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In insecure communication environments where the communication bandwidth is limited, important image data must be compressed and encrypted for transmission. However, existing image compression and encryption algorithms suffer from poor image reconstruction quality and insufficient image encryption security. To address these problems, this paper proposes an image-compression and encryption scheme based on a newly designed hyperchaotic system and two-dimensional compressed sensing (2DCS) technique. In this paper, the chaotic performance of this hyperchaotic system is verified by bifurcation diagrams, Lyapunov diagrams, approximate entropy, and permutation entropy, which have certain advantages over the traditional 2D chaotic system. The new 2D chaotic system as a pseudo-random number generator can completely pass all the test items of NIST. Meanwhile, this paper improves on the existing 2D projected gradient (2DPG) algorithm, which improves the quality of image compression and reconstruction, and can effectively reduce the transmission pressure of image data confidential communication. In addition, a new image encryption algorithm is designed for the new 2D chaotic system, and the security of the algorithm is verified by experiments such as key space size analysis and encrypted image information entropy. [ABSTRACT FROM AUTHOR]

In this paper, the qualitative behaviors of an important four-dimensional Lorenz system with wild pseudohyperbolic attractor that proposed in (Gonchenko et al 2021 Nonlinearity 34 2018–47) are considered. Here, we prove that the four-dimensional Lorenz system with varying parameters is global bounded according to Lyapunov's direct method. Furthermore, we provide a collection of global absorbing sets, where in addition we obtain the rate of the trajectories going from the exterior to the global absorbing set. In particular, we solve the critical case k → 0 + that cannot be resolved by using the previous methods. The fundamental qualitative behaviors are analyzed theoretically and numerically. We present bifurcation diagrams to further explore the complicated dynamical behaviors of this system. The period-doubling bifurcation phenomenon is found. To illustrate the efficiency of our method, we present numerical simulations to show the validity of our research results. Finally, we present some applications of our research results in this paper. [ABSTRACT FROM AUTHOR]

This paper deals with two topics: roll predictions of marine vessels with machine-learning methods and parameter estimation of unknown ocean disturbances when the amplitude, frequency, offset, and phase are difficult to estimate. This paper aims to prevent the risky roll motions of marine vessels exposed to harsh circumstances. First of all, this study demonstrates complex dynamic phenomena by utilising a bifurcation diagram, Lyapunov exponents, and a Poincare section. Without any observers, an adaptive identification applies these four parameters to the globally exponential convergence using linear second-order filters and parameter estimation errors. Then, a backstepping controller is employed to make an exponential convergence of the state variables to zero. Finally, this work presents the prediction of roll motion using reservoir computing (RC). As a result, the RC process shows good performance for chaotic time series prediction in future states. Thus, the poor predictability of Lyapunov exponents may be overcome to a certain extent, with the help of machine learning. Numerical simulations validate the dynamic behaviour and the efficacy of the proposed scheme. [ABSTRACT FROM AUTHOR]

The bounded confidence model assumes simple continuous opinion dynamics in which agents ignore opinions which are too far from their own. The two initial variants—Hegselmann–Krause (HK) and Deffuant–Weisbuch (DW)—of the model have attracted significant attention since the early 2000s. This paper revisits the version of the HK model applied to a probability distribution, earlier studied by Jan Lorenz. It shows that the bifurcation diagram depends on the parity of the size of the discretisation and that adding a small noise to the initial conditions leads to complex transitions involving several phases. [ABSTRACT FROM AUTHOR]

In this paper, we consider the influence of a nonlinear contact rate caused by multiple contacts in classical SIR model. In this paper, we unversal unfolding a nilpotent cusp singularity in such systems through normal form theory, we reveal that the system undergoes a Bogdanov-Takens bifurcation with codimension 2. During the bifurcation process, numerous lower codimension bifurcations may emerge simultaneously, such as saddle-node and Hopf bifurcations with codimension 1. Finally, employing the Matcont and Phase Plane software, we construct bifurcation diagrams and topological phase portraits. Additionally, we emphasize the role of symmetry in our analysis. By considering the inherent symmetries in the system, we provide a more comprehensive understanding of the dynamical behavior. Our findings suggest that if this occurrence rate is applied to the SIR model, it would yield different dynamical phenomena compared to those obtained by reducing a 3-dimensional dynamical model to a planar system by neglecting the disease mortality rate, which results in a stable nilpotent cusp singularity with codimension 2. We found that in SIR models with the same occurrence rate, both stable and unstable Bogdanov-Takens bifurcations occur, meaning both stable and unstable limit cycles appear in this system. [ABSTRACT FROM AUTHOR]

To accurately depict inventory management over time, this paper introduces a fractional inventory management model that builds upon the existing classical inventory management framework. According to the definition of fractional difference equation, the numerical solution and phase diagram of an inventory management system are obtained by MATLAB simulation. The influence of parameters on the nonlinear behavior of the system is analyzed by a bifurcation diagram and largest Lyapunov exponent (LLE). Combined with the related indexes of time series, the complex characteristics of a quantization system are analyzed using spectral entropy and C0. This study concluded that the changing law of system complexity is consistent with the LLE of the system. By analyzing the influence of order on the system, it is found that the inventory changes will be periodic in some areas when the system is fractional, which is close to the actual conditions of the company's inventory situation. The research results of this paper provide useful information for inventory managers to implement inventory and facility management strategies. [ABSTRACT FROM AUTHOR]

