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We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation theorem to prove that there exists a connected branch of positive solutions bifurcating from infinity when the parameter goes to zero. Moreover, if the nonlinearity satisfies additional conditions near zero, we establish a global bifurcation result, and discuss the number of positive solution(s) with respect to the parameter using bifurcation theory and degree theory. [ABSTRACT FROM AUTHOR]

The global bifurcation diagrams for two different one-parametric perturbations ( + λ x and + λ x 2 ) of a dissipative scalar nonautonomous ordinary differential equation x ′ = f (t , x) are described assuming that 0 is a constant solution, that f is recurrent in t, and that its first derivative with respect to x is a strictly concave function. The use of the skewproduct formalism allows us to identify bifurcations with changes in the number of minimal sets and in the shape of the global attractor. In the case of perturbation + λ x , a so-called generalized pitchfork bifurcation may arise, with the particularity of lack of an analogue in autonomous dynamics. This new bifurcation pattern is extensively investigated in this work. [ABSTRACT FROM AUTHOR]

This article explores the nonlinear dynamical analysis and control of a 2 − DOF system that emulates the lateral vibration of an asymmetric rotor model under external, multi-parametric, and mixed excitation. The linear integral resonant controller (L I R C) has been coupled to the rotor as an active damper through a magnetic actuator. The complete mathematical model, governing the nonlinear interaction among the rotor, controller, and actuator, is derived based on electromagnetic theory and the principle of solid mechanics. This results in a discontinuous 2 − DOF system coupled with two 1 / 2 − DOF systems, incorporating the rub-impact effect between the rotor and stator. The complicated mathematical model is investigated using analytical techniques, employing the perturbation method, and validated numerically through time response, basins of attraction, bifurcation diagrams, 0 − 1 chaotic test, and Poincaré return map. The main findings indicate that the asymmetric system model may exhibit nonzero bistable forward whirling motion under external excitation. Additionally, it can whirl either forward or backward under multi-parametric excitation, besides the trivial stable solution. Furthermore, in the case of mixed excitation, the rotor displays nontrivial tristable solutions, with two corresponding to forward whirling orbits and the other one corresponding to backward whirling oscillation. These findings are validated through the establishment of different basins of attraction. Finally, the performance of the L I R C in mitigating rotor vibrations and averting nonlinear catastrophic bifurcations under various excitation conditions. Furthermore, the rotor's dynamical behavior and stability are explored in the event of an abrupt failure of one of the connected controllers. The outcomes demonstrate that the proposed L I R C effectively eliminates dangerous nonlinearities, steering the system to respond akin to a linear system with controllable oscillation amplitudes. However, the sudden controller failure induces a local rub-impact effect, leading to a nonlocal quasiperiodic oscillation and restoring the dominance of the nonlinearities on the system's response. [ABSTRACT FROM AUTHOR]

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

We consider the Paneitz type equation Δ 2 u − α Δ u + β (u − u q) = 0 on a closed Riemannian manifold (M n , g) of dimension n ≥ 3. We reduce the equation to a fourth order ordinary differential equation assuming that (M , g) admits a proper isoparametric function. Assuming that q > 1 , α and β are positive and α 2 > 4 β , we prove that the global nonconstant solutions of this ordinary differential equation only have nondegenerate critical points. Applying global bifurcation theory we then prove multiplicity results for positive solutions of the equation when q < p ⁎ , where p ⁎ = n + 4 n − 4 if n > 4 and p ⁎ = ∞ if n = 3 , 4. As an application and motivation we prove multiplicity results for conformal constant Q -curvature metrics. For example, consider closed positive Einstein manifolds (M n , g) and (X m , h) of dimensions n , m ≥ 3. Assuming that M admits a proper isoparametric function (with a symmetry condition) we prove that as δ > 0 gets close to 0, the number of constant Q -curvature metrics conformal to g δ = g + δ h goes to infinity. [ABSTRACT FROM AUTHOR]

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

ABSTRACT This paper investigates the occurrence of extreme events (EEs) in a Sprott‐like autonomous third‐order nonlinear system exhibiting non‐hyperbolic nature. The presence of non‐hyperbolicity in the system leads to various dynamic behaviors such as quasi‐periodicity, multistability, crisis, and intermittency. In our study, we analyzed the large‐amplitude intermittent chaotic oscillations using time series analysis, one‐parameter bifurcation diagram, Lyapunov spectra, and two‐parameter phase diagram. We confirmed the existence of EEs statistically using probability distribution functions (PDFs). To validate the numerical results, we performed circuit simulation studies using OrCAD PSpice as well as real‐time experimental observations using hardware implementation of the electronic circuit. In our studies, we find that the numerical, simulation, and experimental results are in agreement with each other. [ABSTRACT FROM AUTHOR]

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

This paper studies the bounded confidence model on growing fully-mixed populations. In this model, in addition to the usual opinion clusters, significant secondary clusters of smaller size appear systematically, while those secondary clusters appear erratically and include much fewer agents when the population is fixed. Through simulations, we derive the bifurcation diagram of the growing population model and compare it to the diagram obtained with an evolving probability density instead of agents, and with their equivalent having a fixed population. Our tests, when changing the usual bounded confidence function into a smooth bounded confidence function, suggest that these secondary clusters are mainly generated by a different mechanism when the population is growing than when it is fixed. [ABSTRACT FROM AUTHOR]

