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This study investigated how data series length and gaps in human kinematic data impact the accuracy of Lyapunov exponents (LyE) calculations with and without cubic spline interpolation. Kinematic time series were manipulated to create various data series lengths (28% and 100% of original) and gap durations (0.05–0.20 s). Longer gaps generally resulted in significantly higher LyE% error values in each plane in noninterpolated data. During cubic spline interpolation, only the 0.20-second gap in frontal plane data resulted in a significantly higher LyE% error. Data series length did not significantly affect LyE% error in noninterpolated data. During cubic spline interpolation, sagittal plane LyE% errors were significantly higher at shorter versus longer data series lengths. These findings suggest that not interpolating gaps in data could lead to erroneously high LyE values and mischaracterization of movement variability. When applying cubic spline, a long gap length (0.20 s) in the frontal plane or a short sagittal plane data series length (1000 data points) could also lead to erroneously high LyE values and mischaracterization of movement variability. These insights emphasize the necessity of detailed reporting on gap durations, data series lengths, and interpolation techniques when characterizing human movement variability using LyE values. [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]

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]

The dynamics of a four-dimensional fractional-order (FO) dynamical system from the viewpoint of spectral entropy (SE), C 0 complexity, and algorithm 0–1 are presented in detail in this article. The efficiency of these algorithms in the existence of chaos for FO systems has been investigated as well as other methods such as Lyapunov exponents, Lyapunov dimension, and bifurcation diagrams. With Hamilton's energy analysis for the 4D FO system, it is found that chaotic behavior is more dependent on energy consumption. Therefore, it is necessary to design a negative feedback control to reduce energy consumption and suppress chaotic behavior. Finally, we obtain the global Mittag-Leffler positive invariant sets (GMLPISs) and global Mittag-Leffler attractive sets (GMLASs) of the introduced system. Numerical results indicate the effectiveness of complexity and chaos detection methods as well as bound calculation. [ABSTRACT FROM AUTHOR]

Memristives provide a high degree of non-linearity to the model. This property has led to many studies focusing on developing memristive models to provide more non-linearity. This article studies a novel fractional discrete memristive system with incommensurate orders using ϑ i -th Caputo-like operator. Bifurcation, phase portraits and the computation of the maximum Lyapunov Exponent (L E m a x) are used to demonstrate their impact on the system's dynamics. Furthermore, we employ the sample entropy approach (SampEn), C 0 complexity and the 0-1 test to quantify complexity and validate chaos in the incommensurate system. Studies indicate that the discrete memristive system with incommensurate fractional orders manifests diverse dynamical behaviors, including hidden chaos, symmetry, and asymmetry attractors, which are influenced by the incommensurate derivative values. Moreover, a 2D non-linear controller is presented to stabilize and synchronize the novel system. The work results are provided by numerical simulation obtained using MATLAB R2024a codes. [ABSTRACT FROM AUTHOR]

In this study, a chaos theory-based characterization method is proposed to address the nonlinear behavior of acoustic emission (AE) signals during the startup and shutdown phases of dry gas seals. AE signals were collected through a controlled experiment at three distinct phases: startup, normal operation, and shutdown. Analysis of these signals identified a transition speed of 350 r/min between the mixed lubrication (ML) and hydrodynamic lubrication (HL) states. The maximum Lyapunov exponent, correlation dimension, K-entropy, and attractors of the AE signals throughout the operation of the dry gas seal are calculated and analyzed. The findings indicate that the chaotic features of these signals reflect the friction state of the seal system. Specifically, when the maximum Lyapunov exponent is greater than zero, the system exhibits chaotic behavior. The correlation dimension and K-entropy first increase and then decrease in boundary and hybrid lubrication states, while remaining stable in the hydrodynamic lubrication state. Attractors exhibit clustering in boundary lubrication and dispersion in mixed lubrication states. The proposed method achieves an accuracy of 98.6% in recognizing the friction states of dry gas seals. Therefore, the maximum Lyapunov exponent, correlation dimension, and K-entropy are reliable tools for characterizing friction states, while attractors serve as a complementary diagnostic feature. This approach provides a novel framework for utilizing AE signals to evaluate the friction states of dry gas seals. [ABSTRACT FROM AUTHOR]

Purpose: This paper mainly studies the quasi-periodic bursting of a kind of Duffing–Van der Pol system characterized by two disproportionate periodic frequencies. It aims to further improve the quasi-periodic bursting oscillation mechanism and provide theoretical guidance for practical application. Methods: This paper mainly uses the frequency truncation method and fast-slow analysis method. First, by comparing the Lyapunov exponent spectra under different u 1 , the periodic, quasi-periodic and chaotic states of the system are analyzed. The frequency truncation method is employed to convert the system into a fast-slow system characterized by a single slow variable. Then, the dynamic behaviors of the system are analyzed when the frequency ratios are π and 3 . By means of the Lyapunov exponent, spectrum diagram, phase diagram, etc., the dynamic behavior is analyzed, and the quasi-periodic bursting phenomenon is discussed. Finally, by means of the Lyapunov exponential spectrum, the state change of the system from chaos to quasi-periodic and then to periodic is analyzed. Results: By comparing the Lyapunov exponent spectra under different u 1 , the periodic, quasi-periodic and chaotic states are obtained. Using the Lyapunov exponent, spectrum diagram, phase diagram and time history diagram, the dynamic behavior is determined, and the transition mechanism of quasi-periodic bursting oscillation is discussed. In virtue of the Lyapunov exponent spectrum, the state change of the system from chaos to quasi-periodic and then to periodic is analyzed. Conclusions: Compared with the bursting oscillation with integer excitation ratio, when the excitation ratio is irrational, the system needs a certain method to transform and then analyze its dynamic behavior. In this paper, the frequency truncation method is used to transform the system, and then the quasi-periodic bursting phenomenon is studied, which will be helpful to the study of the bursting phenomenon of double-excitation coupled systems. [ABSTRACT FROM AUTHOR]