In the paper we constructed the new wheat growth model, based on the generalized logistic equation. Starting from the theoretical framework of the cobweb model, we adapted generalized logistic equation to better fit the real data of wheat prices, according to the presented wheat growth model. The aim of the paper is to present how logistic and generalized logistic equations can be used for both prediction of wheat prices and for the wheat price stability analysis. Data analysis showed better performances of the generalized logistic map in comparation with the conventional logistic map as a main result of this paper. For estimated parameters of the model the bifurcation diagrams also have been presented to show stability of wheat price over time. The conclusion is that the proposed model can be useful in predicting future wheat prices in the short-run period, as well as for the analysis of stability in conditions of uncertainty, which is also a recommendation for the application of the model in the future research. [ABSTRACT FROM AUTHOR]

We consider the Schrödinger operator on the plane with bounded potential , where is a real potential, and are compactly supported complex potentials, and is a small parameter, assuming that the lower part of the spectrum of the one-dimensional Schrödinger operator consists of a pair of isolated eigenvalues and the essential spectrum of the operator has a virtual level at its lower edge and a spectral singularity inside. Additionally, we assume that there is a certain superposition of eigenvalues of the operator with the virtual level and spectral singularity of the operator ; this leads to the emergence of a special threshold in the essential spectrum of the perturbed operator, with the perturbation leading to a bifurcation of this threshold into eigenvalues and resonances with multiplicity doubling. The bifurcation scenario described in this paper is qualitatively different from the previously known ones. [ABSTRACT FROM AUTHOR]

Spiking networks, as the third generation of neural networks, are of great interest today due to their low power consumption in cognitive processes. This important characteristic has caused the hardware implementation techniques of spiking networks in the form of neuromorphic systems attract a lot of attention. For the first time, the focus is on the digital implementation based on CORDIC approximation of the Hindmarsh–Rose (HR) neuron so that the hardware implementation cost is lower than previous studies. If the digital design of a neuron is done efficient, the possibility of implementing a population of neurons is provided for the feasibility of low-consumption implementation of high-level cognitive processes in hardware, which is considered in this paper through edge detector, noise removal and image magnification spiking networks based on the proposed CORDIC_HR model. While using less hardware resources, the proposed HR neuron model follows the behavior of the original neuron model in the time domain with much less error than previous study. Also, the complex nonlinear behavior of the original and the proposed model of HR neuron through the bifurcation diagram, phase space and nullcline space analysis under different system parameters was investigated and the good follow-up of the proposed model was confirmed from the original model. In addition to the fact that the individual behavior of the original and the proposed neurons is the same, the functional and behavioral performance of the randomly connected neuronal population of original and proposed neuron model is equal. In general, the main contribution of the paper is in presenting an efficient hardware model, which consumes less hardware resources, follows the behavior of the original model with high accuracy, and has an acceptable performance in image processing applications such as noise removal and edge detection. [ABSTRACT FROM AUTHOR]

In this paper, an improved two-dimensional chaotic mapping is constructed by introducing a Logistic mapping and a Sine mapping. By studying its Lyapunov and bifurcation diagrams, it is proved that the proposed chaotic mapping has better chaotic properties. A new four-dimensional chaotic system is also proposed, and the richer dynamical behavior of the new system is verified by studying the Lyapunov exponent and bifurcation, SE and C0 complexity, and coexisting attractors of the new system. Then, a new color image encryption algorithm is proposed based on an improved two-dimensional chaotic mapping and a new four-dimensional chaotic system and DNA coding. The improved two-dimensional chaotic mapping and the new four-dimensional chaotic system increase the complexity and randomness of the system, giving the encryption algorithm a higher level of security. In this algorithm, the pixel correlation of the three channels of RGB is first broken by ascending arrangement of disorder, the chaotic sequence generated by the iteration of the four-dimensional chaotic system is applied in the DNA encoding encryption. Finally, a comprehensive evaluation of the algorithm performance is presented. In particular, statistical and information entropy analyses verify that the algorithm in this paper is resistant to statistical attacks based on the frequency of pixel values. The results obtained for the resistance to differential attacks are close to the ideal value, indicating that the algorithm has good resistance to plaintext attacks. By calculating the key space and analysing the key sensitivity, it is verified that the algorithm has a strong resistance to brute-force exhaustive attacks. The decryption algorithm can recover the original image by adding noise to the ciphertext image or cropping part of the region, which proves that the algorithm has strong resistance to interference and strong robustness. Test experiments show that this algorithm has high security and is competitive with other algorithms. [ABSTRACT FROM AUTHOR]

In this paper, a three dimensional discrete eco-epidemiological model with Holling type-III functional response is proposed. Boundedness of the solutions of the system is analyzed. Existence condition and stability of all fixed points are discussed for the proposed model. Furthermore, we obtained the transcritical bifurcation surfaces of the system by bifurcation theory. Based on the explicit criteria for the Neimark Sacker bifurcation and flip bifurcation, we obtained that the system undergoes these two types of bifurcations at the positive fixed point. Then we apply a hybrid control strategy that based on both parameter perturbation and a state feedback strategy to control the Neimark-Sacker bifurcation. Finally, some numerical simulations are carried out to support the analytical results. [ABSTRACT FROM AUTHOR]

