim circuit simulation software and FPGA hardware, respectively. [ABSTRACT FROM AUTHOR]