Two different control methods are proposed in this paper to effectively control the chaotic phenomenon of nonlinear dynamical system. One is a new Hamilton energy feedback control method based on Helmholtz's theorem, which reduces the Lyapunov exponents value of the system by adjusting the feedback gain for controlling chaos. The other is to control the chaos of the system by using delayed feedback control method. Based on this method, we consider the local asymptotic stability of the equilibrium point of the system, and give conditions for the existence of the Hopf bifurcation of the system and the stability domain of the delay parameters. By using the centre manifold theorem and the Poincare normal form method, specific formulas for determining the direction of Hopf bifurcation and the stability of the bifurcation periodic solutions are derived. Finally, the simulation results show that chaos can be controlled by choosing appropriate time-delay parameters. [ABSTRACT FROM AUTHOR]

This paper offers a review while also studying yet unexplored features of the area of chaotic systems subjected to parameter drift of non-negligible rate, an area where the methods of traditional chaos theory are not applicable. Notably, periodic orbit expansion cannot be applied since no periodic orbits exist, nor do long-time limits, since for drifting physical processes the observational time can only be finite. This means that traditional Lyapunov-exponents are also ill-defined. Furthermore, such systems are non-ergodic, time and ensemble averages are different, the ensemble approach being superior to the single-trajectory view. In general, attractors and phase portraits are time-dependent in a non-periodic fashion. We describe the use of general methods which remain nevertheless applicable in such systems. In the phase space, the analysis is based on stable and unstable foliations, their intersections defining a Smale horseshoe, and the intersection points can be identified with the chaotic set governing the core of the drifting chaotic dynamics. Because of the drift, foliations and chaotic sets are also time-dependent, snapshot objects. We give a formal description for the time-dependent natural measure, illustrated by numerical examples. As a quantitative indicator for the strength of chaos, the so-called ensemble-averaged pairwise distance (EAPD) can be evaluated at any time instant. The derivative of this function can be considered the instantaneous (largest) Lyapunov exponent. We show that snapshot chaotic saddles, the central concept of transient chaos, can be identified in drifting systems as the intersections of the foliations, possessing a time-dependent escape rate in general. In dissipative systems, we find that the snapshot attractor coincides with the unstable foliation, and can consist of more than one component. These are a chaotic one, an extended snapshot chaotic saddle, and multiple regular time-dependent attractor points. When constructing the time-dependent basins of attraction of the attractor points, we find that the basin boundaries are time-dependent and fractal-like, containing the stable foliation, and that they can even exhibit Wada properties. In the Hamiltonian case, we study the phenomenon of the break-up of tori due to the drift in terms of both foliations and EAPD functions. We find that time-dependent versions of chaotic seas are not always fully chaotic, they can contain non-chaotic regions. Within such regions we identify time-dependent non-hyperbolic regions, the analogs of sticky zones of classical Hamiltonian phase spaces. We provide approximate formulas for the information dimension of snapshot objects, based on time-dependent Lyapunov exponents and escape rates. Besides these results, we also give possible applications of our methods e.g. in climate science and in the area of Lagrangian Coherent Structures. • The general framework of chaos theory is extended to systems with parameter drift. • Time-dependent Lyapunov exponents, escape rates, measures and dimensions are defined. • Time-dependent chaotic attractors are related to time-dependent chaotic saddles. • Time-dependent chaotic seas contain non-chaotic and non-hyperbolic sub-regions. [ABSTRACT FROM AUTHOR]

Different species in the environment compete with one another for shared resources and suitable habitats. As a result, it becomes crucial to identify the essential components involved in preserving biodiversity. This paper describes the emergence of a wide range of temporal and spatiotemporal dynamics of an intraguild predation (IGP) model with intraspecies competition and Holling type-II response function between the IG-prey and IG-predator. To explore the dynamics of the temporal model, we study the local stability of all the biologically feasible steady states, global stability of the interior steady state, and Hopf bifurcation. Performing partial rank correlation coefficient (PRCC) sensitivity analysis, we picked more sensitive parameters and plotted the stability regions in two-parameter spaces. The dynamics exhibited by the system can demonstrate some important hallmarks noticed in the environments, such as the existence and nonexistence of oscillatory behavior in the IGP model. The analytical formulations for the stability and bifurcation of the coexistence of the steady-state are not easy to obtain, but we have verified numerically that the limit cycle bifurcates from the coexistence of the steady-state and there eventually emerges a chaotic behavior through period-doubling bifurcations. By plotting the largest Lyapunov exponent (LLE), we have verified that the temporal system experiences chaotic behavior. It is demonstrated that in the case of a diffusive system, diffusion may cause the coexistence of the steady state, which is otherwise stable, to become unstable. Criteria for the occurrence of Turing instability connected to the IGP model are derived. Our numerical illustrations demonstrate that diffusion-driven instability develops different stationary patterns based on the different initial conditions and diffusion parameters. The spatiotemporal patterns are observed in the Turing, Hopf–Turing, and Hopf regions. [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]