The single-phase photovoltaic energy storage inverter represents a pivotal component within photovoltaic energy storage systems. Its operational dynamics are often intricate due to its inherent characteristics and the prevalent usage of nonlinear switching elements, leading to nonlinear characteristic bifurcation such as bifurcation and chaos. In this paper, a deep investigation of a single-phase H-bridge photovoltaic energy storage inverter under proportional–integral (PI) control is made, and a sinusoidal delayed feedback control (SDFC) strategy to mitigate the nonlinear characteristics is proposed. A frequency domain mapping model of the system is established, then, by analyzing the Jacobian matrix and equilibrium points, the bifurcation diagram is formed, and finally, the stable operational domains are determined under double and triple bifurcation parameters. Through simulation experiments, the efficacy of this strategy is validated. The findings show that through the control strategy, the stable operational envelope of the inverter can be greatly expanded and the nonlinear dynamic phenomena can be notably suppressed. [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]

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]

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]

In this paper, we investigate a Leslie-type predator-prey model that incorporates prey harvesting and group defense, leading to a modified functional response. Our analysis focuses on the existence and stability of the system's equilibria, which are essential for the coexistence of predator and prey populations and the maintenance of ecological balance. We identify the maximum sustainable yield, a critical factor for achieving this balance. Through a thorough examination of positive equilibrium stability, we determine the conditions and initial values that promote the survival of both species. We delve into the system's dynamics by analyzing saddle-node and Hopf bifurcations, which are crucial for understanding the system transitions between various states. To evaluate the stability of the Hopf bifurcation, we calculate the first Lyapunov exponent and offer a quantitative assessment of the system's stability. Furthermore, we explore the Bogdanov-Takens (BT) bifurcation, a co-dimension 2 scenario, by employing a universal unfolding technique near the cusp point. This method simplifies the complex dynamics and reveals the conditions that trigger such bifurcations. To substantiate our theoretical findings, we conduct numerical simulations, which serve as a practical validation of the model predictions. These simulations not only confirm the theoretical results but also showcase the potential of the model for predicting real-world ecological scenarios. This in-depth analysis contributes to a nuanced understanding of the dynamics within predator-prey interactions and advances the field of ecological modeling. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Kirchhoff type problems: {−(∫Ω |∇u|²dx)∆u = λu³ + g(u, λ), in Ω, u = 0, on ∂Ω, where λ is a parameter. Under some natural hypotheses on g and Ω, we establish a unilateral global bifurcation result from interval for the above problem. By applying the above result, under some suitable assumptions on nonlinearity, we shall investigate the existence of one-sign solutions for a class of Kirchhoff type problems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the sustained periodic oscillation phenomenon in a three-species delayed predation system with Holling type II functional response and age structure in top predator. The top predator fertility function is regarded as a piecewise continuous smooth function with regard to their maturation period τ 2 . The complicated dynamic behavior is proved by integrated semigroup theory and Hopf bifurcation theorem for semilinear equations with non-dense domain. Through qualitative analysis and bifurcation study of the system, we yield that this system has a nontrivial periodic solution that bifurcates from the positive equilibrium age distribution when bifurcation parameter τ passes through some critical values. Numerical simulations are also provided to illustrate theoretically analytical results. [ABSTRACT FROM AUTHOR]

In this paper, a bogie system with time delay is established. The centre manifold theorem and norm form theory are used to reduce the infinite dimensional delay system to the finite dimensional ordinary differential system, and the bifurcation behaviour of the bogie system is analysed. The MatCont toolkit was used to verify the subcritical bifurcation in the system. In addition, the influence of parameters on the bifurcation diagram of the system and the influence of parameters on the critical speed of the bogie system during the hunting motion are also analysed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the use of Lambert W functions in the analysis of nonlinear dynamics and bifurcations of a new two-dimensional γ -Ricker population model. Through the use of such transcendental functions, it is possible to study the fixed points and the respective eigenvalues of this exponential diffeomorphism as analytical expressions. Consequently, the maximum number of fixed points is proved, depending on whether the Allee effect parameter γ is even or odd. In addition, the analysis of the bifurcation structure of this γ -Ricker diffeomorphism, also taking into account the parity of the Allee effect parameter, demonstrates the results established using the Lambert W functions. Numerical studies are included to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the possibilities of complex nonlinear dynamic signal generation using a simple photorefractive two-wave mixing system without any feedback using numerical simulations. The novel idea is to apply a sinusoidal electric field to the system inroder to extract nonlinear dynamic behavior. The mathematical model of the system was constructed using Kogelnick's coupled wave equations and Kukhtarev's material equation. The spatio-temporal evolution of the system was simulated in discrete steps numerically. The temporal evolution of the output light intensity exhibits period doubling, behavior which is a characteristic feature of complex nonlinear dynamic systems. A bifurcation diagram and Lyapunov exponentials confirm the presence of the period-doubling route to chaos in the system. The observed complex signal pattern varies uniformly with respect to the amplitude of the applied field, indicating a controllable nonlinear dynamic behavior. Such a system can be very useful in applications such as photonic reservoir computing, in-materio computing, photonic neuromorphic networks, complex signal detection, and time series prediction. [ABSTRACT FROM AUTHOR]

In this paper, the characteristics of absolute value memristors are verified through the circuit implementation and construction of a chaotic system with a conditional symmetric fractional-order memristor. The dynamic behavior of fractional-order memristor systems is explored using fractional-order calculus theory and the Adomian Decomposition Method (ADM). Concurrently, the investigation probes into the existence of coexisting symmetric attractors, multiple coexisting bifurcation diagrams, and Lyapunov exponent spectra (LEs) utilizing system parameters as variables. Additionally, the system demonstrates an intriguing phenomenon known as offset boosting, where the embedding of an offset can adjust the position and size of the system's attractors. To ensure the practical applicability of these findings, a fractional-order sliding mode synchronization control scheme, inspired by integer-order sliding mode theory, is designed. The rationality and feasibility of this scheme are validated through a theoretical analysis and numerical simulation. [ABSTRACT FROM AUTHOR]

10.3390/sym13020341 14 Grassi G. Chaos in the Real World: Recent Applications to Communications, Computing, Distributed Sensing, Robotic Motion, Bio-Impedance Modelling and Encryption Systems. M. Berezowski and M. Lawnik, in the paper "Homotopic Parametric Continuation Method for Determining Stationary States of Chemical Reactors with Dispersion" [[6]], have illustrated an approach that combines the homotopy method and the parametric continuation method to solve a system of nonlinear differential equations. Referring to chaotic systems, it is well-known that they are nonlinear dynamical systems, which are distinguished by sensitive dependence on initial conditions and by having evolution through phase space that appears to be quite random [[1]]. Additionally, G. Grassi has reviewed the applications of chaos in robotic motion, in chaos-based logic gates for computing and in the hardware implementations of encryption systems [[14]]. [Extracted from the article]

Summary: In this paper, a new fifth‐order eight‐wing chaotic system is proposed on the basis of a five‐dimensional four‐wing chaotic system. The resulting new system is used as a basis for introducing a method that can control both the number of wings of the system and the direction of the change in the number of wings. Add a rounding function with a cosine function or a sine function to the system equation. The number of wings of the control system is controlled by changing the parameter values of the rounding function. This method can significantly increase the complexity of the system when applied in multiple directions of the system equations. And this method can achieve the purpose of controlling the wing number of the whole system by changing only one parameter. By observing the phase diagram and bifurcation diagram of the system, it is found that it has more complex chaotic properties and rich dynamical behavior and there are coexisting attractors. The chaotic properties of the system are confirmed by observing the Poincaré cross section and the 0–1 test results of the system. Afterward, the simulation circuit of the fifth‐order system is built in the simulation software Multisim. And the obtained phase diagram is consistent with the results obtained in MATLAB which confirmed the realizability of the system. An FPGA hardware implementation of this fifth‐order eight‐wing system has also been completed at the end of the paper to demonstrate the implementability of the system. And the results are highly consistent with those obtained in the MATLAB software. The fifth‐order chaotic system is combined with the dislocation algorithm and diffusion algorithm to apply to the image encryption system. The encryption system contains a variety of encryption algorithms for secondary encryption of images including Arnold dislocation and repeat‐free scrambling algorithm. The diffusion algorithm includes the addition of the modulus length diffusion and multiplicative operation diffusion based on the GF(257) domain. After a series of performance analysis, the encryption system is proved to have good secrecy effect and great application value in the image encryption neighborhood. [ABSTRACT FROM AUTHOR]

In this paper, we discussed the impossibility of achieving zero-covid cases per day for all time with the help of fuzzy theory, while how a single case can trigger chaotic situation in the nearby city is elaborated using multi-scrolls. To accomplish this goal, we consider the number of new cases per day; x 1 to be the preferred state variable by restricting its value to the interval (0, 1). One can need to think of x 1 as a member of a fuzzy set and provide that set with appropriate membership functions. Moreover, how a single incident in one city can spread chaos to other cities is also addressed at length, using multi-scroll attractors and the signal excitation function. In addition, a bifurcation diagram of daily new instances vs the parameter α 1 is shown, elaborating that daily new cases may show a decrease under strict rules and regulations, but can again lead to chaos. Apart from biologist, this paper can play vital role for engineers as well in a sense that, a signal function can be embedded in non-symmetric systems for the creation of multi-scroll attractors in all directions using a generalized algorithm that has been designed in the current work. Finally, it is our future target to show that the covid is leading towards influenza and will be no more dangerous as was in the past. [ABSTRACT FROM AUTHOR]