The phenomenon of adversarial examples is one of the most attractive topics in machine learning research these days. These are particular cases that are able to mislead neural networks, with critical consequences. For this reason, different approaches are considered to tackle the problem. On the one side, defense mechanisms, such as AutoEncoder-based methods, are able to learn from the distribution of adversarial perturbations to detect them. On the other side, chaos theory and Lyapunov exponents (LEs) have also been shown to be useful to characterize them. This work proposes the combination of both domains. The proposed method employs these exponents to add more information to the loss function that is used during an AutoEncoder training process. As a result, this method achieves a general improvement in adversarial examples detection performance for a wide variety of attack methods. [ABSTRACT FROM AUTHOR]

• Both maximum Lyapunov exponents and correlation dimensions were significantly greater during heigher sevoflurane anaesthesia. • Approximate entropy conversely decrease along with deepening of sevoflurane anesthesia. • Chaotic nature of EEG might increase as anaesthesia deepens, despite the decrease in randomness as reflected in entropy. In conscious states, the electrodynamics of the cortex are reported to work near a critical point or phase transition of chaotic dynamics, known as the edge-of-chaos, representing a boundary between stability and chaos. Transitions away from this boundary disrupt cortical information processing and induce a loss of consciousness. The entropy of the electroencephalogram (EEG) is known to decrease as the level of anesthesia deepens. However, whether the chaotic dynamics of electroencephalographic activity shift from this boundary to the side of stability or the side of chaotic enhancement during anesthesia-induced loss of consciousness remains poorly understood. We investigated the chaotic properties of EEGs at two different depths of clinical anesthesia using the maximum Lyapunov exponent, which is mathematically regarded as a formal measure of chaotic nature, using the Rosenstein algorithm. In 14 adult patients, 12 s of electroencephalographic signals were selected during two depths of clinical anesthesia (sevoflurane concentration 2% as relatively deep anesthesia, sevoflurane concentration 0.6% as relatively shallow anesthesia). Lyapunov exponents, correlation dimensions and approximate entropy were calculated from these electroencephalographic signals. As a result, maximum Lyapunov exponent was generally positive during sevoflurane anesthesia, and both maximum Lyapunov exponents and correlation dimensions were significantly greater during deep anesthesia than during shallow anesthesia despite reductions in approximate entropy. The chaotic nature of the EEG might be increased at clinically deeper inhalational anesthesia, despite the decrease in randomness as reflected in the decreased entropy, suggesting a shift to the side of chaotic enhancement under anesthesia. [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 research explores the incorporation of variable order (VO) fractional calculus into the hyperchaotic Lorenz system and studies various chaotic features and attractors. Initially, we propose a variable fractional order hyperchaotic Lorenz system and numerically solve it. The solutions are obtained for multiple choices of control parameters, and these results serve as reference solutions for exploring chaos with the artificial intelligence tool radial basis function network (RBFN). We rebuild phase spaces and trajectories of system states to exhibit chaotic behavior at various levels. To further assess the sensitivity of chaotic attractors, Lyapunov exponents are calculated. The efficacy of the designed computational RBFN is validated through the RMSE and extensive error analysis. The proposed research on AI capabilities aims to introduce an innovative methodology for modeling and analyzing hyperchaotic dynamical systems with variable orders. [ABSTRACT FROM AUTHOR]

The stochastic dynamical ρ4 equation is utilized as a robust framework for modeling the behavior of complex systems characterized by randomness and nonlinearity, with applications spanning various scientific fields. The aim of this paper is to employ an analytical method to identify stochastic traveling wave solutions of the dynamical ρ4 equation. Novel hyperbolic and rational functions are investigated through this method. A Galilean transformation is applied to reformulate the model into a planar dynamical system, which enables a comprehensive qualitative analysis. Additionally, the emergence of chaotic and quasi-periodic patterns following the introduction of a perturbation term is addressed. Simulation results indicate that significant changes in the systems’ dynamic behavior are caused by adjusting the amplitude and frequency parameters. Our findings indicate the impact of the method on system dynamics and its efficacy in analyzing solitons and phase behavior in nonlinear models. These discoveries provide fresh perspectives on how the suggested method can lead to notable shifts in the systems’ dynamic behavior. The effectiveness and practicality of the proposed methodology in scrutinizing soliton solutions and phase visualizations across diverse nonlinear models are underscored by these revelations. [ABSTRACT FROM AUTHOR]

We utilize Lyapunov exponents to quantitatively assess the hyperchaos and categorize the limit sets of complex dynamical systems. While there are numerous methods for computing Lyapunov exponents in integer-order systems, these methods are not suitable for fractional-order systems because of the nonlocal characteristics of fractional-order derivatives. This paper introduces innovative eight-dimensional chaotic systems that investigate fractional-order dynamics. These systems exploit the memory effect inherent in the Grünwald–Letnikov (G-L) derivative. This approach enhances the system's applicability and compatibility with traditional integer-order systems. An 8D Chen's fractional-order system is utilized to showcase the effectiveness of the presented methodology for hyperchaotic systems. The simulation results demonstrate that the proposed algorithm outperforms existing algorithms in both accuracy and precision. Moreover, the study utilizes the 0–1 Test for Chaos, Kolmogorov–Sinai (KS) entropy, the Kaplan–Yorke dimension, and the Perron Effect to analyze the proposed eight-dimensional fractional-order system. These additional metrics offer a thorough insight into the system's chaotic behavior and stability characteristics. [ABSTRACT FROM AUTHOR]