Since coronary artery disease is the leading global cause of mortality and morbidity, this paper investigates the chaos suppression of coronary artery systems. The motivation of the paper is to discuss and analyze coronary artery disease in the field of dynamics. Firstly, the mathematic model of coronary artery systems is formulated and the properties of this model are illustrated by bifurcation diagram, information entropy analysis, phase plane trajectory, and Poincaré section. With regard to the uncertainties of coronary artery systems, the disturbance observer technique is adopted. Meanwhile, the smooth second-order sliding mode controller is designed to suppress the chaos phenomenon. In light of the combination of the controller and observer, the stability of such a closed-loop system is proven in the sense of Lyapunov. Finally, some numerical simulations demonstrate the feasibility and validity of such design. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to construct a multiwing chaotic system that has hidden attractors with multiple stable equilibrium points. Because the multiwing hidden attractors chaotic systems are safer and have more dynamic behaviors, it is necessary to construct such a system to meet the needs of developing engineering. Design/methodology/approach: By introducing a multilevel pulse function into a three-dimensional chaotic system with two stable node–foci equilibrium points, a hidden multiwing attractor with multiple stable equilibrium points can be generated. The switching behavior of a hidden four-wing attractor is studied by phase portraits and time series. The dynamical properties of the multiwing attractor are analyzed via the Poincaré map, Lyapunov exponent spectrum and bifurcation diagram. Furthermore, the hardware experiment of the proposed four-wing hidden attractors was carried out. Findings: Not only unstable equilibrium points can produce multiwing attractors but stable node–foci equilibrium points can also produce multiwing attractors. And this system can obtain 2N + 2-wing attractors as the stage pulse of the multilevel pulse function is N. Moreover, the hardware experiment matches the simulation results well. Originality/value: This paper constructs a new multiwing chaotic system by enlarging the number of stable node–foci equilibrium points. In addition, it is a nonautonomous system that is more suitable for practical projects. And the hardware experiment is also given in this article which has not been seen before. So, this paper promotes the development of hidden multiwing chaotic attractors in nonautonomous systems and makes sense for applications. [ABSTRACT FROM AUTHOR]

With the deep extension of coal mining in China, fault water inrush has become one of the major disasters threatening the safety production of coal mine. Based on the control equations of steady state and non-Darcy seepage in fractured rock mass, the multi-parameter nonlinear dynamic seepage equations of fractured rock mass are established in this paper. Based on the nonlinear dynamics theory, the function of the state variable in the system is derived, and the influence of the gradual change of non-Darcy flow factors on the structural stability of seepage system is studied. The research achievements show that there are three branches in the equilibrium state of the seepage system. Specifically, the stability of the equilibrium state changes abruptly near the limit parameter. The seepage dynamic system of fractured rock mass has the delayed bifurcation, and the coal mine disaster such as fault water inrush occurs easily at the bifurcation point. The research results are of great significance to enrich the theory of fault water inrush in coal mine, and to reveal the disastrous mechanism of fault water inrush and guide its prevention and control technology in coal mine, which can provide the theoretical reference for predicting the water seepage stability in fractured rock mass. [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. [ABSTRACT FROM AUTHOR]

This paper examines the dynamic behavior of a particular category of discrete predator-prey system that feature both fear effect and refuge, using both analytical and numerical methods. The critical coefficients and properties of bifurcating periodic solutions for Flip and Hopf bifurcations are computed using the center manifold theorem and bifurcation theory. Additionally, numerical simulations are employed to illustrate the bifurcation phenomenon and chaos characteristics. The results demonstrate that period-doubling and Hopf bifurcations are two typical routes to generate chaos, as evidenced by the calculation of the maximum Lyapunov exponents near the critical bifurcation points. Finally, a feedback control method is suggested, utilizing feedback of system states and perturbation of feedback parameters, to efficiently manage the bifurcations and chaotic attractors of the discrete predator-prey model. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behaviors of a discrete-time fractional-order population model are considered. The stability analysis and the topological classification of the model at the fixed point have been investigated. It is shown that the model undergoes flip and Neimark–Sacker bifurcations around the co-existence fixed point by using the bifurcation and the normal form theory. These bifurcations lead to chaos when the parameter changes at critical point. In order to control chaotic behavior in the model result from Neimark–Sacker bifurcation, the OGY feedback method has been used. Furthermore, some numerical simulations, including bifurcation diagrams, phase portraits and maximum Lyapunov exponents of the presented model are plotted to support the correctness of the analytical results. The positive Lyapunov exponents demonstrate that chaotic behavior exists in the considered model. [ABSTRACT FROM AUTHOR]

To improve the salient region security and transmission efficiency of remote sensing images, a multiple remote sensing image (MRSIs) encryption scheme based on saliency extraction and magic cube circular motion is proposed in this paper. The scheme provides two tiers of privacy protection for salient regions (airport locations) in remote sensing images. First, a 4D improved discrete tabu learning neuron (4D-IDTLN) chaotic system is proposed. The analysis of phase diagrams, the Lyapunov exponents (LEs) spectrum and bifurcation diagrams of the system show that the system exhibits rich dynamic behaviors. Second, the salient regions of remote sensing images are classified and extracted via knowledge-oriented saliency (KOS) and vision-oriented saliency (VOS) techniques to create a mask contour positioning model (MCPM) for the salient regions, which is then encrypted. Then, the MRSIs are fused into a cube, which is encrypted using magic cube circular motion and chaotic sequences and further encrypted using closed-loop diffusion. Finally, the security of the proposed encryption scheme is evaluated. The results indicate that the scheme provides higher security and better transmission efficiency for MRSIs. [ABSTRACT FROM AUTHOR]

The tapping-mode atomic force microscope (TM-AFM) is widely used today; however, improper matching between the operating medium and the sampling time may lead to inaccurate measurement results. The relationship between the damping coefficient and the steady state of the TM-AFM microcantilever is investigated in this paper using multiple stability theory. Firstly, the effects of changes in dimensionless linear damping coefficients and dimensionless piezoelectric film damping coefficients on the motion stability of the system are examined using bifurcation diagrams, phase trajectories, and domains of attraction. Subsequently, the degrees of effect of the two damping coefficients on the stability of the system are compared. Finally, the bi-parametric bifurcation characteristics of the system under a specific number of iterative cycles are investigated using the bi-parametric bifurcation diagram in conjunction with the actual working conditions, and the boundary conditions for the transition of the system's motion from an unstable state to a stable state are obtained. The results of the study show that to ensure the accuracy and reliability of the individual measurement data in 500 iteration cycles, the dimensionless linear damping coefficient must be greater than 0.01014. Our results will provide valuable references for TM-AFM measurement media selection, improving TM-AFM imaging quality, measurement accuracy and maneuverability, and TM-AFM troubleshooting. [ABSTRACT FROM AUTHOR]

This paper concerns a discrete-time phytoplankton–zooplankton model in which the effects of toxin produced by phytoplankton and refuges provided for phytoplankton on the interactions between phytoplankton and zooplankton are considered. We first discuss the existence and stability of fixed points. When two parameters change, it is shown that there exist some codimension-two bifurcations, including fold-flip bifurcation and strong resonance bifurcations. Moreover, we give the corresponding bifurcation diagrams, phase diagrams and maximum Lyapunov exponent diagrams. The results show that phytoplankton toxin and refuges play important roles in the occurrence and termination of algal blooms in freshwater lakes. [ABSTRACT FROM AUTHOR]

The paper is organized to study some mathematical properties and dynamics of a simple Cournot duopoly game based on a computed quadratic cost. The time evolution of this game is described by a two-dimensional noninvertible discrete time map using the bounded rationality mechanism. For this map, some dynamic characteristics such as multistability and synchronization are investigated. Its equilibrium points are obtained for the asymmetric case, and their conditions of stability are obtained. Our results investigate that the Nash equilibrium point may be unstable due to flip bifurcation and under certain parameter values, and Neimark–Sacker bifurcation is born after the period-4 cycle. Through some restrictions, the coordinate axes of the map construct an invariant manifold, and therefore, their dynamics can be analyzed by using a one-dimensional map. In the symmetric case, both firms behave identically, and this implies that the diagonal set forms an invariant manifold, and hence the synchronization phenomena take place. Furthermore, the global bifurcation of the map is confirmed through contact between critical curves and the boundaries of infeasible domains. [ABSTRACT FROM AUTHOR]

In this paper, the singularly perturbed modified Gardner equation is considered. Firstly, for the unperturbed equation, under certain parameter conditions, we obtain the exact expressions of kink wave solution and antikink wave solution by using the bifurcation method of dynamical systems. Then, the persistence of the kink and antikink wave solutions of the perturbed modified Gardner equation is studied by exploiting the geometric singular perturbation theory and the Melnikov function method. When the perturbation parameter is sufficiently small, we obtain the sufficient conditions to guarantee the existence of kink and antikink wave solutions. [ABSTRACT FROM AUTHOR]

In order to reveal the general escape mechanism of non-hyperbolic chaotic invariant sets under both weak noise limit and finite noise intensity, the simplest example of Hénon map, which represents the stretching and folding of the phase space, is taken to study the escape mechanism under two kinds of global bifurcation: the fractal boundary crisis and the attractor contact crisis. In this paper, we revealed the general exit mechanism by analyzing the escape paths derived from the shooting method and Monte Carlo simulation. Finally, to further demonstrate universality, an example of high-dimensional differential dynamical system, namely the transient chaos Duffing oscillator, is examined, which underscores the main idea of this paper analyzing specific deterministic structures not only enhances the comprehensibility of the escape process, but also allows for predictions of general escape behavior under weak noise intensity. [ABSTRACT FROM AUTHOR]

The linear and weakly nonlinear stability analyses of Darcy–Bénard convection of a Newtonian fluid experiencing a uniform vertical cross-flow is investigated in the paper for various aspect ratios. Making use of the Maclaurin series representation, an expression for axial eigenfunctions is obtained with the radial eigenfunction being a Bessel function of first kind. These eigenfunctions are influenced by the Peclet number, Pe, the non-dimensional number that signifies the rate of vertical cross-flow. The modified-Vadasz-Lorenz model obtained in this paper has newly defined non-dimensional parameters that capture the influence of vertical cross-flow. From the linear stability analysis, it is found that the effect of introducing vertical cross-flow is to stabilize the system. Using a weakly nonlinear stability analysis, the closed-form expression of the Hopf-Rayleigh number as a function of Pe is obtained. Furthermore, the behavior of the modified-Vadasz–Lorenz model is analyzed using the largest Lyapunov exponent and the bifurcation diagram. This gives information about the intensity of chaos and occurrence of the periodic motion. We observe that the influence of vertical cross-flow is to increase the value of the Hopf–Rayleigh number and thereby to delay the onset of chaos. Furthermore, the appearance of the first periodic point is preponed due to the vertical cross-flow. As the rate of vertical cross-flow increases, the intensity of chaos decreases, thereby indicating that the effect of introducing vertical cross-flow is to suppress chaos. [ABSTRACT FROM AUTHOR]