The Maximum Lyapunov Exponent(MLE) measures the sensitivity of a chaotic system to its initial conditions. In chaos theory, systems with positive MLE values are considered chaotic, and larger MLE values generally indicate more robust chaos. A higher MLE suggests a more chaotic and complex behavior, which could be beneficial for encryption as it might provide enhanced security due to increased sensitivity to initial conditions and resistance to attacks. This paper searches for whether a 1D seed chaotic map or a 1D integrated chaotic map is preferred for image encryption. Does the MLE have any role? Firstly, three 1D integrated chaotic maps were designed: Sine-Cubic Integrated Map(SCIM), Tent-Logistic Integrated Map(TLIM), and Sine-Logistic Integrated Map(SLIM). These integrated chaotic maps are designed using the four available seed maps: sine, logistics, cubic, and tent. Thus, we have considered seven 1D chaotic maps to analyse and answer the question. Secondly, image encryption and decryption are performed using the considered seven 1D chaotic maps, one after the other, and the security measures of the encrypted image are analysed using various available tools. The image encryption is performed using block shuffling as diffusion and bit-Xor operation as the confusion process. A comparative analysis is performed using the six quantitative security analysis tools. According to the encryption correlation coefficient value of 0.0017, the Pick Signal-To-Noise Ratio(PSNR) value of 9.204, the Mean Square Error(MSE) value of 7809.1, the Number of Pixel Change Rate(NPCR) value of 95.3903, the Unified Average Change Intensity(UACI) value of 33.3676, and the information entropy value of 7.9635, the sine map is ranked first in security. The comparative analysis result reveals that seed maps give better encrypted image security than integrated chaotic maps. Therefore, integrating 1D chaotic maps is not guaranteed to get a better-secured encrypted image. Further analysis is made to understand if the MLE directly impacts the security of the encrypted process. It is found that the integrated chaotic maps provide a higher MLE. However, in this analysis, we couldn't observe the direct relationship between the MLE and the security of the encrypted image. This suggests that other factors beyond just MLE contribute to the security of the encryption process. [ABSTRACT FROM AUTHOR]

This research explores the complex dynamics of a Novel Four-Dimensional Fractional Supply Chain System (NFDFSCS) that integrates a quadratic interaction term involving the actual demand of customers and the inventory level of distributors. The introduction of the quadratic term results in significantly larger maximal Lyapunov exponents (MLE) compared to the original model, indicating increased system complexity. The existence, uniqueness, and Ulam–Hyers stability of the proposed system are verified. Additionally, we establish the global Mittag-Leffler attractive set (MLAS) and Mittag-Leffler positive invariant set (MLPIS) for the system. Numerical simulations and MATLAB phase portraits demonstrate the chaotic nature of the proposed system. Furthermore, a dynamical analysis achieves verification via the Lyapunov exponents, a bifurcation diagram, a 0–1 test, and a complexity analysis. A new numerical approximation method is proposed to solve non-linear fractional differential equations, utilizing fractional differentiation with a non-singular and non-local kernel. These numerical simulations illustrate the primary findings, showing that both external and internal factors can accelerate the process. Furthermore, a robust control scheme is designed to stabilize the system in finite time, effectively suppressing chaotic behaviors. The theoretical findings are supported by the numerical results, highlighting the effectiveness of the control strategy and its potential application in real-world supply chain management (SCM). [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]

The design of chaotic systems with complex dynamic behaviors has always been a key aspect of chaos theory in engineering applications. This study introduces a novel fractional-order system characterized by hidden dynamics, hyperchaotic behavior, and multi-scroll attractors. By employing fractional calculus, the system's order is extended beyond integer values, providing a richer dynamic behavior. The system's hidden dynamics are revealed through detailed numerical simulations and theoretical analysis, demonstrating complex attractors and bifurcations. The hyperchaotic nature of the system is verified through Lyapunov exponents and phase portraits, showing multiple positive exponents that indicate a higher degree of unpredictability and complexity. Additionally, the system's multi-scroll attractors are analyzed, showcasing their potential for secure communication and encryption applications. The fractional-order approach enhances the system's flexibility and adaptability, making it suitable for a wide range of practical uses, including secure data transmission, image encryption, and complex signal processing. Finally, based on the proposed fractional-order system, we designed a simple and efficient medical image encryption scheme and analyzed its security performance. Experimental results validate the theoretical findings, confirming the system's robustness and effectiveness in generating complex chaotic behaviors. [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 this paper, the qualitative properties of a physically extended six-dimensional Lorenz system, with additional physical terms describing rotation and density, which was proposed in [Moon et al., 2019] have been investigated. The dissipation, invariance, Lyapunov exponents, Kaplan–Yorke dimension, ultimate boundedness and global attractivity of this six-dimensional Lorenz system have been discussed in detail according to the chaotic systems theory. We find that this system exhibits chaos phenomena for a new set of parameters. It is well known that the general method for studying the bounds of a chaotic system is to construct a suitable Lyapunov-like function (or the generalized positive definite and radically unbounded Lyapunov function). However, the higher the dimension of a chaotic system, the more difficult it is to construct the Lyapunov-like function. The innovation of this paper is that we first construct the suitable Lyapunov-like function for this six-dimensional Lorenz system, and then we prove that this system is not only globally bounded for varying parameters, but it also gives a collection of global absorbing sets for this system with respect to all parameters of this system according to Lyapunov's direct method and the optimization method. Furthermore, we obtain the rate of the trajectories going from the exterior to the global absorbing set. Some numerical simulations are presented to validate our research results. Finally, we give a direct application of the results obtained in this paper. According to the results of this paper, we can conclude that the equilibrium point O (0 , 0 , 0 , 0 , 0 , 0) of this system is globally exponentially stable. [ABSTRACT FROM AUTHOR]