In this paper, we study the q-deformed logistic map in Mann orbit (superior orbit) which is a two-step fixed point iterative algorithm. The main aim of this paper is to investigate the whole dynamical behavior of the proposed map through various techniques such as fixed point and stability approach, time-series analysis, bifurcation plot, Lyapunov exponent and cobweb diagram. We notice that the chaotic behavior of q-deformed logistic map can be controlled by choosing control parameters carefully. The convergence and stability range of the map can be increased substantially. Moreover, with the help of bifurcation diagrams, we prove that the stability performance of this map is larger than that of existing other one dimensional chaotic maps. This map may have better applications than that of classical logistic map in various situations as its stability performance is larger. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the spatiotemporal pattern formations and bifurcations for a nondegenerate reaction-diffusion rabies SIR model which was used to explain the epidemiological patterns observed in Europe. First, by using the iteration methods, we are able to show the global existence and boundedness of in-time solutions of the parabolic system. Second, for the ODEs, we analytically prove the phenomena observed by Anderson et al. [Nature, 289 (1981), pp. 765-771]: if the carrying capacity k is smaller than some positive k*, then rabies eventually dies out; if k is larger than k*, then the rabies prevails. Moreover, if k* < k < k for some positive ... > k*, then the endemic equilibrium solution is (locally asymptotically) stable, while it is unstable if k > ... In particular, at k = ..., the loss of the stability of the endemic equilibrium leads to a Hopf bifurcation. Finally, for the PDEs, we derive sufficient conditions on the diffusion rates so that under these conditions, Turing instability of both the endemic equilibrium solution and the Hopf bifurcating spatially homogeneous periodic solutions can occur. Once Turing instability of the solution (equilibrium or periodic solution) occurs, it is observed numerically that the system might have new spatiotemporal patterns. [ABSTRACT FROM AUTHOR]

This paper studies the dynamic analysis, circuit implementation and circuit synchronization of a class of fractional-order memristor chaotic system based on Chua's circuit. First, the dynamic characteristics of the system was analyzed with the common chaotic description methods, such as Lyapunov exponent spectrum(LEs), bifurcation diagram, Poincaré diagram, phase trajectory diagram and Complexity spectral entropy (Complexity SE). It can be found that the small change of the initial value can make the dynamic behavior of the system change between the periodic window and the chaotic region. When fractional order q and system parameters are used as a variable, the dynamic behavior of the system can be changed. Second, the system circuit model was modeled and simulated, and then the consistency with the numerical simulation was verified through the implementation of the circuit. Finally, through the design of fractional-order memristor circuit, the circuit synchronization of the system was realized. Therefore, this paper not only studies the dynamic behavior of the system through numerical analysis. Moreover, the effectiveness of the system design is further illustrated by the implementation of analog circuit. [ABSTRACT FROM AUTHOR]

This paper develops a spatial platform for economic agglomerations that can represent a hierarchical structure of cities, towns, and so on. A global system models geographical distribution of a system of cities, while population size and local geography of each city are modeled by an individual square lattice network. The mechanism of economic agglomeration is described by economic geography models with the replicator dynamics. As a major theoretical contribution of this paper, we elucidate the bifurcation mechanism of the mono-centric distribution in a square domain. Such bifurcation expresses how satellite places appear around a large city. This bifurcation behavior, called sustaining bifurcation, is different from symmetry-breaking bifurcation studied in nonlinear mathematics, and is not given much attention in economic geography, despite its importance. An air-network of cities with the seven largest hub airports in USA is employed as a realistic example, and is modeled as a global–local system comprising a series of local square lattices with different sizes connected by an equidistant economy. This system for an economic geography model displays successive sustaining bifurcations occurring many times leading to gradual emergence of satellite places around a large city. [ABSTRACT FROM AUTHOR]

This paper analytically and numerically investigates the dynamical characteristics of a fractional Duffing–van der Pol oscillator with two periodic excitations and the distributed time delay. First, we consider the pitchfork bifurcation of the system driven by both a high-frequency parametric excitation and a low-frequency external excitation. Utilizing the method of direct partition of motion, the original system is transformed into an effective integer-order slow system, and the supercritical and subcritical pitchfork bifurcations are observed in this case. Then, we study the chaotic behavior of the system when the two excitation frequencies are equal. The necessary condition for the existence of the horseshoe chaos from the homoclinic bifurcation is obtained based on the Melnikov method. Besides, the parameters effects on the routes to chaos of the system are detected by bifurcation diagrams, largest Lyapunov exponents, phase portraits, and Poincaré maps. It has been confirmed that the theoretical predictions achieve a high coincidence with the numerical results. The techniques in this paper can be applied to explore the underlying bifurcation and chaotic dynamics of fractional-order models. [ABSTRACT FROM AUTHOR]