Chaotic systems depict complex dynamics, thanks to their nonlinear behaviors. With recent studies on fractional-order nonlinear systems, it is deduced that fractional-order analysis of a chaotic system enriches its dynamic behavior. Therefore, the investigation of the chaotic behavior of a 4D memristive Chen system is aimed in this study by taking the order of the system as fractional. The nonlinear behavior of the system is observed numerically by comparing the fractional-order bifurcation diagrams and Lyapunov Exponents Spectra with 2D phase portraits. Based on these analyses, two different fractional orders (i.e., q = 0.948 and q = 0.97) are determined where the 4D memristive system shows chaotic behavior. Furthermore, a single state fractional-order sliding mode controller (FOSMC) is designed to maintain the states of the fractional-order memristive chaotic system on the equilibrium points. Then, control method results are obtained by both numerical simulations and different illustrative experiments of microcontroller-based realization. As expected, voltage outputs of the microcontroller-based realization are in good agreement with the time series of numerical simulations. [ABSTRACT FROM AUTHOR]

In the actual printing production process, the printing electronic motion membrane is susceptible to transverse vibration caused by interference from air drag, temperature changes, and other external conditions, resulting in membrane wrinkles, slippage, and other phenomena in the transmission process. We studied the bifurcation and chaos movement properties of anisotropic membranes under air drag and temperature. According to D'Alembert's theory and von Kármán's principle, the nonlinear dynamic differential formulas of axially moving anisotropic membranes with gas-thermal-elastic coupling are established. The Galerkin method is applied to discretize the formulas to obtain the state equation of the system. Finally, numerical simulations are performed by applying the fourth-order Runge–Kutta method to analyze the bifurcation and chaos of the system in terms of orthotropic coefficient, dimensionless air drags, and dimensionless temperature. The bifurcation diagrams, Lyapunov exponent diagrams, displacement time-history diagrams, phase-trajectory plane diagrams, and Poincaré diagrams of the membrane system are obtained. The results show that the anisotropic coefficient, dimensionless air drag, and dimensionless temperature significantly impact the investigated nonlinear dynamic of the anisotropic membrane, which provides a theoretical basis for production efficiency and high-quality printing equipment. [ABSTRACT FROM AUTHOR]

We carry out an evolutionary study of the COVID-19 pandemic, focusing on the case of Lebanon. The disease spread exhibits four eruption phases or waves. Chaos theory tools point toward a correlation of events, essentially obeying a quasi-deterministic chaotic regime. The analysis of the time series yields a largest Lyapunov exponent of 0.263, indicative of a chaotic trend. The review of past and recent analyses and modeling of pandemics could assist in the predictabilty of their course of evolution, effective management and decision making for health authorities. [ABSTRACT FROM AUTHOR]

Home area network (HAN) devices send the electricity consumption and other important data to the smart meter that must remain confidential from other devices. This paper proposes a novel one-dimensional hybrid chaotic map. The proposed map shows excellent chaotic properties when analyzed by bifurcation diagram, Lyapunov exponent & Shannon entropy. We further design an encryption strategy for data transfers between the smart meter and HAN devices. The proposed encryption scheme uses the existing lightweight key management in advanced metering infrastructure (LKM-AMI) architecture for data transfers, in which the encrypted data is transferred through an insecure channel and private keys are provided by trusted third party (TTP) through secure channels. The 2-way communication between HAN devices and the smart meter sends messages that are encrypted by using the proposed novel hybrid one-dimensional chaotic map. The encryption strategy mainly consists of three steps. In the first step, the seed and the control parameters are initialized. The second phase generates two intermediate keys using the proposed hybrid chaotic map. In the last phase, we encrypt the message by applying permutation followed by diffusion using intermediate keys. The proposed encryption strategy is resistant to various attacks. [ABSTRACT FROM AUTHOR]

Chaos theory has been widely utilized in password design, resulting in an encryption algorithm that exhibits strong security and high efficiency. However, rapid advancements in cryptanalysis technology have rendered single system generated sequences susceptible to tracking and simulation, compromising encryption algorithm security. To address this issue, we propose an image encryption algorithm based on hybrid heterogeneous time-delay chaotic systems. Our algorithm utilizes a collection of sequences generated by multiple heterogeneous time-delay chaotic systems, rather than sequences from a single chaotic system. Specifically, three sequences are randomly assigned to image pixel scrambling and diffusion operations. Furthermore, the time-delay chaotic system comprises multiple hyperchaotic systems with positive Lyapunov exponents, exhibiting a more complex dynamic behavior than non-delay chaotic systems. Our encryption algorithm is developed by a plurality of time-delay chaotic systems, thereby increasing the key space, enhancing security, and making the encrypted image more difficult to crack. Simulation experiment results verify that our algorithm exhibits superior encryption efficiency and security compared to other encryption algorithms. [ABSTRACT FROM AUTHOR]

Purpose: Planetary gears generate a lot of heat during the meshing transmission process, which results in thermal deformation of the gear teeth and affects the nonlinear dynamic characteristics of the system. It is valuable to study the influence of the thermal effect on the chaotic characteristics of planetary gear systems. Method: Based on the law of thermal deformation, considering the temperature effect and multiple nonlinear factors, including the coupling effects of time-varying meshing stiffness, meshing error, and gear backlash, a nonlinear dynamics model of the planetary gear system is established. The system's differential equation is derived from the gear system dynamics theory and solved using the Runge–Kutta method. The impact law of temperature, time-varying stiffness coefficient, and damping ratio changes on the bifurcation features of the planetary gear system is studied by combining the maximum Lyapunov exponent diagram, bifurcation diagram, time domain diagram, phase diagram, Poincare diagram, and spectrogram. Results: The planetary gear system exhibits chaotic and rich bifurcation characteristics under the coupling effect of multiple nonlinear factors. With the increase in temperature, the system experiences a kinematic process of chaos-four period-two period-single period. As the time-varying stiffness coefficient changes, the system exhibits chaotic, three-periodic, and two-periodic motion states. The effect of the engagement damping ratio change on the nonlinear characteristics of the system is significant when the value of the temperature is in the range of 0 < ∆T < 68 °C. Conclusion: The chaotic motion of the system is weakened under higher temperature and larger damping ratio, which has an obvious effect on improving the system's stability. The pertinent conclusions can be used as future references for gear system development. [ABSTRACT FROM AUTHOR]

The objective of this study was to analyze the complex dynamics of a discrete-time predator-prey system by using the piecewise constant argument technique. The existence and stability of fixed points were examined. It was shown that the system experienced period-doubling (PD) and Neimark-Sacker (NS) bifurcations at the positive fixed point by using the center manifold and bifurcation theory. The management of the system's bifurcating and fluctuating behavior may be controlled via the use of feedback and hybrid control approaches. Both methods were effective in controlling bifurcation and chaos. Furthermore, we used numerical simulations to empirically validate our theoretical findings. The chaotic behaviors of the system were recognized through bifurcation diagrams and maximum Lyapunov exponent graphs. The stability of the positive fixed point within the optimal prey growth rate range A1 < a < A2 was highlighted by our observations. When the value of a falls below a certain threshold A1, it becomes challenging to effectively sustain prey populations in the face of predation, thereby affecting the survival of predators. When the growth rate surpasses a specific threshold denoted as A2, it initiates a phase of rapid expansion. Predators initially benefit from this phase because it supplies them with sufficient food. Subsequently, resource depletion could occur, potentially resulting in long-term consequences for populations of both the predator and prey. Therefore, a moderate amount of prey's growth rate was beneficial for both predator and prey populations. [ABSTRACT FROM AUTHOR]

The objectives of this study are to: (1) develop an approach for extracting representative information from dam behaviour data and (2) diagnose deformation and crack opening behaviour by integrating techniques in transfer entropy (TE), chaos theory and wavelet transform (WT). First, TE is employed to map the information flow between measuring points to select the representative measuring point. The phase space reconstruction of observation data on dam behaviour is implemented, where lag time τ and embedding dimension m are identified by the statistical C–C method. The non-linear structural dynamical behaviour of the dam is then quantified in the form of the largest Lyapunov exponent representing system chaoticity. In addition, a moving window formula is employed to scan along the observation data to ensure an on-line continuous diagnosis. WT may be employed to assist in detecting the time where significant changes occur. A simple 4 degrees of freedom spring-mass-damper (K-M-C) system is employed to illustrate the performance of the proposed approach. The approach is then applied to an actual dam using monitoring data. From the deformation and crack opening behaviour, it is concluded that the studied dam is performing with no significant changes in behaviour over time. Transfer entropy is applied to map the information flow between measuring points. The representative information on structural behaviour of an actual dam is identified. TE and chaos theory are combined to diagnose the dam behaviour. A moving window is adopted to facilitate online statistical diagnosis. [ABSTRACT FROM AUTHOR]

Tinnitus is a heterogeneous condition of hearing a rattling sound when there is no auditory stimulus. This rattling sound is associated with abnormal synchronous oscillations in auditory and non-auditory cortical areas. Since tinnitus is a highly heterogeneous condition with no objective detection criteria, it is necessary to search for indicators that can be compared between and within participants for diagnostic purposes. This study introduces heterogeneous though comparable indicators of tinnitus through investigation of spontaneous fluctuations in resting-state brain dynamics. The proposed approach uses nonlinear measures of chaos theory, to detect tinnitus and cross correlation patterns to reflect many of the previously reported neural correlates of tinnitus. These indicators may serve as effective measures of tinnitus risk even at early ages before any symptom is reported. The approach quantifies differences in oscillatory brain dynamics of tinnitus and normal subjects. It demonstrates that the left temporal areas of subjects with tinnitus exhibit larger lyapunov exponent indicating irregularity of brain dynamics in these regions. More complex dynamics is further recognized in tinnitus cases through entropy. We use this evidence to distinguish tinnitus patients from normal participants. Besides, we illustrate that certain anticorrelation patterns appear in these nonlinear measures across temporal and frontal areas in the brain perhaps corresponding to increased/decreased connectivity in certain brain networks and a shift in the balance of excitation and inhibition in tinnitus. Additionally, the main correlations are lost in tinnitus participants compared to control group suggesting involvement of distinct neural mechanisms in generation and persistence of tinnitus. [ABSTRACT FROM AUTHOR]