A new fractional-order cellular neural network (CNN) system is solved using the Adomian decomposition method (ADM) with the hyperbolic tangent activation function in this paper. The equilibrium point is analyzed in this CNN system. The dynamical behaviors are studied as well, using a phase diagram, bifurcation diagram, Lyapunov Exponent spectrum (LEs), and spectral entropy (SE) complexity algorithm. Changing the template parameters and the order values has an impact on the dynamical behaviors. The results indicate that rich dynamical properties exist in the system, such as hyperchaotic attractors, chaotic attractors, asymptotic periodic loops, complex coexisting attractors, and interesting state transition phenomena. In addition, the digital circuit implementation of this fractional-order CNN system is completed on a digital signal processing (DSP) platform, which proves the accuracy of ADM and the physical feasibility of the CNN system. The study in this paper offers a fundamental theory for the fractional-order CNN system as it applies to secure communication and image encryption. [ABSTRACT FROM AUTHOR]

Further exploration into the influence of a memristor on the behavior of chaotic systems deserves attention. When constructing memristor chaotic systems, it is commonly believed that increasing the number of memristors will lead to better system performance. This paper proposes a class of chaotic maps with different discrete memristors, achieved through internal perturbation based on the Sine map. The I-V curve of the discrete memristor has a symmetrical structure. The dynamic characteristics of the designed system are analyzed using the chaotic attractor phase diagram, Lyapunov exponent (LE) spectrum, and bifurcation diagram. Numerical simulations demonstrate that internal perturbations of discrete memristors enhance the Sine map's chaotic characteristics, expand the chaos range, and improve the ergodicity and LE value. Moreover, the type of discrete memristors has a significant impact on the dynamic characteristics of the system, while the number of discrete memristors has little influence. Therefore, in this paper, a direction for the design of a discrete memristor chaotic system is provided. Finally, a discrete memristor chaotic map with a simple structure and better performance is selected. Based on this, a pseudo-random sequence generator is designed, and the generated sequence passes the National Institute of Standards and Technology (NIST) test. [ABSTRACT FROM AUTHOR]

In this paper, a four-dimensional chaotic system based on a flux-controlled memristor with cosine function is constructed. It has infinitely many equilibria. By changing the initial values x (0) , z (0) and u (0) of the system and keeping the parameters constant, we obtained the distribution of infinitely many single-wing and double-wing attractors along the u -coordinate, which verifies the initial-offset boosting behavior of the system. Then the complex dynamical behavior of the system is studied in detail through the phase portraits of coexisting attractors, the average value of state variables, Lyapunov exponent spectrum, bifurcation diagram, attraction basin and the complexity of spectral entropy (SE). In addition, the simulation of the Multisim circuit is also carried out, and the results of numerical simulation and analog circuit simulation are consistent. Finally, the chaotic sequence generated by the system is applied to image encryption, and according to the performance analysis, the proposed chaotic system has good security performance. [ABSTRACT FROM AUTHOR]

He also wrote about male-male combat and morphology in the Caprinidae (sheep and goats) alone and with his advisor (Schaffer 1968, Reed and Schaffer 1972a, Schaffer and Reed 1972b). Bill moved on to work on nonlinear dynamics in disease systems with his Danish collaborator Lars Olsen for many years (Olsen et al. 1988, Olsen and Schaffer 1990). In their 1986 TREE paper, Bill and his then-student Mark Kot argued that chaotic dynamics were likely widespread in ecology, and that this might necessitate a fundamental reappraisal of many ideas in population and community ecology (Schaffer and Kot 1986). Ecologist William (Bill) M. Schaffer died on 16 January 2021, in Tucson, Arizona, USA. [Extracted from the article]

In this paper, we consider the complicated dynamics of a delay-based photonic reservoir computing system. Since conventional computer architectures are approaching their limit, it is imperative to find new, efficient and fast ways of data processing. Photonic reservoir computing (RC) is a promising way which combines the computational capabilities of recurrent neural networks with high processing speed and energy efficiency of photonics. As the RC system is very promising, we analyze its dynamics so that we can make better use of it. In this paper, we mainly focus on its double Hopf bifurcation. We first analyze the existence of double Hopf bifurcation points. Then we use DDE-BIFTOOL to draw the bifurcation diagrams with respect to two bifurcation parameters, i.e. feedback strength η and delay τ , and give a clear picture of the double Hopf bifurcation points of the system. These figures show stability switches and the existence of double Hopf bifurcation points. Finally, we employ the method of multiple scales to obtain their normal forms, use the method of normal form to unfold and classify their local dynamics. The classification and unfolding of these double Hopf bifurcation points are obtained. Three types of double Hopf bifurcations are found. We verify the results by numerical simulations and find its complicated behavioral dynamics. For example, there exist stable equilibrium, stable periodic and quasi-periodic solutions in distinct regions. The discovered rich dynamical phenomena can help us to choose suitable values of parameters to achieve excellent performance of RC. [ABSTRACT FROM AUTHOR]