This paper introduces a new 3D chaotic attractor in a gene regulatory network. The proposed model has eighteen parameters. Formulas for characteristic numbers of critical points for three-dimensional systems were considered. We show that the three equilibrium points of the new chaotic 3D system are unstable and deduce that the three-dimensional system exhibits chaotic behavior. The possible outcomes of this 3D model were compared with the results of the Chua circuit. The bifurcation structures of the proposed 3D system are investigated numerically, showing periodic solutions and chaotic solutions. Lyapunov exponents and Kaplan-Yorke dimension are calculated. For calculations, the Wolfram Mathematica is used. [ABSTRACT FROM AUTHOR]

Discrete time models, one linear and one non-linear, are investigated, both with a herbivore species that consumes a basal food source species. Results are presented for coexistence of the species and to illustrate chaotic behaviour as parameters are varied in the non-linear model. The results indicate the benefit of fertilization in terms of the region of parameter space for which coexistence occurs. Possible extensions from these models for independent investigations are provided alongside classroom exercises. [ABSTRACT FROM AUTHOR]

Using the sawtooth map as the basis of a coupled map lattice enables simple analytic results to be obtained for the global Lyapunov spectra of a number of standard lattice networks. The results presented can be used to enrich a course on chaos or dynamical systems by providing tractable examples of higher dimensional maps and links to a number of standard results in matrices. A number of suggestions for classroom use are given. [ABSTRACT FROM AUTHOR]

This paper aims to assess the nonlinear vibration of a radial gate induced by flood discharge; the measured acceleration response data of a spillway radial gate are analyzed using the chaos theory. The results show that the vibration responses of the gate at three opening heights present clear chaotic characteristics, and the chaotic characteristics of the lower main beam point are greater than other points. Moreover, the y-direction (vertical) correlation dimensions of the three measuring points on the supporting arm are larger than those of the x-direction (axial) and z-direction (lateral). The vertical vibration of the supporting arm is more complex and presents more uncertainties, which should be paid attention to in the literature. Under three different gate opening heights, the maximum Lyapunov exponent of each measuring point ranges from 0.0246 to 0.0681. In addition, the flow fluctuation load is the main excitation source of the gate vibration chaotic characteristics. [ABSTRACT FROM AUTHOR]

This study explores the application of random matrices to track chaotic dynamics within the Chirikov standard map. Our findings highlight the potential of matrices exhibiting Wishart-like characteristics, combined with statistical insights from their eigenvalue density, as a promising avenue for chaos monitoring. Inspired by a technique originally designed for detecting phase transitions in spin systems, we successfully adapted and applied it to identify analogous transformative patterns in the context of the Chirikov standard map. Leveraging the precision previously demonstrated in localizing critical points within magnetic systems in our prior research, our method accurately pinpoints the Chirikov resonance overlap criterion for the chaos boundary at K ≈ 2.43 , reinforcing its effectiveness. Additionally, we verified our findings by employing a combined approach that incorporates Lyapunov exponents and bifurcation diagrams. Lastly, we demonstrate the adaptability of our technique to other maps, establishing its capability to capture the transition to chaos, as evidenced in the logistic map. [ABSTRACT FROM AUTHOR]

The complexity of hydrological data requires an understanding of its spatiotemporal evolution for effective modeling and prediction. In this study, the underlying spatiotemporal, chaotic, and nonlinear dynamics of rainfall and streamflow data at different timescales in Ceará were assessed by applying chaos theory concepts. This assessment included phase space reconstruction (PSR), correlation dimension, the largest Lyapunov exponent (LLE), nonlinear methods such as recurrence plots, and recurrence quantification analysis techniques. The results indicated that as the timescale increased, the required dimension for PSR decreased for most stations, indicating a shift in the dynamics of the variables. The presence of chaos was confirmed in 78% of the rainfall stations at a monthly timescale by the correlation dimension and positive values of LLE. For streamflow, 73% of monthly data showed indications of chaos using both methods. Values of LLE ranged from 0.02 to 0.24 and − 0.32 to 3.4 for monthly data of rainfall and streamflow, respectively. Most methods showed that the northwestern area of Ceará exhibited higher complexity. Furthermore, the extent of data complexity varied temporally, with the monthly timescale being more complex than the annual timescale. The study revealed that long-term predictions for streamflow may be ineffective for water resources in the region. The results suggested a potential for rainfall predictions up to six years in advance. These findings have implications for developing an integrated water management plan in the region and provide insight into how hydrological variables evolve over time and space. [ABSTRACT FROM AUTHOR]

This paper discusses the complex dynamic behavior of a novel chaotic system, which was firstly established by introducing a memristor into a similar Chen's system. Then by choosing a as the key parameter, we analyze the stability of memristor system based on eigenvalue theory. It is also found that when a cross some critical values, the system can exhibit Neimark–Sacker bifurcation and chaos behaviors. Some numerical simulations including phase diagrams and maximum Lyapunov exponent graph of the memristor-based systems are presented to verify the existence of chaos attractors. Finally, to make the results of this paper useful in the actual situation, such as the design of chaos security algorithm, analog electronic circuit of memristor chaotic system is designed. [ABSTRACT FROM AUTHOR]

This paper presents the temporal and spatiotemporal dynamics of a delayed prey–predator system with a variable carrying capacity. Prey and predator interact via a Holling type-II functional response. A detailed dynamical analysis, including well-posedness and the possibility of coexistence equilibria, has been performed for the temporal system. Local and global stability behavior of the co-existence equilibrium is discussed. Bistability behavior between two coexistence equilibria is demonstrated. The system undergoes a Hopf bifurcation with respect to the parameter β , which affects the carrying capacity of the prey species. The delayed system exhibits chaotic behavior. A maximal Lyapunov exponent and sensitivity analysis are done to confirm the chaotic dynamics. In the spatiotemporal system, the conditions for Turing instability are derived. Furthermore, we analyzed the Turing pattern formation for different diffusivity coefficients for a two-dimensional spatial domain. Moreover, we investigated the spatiotemporal dynamics incorporating two discrete delays. The effect of the delay parameters in the transition of the Turing patterns is depicted. Various Turing patterns, such as hot-spot, coldspot, patchy, and labyrinth, are obtained in the case of a two-dimensional spatial domain. This study shows that the parameter β and the delay parameters significantly instigate the intriguing system dynamics and provide new insights into population dynamics. Furthermore, extensive numerical simulations are carried out to validate the analytical findings. The findings in this article may help evaluate the biological revelations obtained from research on interactions between the species. [ABSTRACT FROM AUTHOR]

The dynamics of a discrete Holling–Tanner model with Beddington–DeAngelis functional response is studied. The permanence and local stability of fixed points for the model are derived. The center manifold theorem and bifurcation theory are used to show that the model can undergo flip and Hopf bifurcations. Codimension-two bifurcation associated with 1:2 resonance is analyzed by applying the bifurcation theory. Numerical simulations are performed not only to verify the correctness of theoretical analysis but to explore complex dynamical behaviors such as period-6, 7, 10, 12 orbits, a cascade of period-doubling, quasi-periodic orbits, and the chaotic sets. The maximum Lyapunov exponents validate the chaotic dynamical behaviors of the system. The feedback control method is considered to stabilize the chaotic orbits. These complex dynamical behaviors imply that the coexistence of predator and prey may produce very complex patterns. [ABSTRACT FROM AUTHOR]

Dynamical chaos is a fundamental manifestation of gravity in astrophysical, many-body systems. The spectrum of Lyapunov exponents quantifies the associated exponential response to small perturbations. Analytical derivations of these exponents are critical for understanding the stability and predictability of observed systems. This paper presents a new model for chaos in systems with eccentric and crossing orbits. Here, exponential divergence is not a continuous process but rather the cumulative effect of an ever-increasing linear response driven by discrete events at regular intervals, i.e. punctuated chaos. We show that long-lived systems with punctuated chaos can magnify Planck length perturbations to astronomical scales within their lifetime, rendering them fundamentally indeterministic. [ABSTRACT FROM AUTHOR]

This article introduces a novel model of asymmetric neural networks combined with fractional difference memristors, which has both theoretical and practical implications in the rapidly evolving field of computational intelligence. The proposed model includes two types of fractional difference memristor elements: one with hyperbolic tangent memductance and the other with periodic memductance and memristor state described by sine functions. The authenticity of the constructed memristor is confirmed through fingerprint verification. The research extensively investigates the dynamics of a coupled neural network model, analyzing its stability at equilibrium states, studying bifurcation diagrams, and calculating the largest Lyapunov exponents. The results suggest that when incorporating sine memristors, the model demonstrates coexisting state variables depending on the initial conditions, revealing the emergence of multi-layer attractors. The article further demonstrates how the memristor state shifts through numerical simulations with varying memductance values. Notably, the study emphasizes the crucial role of memductance (synaptic weight) in determining the complex dynamical characteristics of neural network systems. To support the analytical results and demonstrate the chaotic response of state variables, the article includes appropriate numerical simulations. These simulations effectively validate the presented findings and provide concrete evidence of the system's chaotic behavior. [ABSTRACT FROM AUTHOR]

With the development of sharing economy, a joint distribution mode has been increasingly adopted as the preferred cooperation mode of third-party logistics enterprises to achieve the efficient, resource-saving, and profit-optimal business goals of enterprises. In the joint distribution mode, the distribution price is one of key factors that influences the operation of the joint distribution. Thus, to acquire the optimal pricing for the logistics enterprises, we establish a pricing model based on dynamic game theory for a joint distribution system including one joint distribution company and two express enterprises. In the proposed model, two dimensions of games exist simultaneously, including the game between express competitors and the game between express and distribution enterprises. The multidimensional game leads to more complex system characteristics. Through the stability analysis, we find the Nash equilibrium point and its stability conditions. Numerical simulations are conducted to investigate the complex dynamical behaviors of the game model, such as the system stability region, the bifurcation diagram, the largest Lyapunov exponent, strange attractors, etc. The simulation results indicate that different price adjustment speeds and ranges have a significant impact on the system stability and the profits of all participants in the game. The parameter adjustment control can well dominate the chaotic behaviors of the system. Enterprises should make pricing decisions based on their market positions to promote the continuous and stable development of the operation mode of the multi-agent joint sharing distribution center. [ABSTRACT FROM AUTHOR]

Adaptive network models arise when describing processes in a wide range of fields and are characterized by some specific effects. One of them is mixed dynamics, which is the third type of chaos in addition to the conservative and dissipative types. In this work, we consider a more complex type of connections between network elements—simplex, or higher-order adaptive interactions. Using numerical simulation methods, we analyze various characteristics of mixed dynamics and compare them with the case of pairwise couplings. We found that mixed dynamics in the case of simplex interactions is characterized by a very high similarity of a chaotic attractor to a chaotic repeller, as well as a stronger closeness of the sum of the Lyapunov exponents of the attractor and repeller to zero. This means that in the case of three elements, the conservative properties of the system are more pronounced than in the case of two. [ABSTRACT FROM AUTHOR]