In recent years, conventional logistic maps have been applied across various fields including modeling and security, owing to their versatility and utility. However, their reliance on a single modifiable parameter limits their adaptability. This paper aims to explore generalized logistic maps with arbitrary powers, which offer greater flexibility compared to the standard logistic map. By introducing additional parameters in the form of arbitrary powers, these maps exhibit increased degrees of freedom, thus expanding their applicability across a wider spectrum of scenarios. Consequently, the conventional logistic map emerges as a specific instance within the proposed framework. The inclusion of arbitrary powers enriches system dynamics, enabling a more nuanced exploration of system behavior in diverse contexts. Through a series of illustrations, this study investigates the influence of arbitrary powers and equation parameters on equilibrium points, their positions, stability conditions, basin of attraction, and bifurcation diagrams, including the emergence of chaotic behavior. [ABSTRACT FROM AUTHOR]

Aeroengine power turbine rotor is a typical nonlinear rotor system, in which the interference connection between long shaft and short shaft has a prominent influence on the coupling vibration of the system, especially at high speed, the nonlinear stiffness change of interference contact is more obvious. Therefore, the paper considers the change of the contact characteristics of the long shaft and the short shaft of the turbine rotor, and the nonlinear restoring force of the bearing to analyze the dynamic characteristics of the rotor. Firstly, according to the theory of elasticity, the contact parameters under different speed states are calculated, the contact stiffness model under interference fit is established, and the relationship between contact stiffness and speed is analyzed. Secondly, according to the Hertz contact theory, a bearing nonlinear restoring force model considering time-varying characteristics is established. Finally, the dynamic model of the bearing-flexible rotor coupling considering interference contact is established. The influence of contact parameters and unbalance parameters on the nonlinear vibration characteristics of the rotor is analyzed by amplitude-frequency curve, bifurcation diagram, time-domain waveform, spectrum diagram, trajectory diagram and Poincaré diagram. In addition, the simulation results are compared with the experiment results of the rotor test rig to verify the accuracy of the established dynamic model. By analyzing the influence of different parameters such as interference magnitude, unbalance position, unbalance magnitude and unbalance couple on the nonlinear vibration of the rotor, the basis for the design and vibration control of the cantilever turbine rotor is provided. [ABSTRACT FROM AUTHOR]

This study aims to investigate a financial system consisting of four ordinary differential equations associated with the rate of interest, investment demand, price index, and the density of financial information gained by the population. The equilibrium and local stability of the system are investigated numerically. The impact of saving amounts and the rate of investment demand increases after getting financial information on the system are discussed. The findings of the study are verified graphically. It is found that the system becomes stable if the rate of investment demand increases after getting financial information kept at a certain level, such that the savings amount is maintained at a higher level. Also, the bifurcation diagrams of the system for various significant parameters that affect the system's stability have been depicted. [ABSTRACT FROM AUTHOR]

This paper proposed two new generalizations of the logistic function, each drawing on non-extensive thermodynamics, the q-logistic Equation and the logistic Equation of arbitrary order, respectively. It demonstrated the impact of chaos theory by integrating it with logistics Equations and revealed how minor parameter variations will change system behavior from deterministic to non-deterministic behavior. Moreover, this work presented BifDraw – a Python program for drawing bifurcation diagrams using classical logistic function and its generalizations illustrating the diversity of the system’s response to the changes in the conditions. The research gave a pivotal role to the place of the logistic Equation in chaos theory by looking at its complicated dynamics and offering new generalizations that may be new in terms of thermodynamic basic states and entropy. Also, the paper investigated dynamics nature of the Equations and bifurcation diagrams in it which present complexity and the surprising dynamic systems features. The development of the BifDraw tool exemplifies the practical application of theoretical concepts, facilitating further exploration and understanding of logistic Equations within chaos theory. This study not only deepens the comprehension of logistic Equations and chaos theory, but also introduces practical tools for visualizing and analyzing their behaviors. [ABSTRACT FROM AUTHOR]

The hydraulic turbocharger plays a vital role in harnessing the energy stored in brine within reverse osmosis desalination plants. To optimize the efficiency and durability of this equipment, it is crucial to develop accurate dynamic models of the turbocharger rotor. An improved understanding of rotor dynamics enables the integration of innovative technologies such as Hydraulic Energy Management Integration, effectively enhancing efficiencies in systems characterized by small capacities and high rotational speeds. This study presents a dynamic modeling methodology for the hydraulic turbocharger. The analysis involves approximating the turbocharger rotor with an equivalent finite element shaft line model. Verification of the model's natural frequency is conducted using three-dimensional finite element analysis, employing the ANSYS modal analysis module. Computational fluid dynamics is employed to evaluate the fluid forces, while the Reynolds equation is utilized to assess the journal bearing forces. The resulting model is employed to investigate the nonlinear dynamics of the rotor, examining the impact of various system parameters, including rotational speed, unbalance forces, and shaft geometrical parameters. The results highlight the significance of balancing the turbine and pump disks for optimal performance. Furthermore, the research demonstrates that increasing the shaft length reduces the rotor's threshold speed, while increasing the shaft diameter initially raises the threshold speed until it reaches a critical value. Beyond this critical value, further increases in shaft diameter lead to a decrease in the threshold speed. [ABSTRACT FROM AUTHOR]

Due to spatial scarcity and uncertainties in sediment data, initial and boundary conditions in deep-time climate simulations are not well constrained. On the other hand, depending on these conditions, feedback mechanisms in the climate system compete and balance differently. This opens up the possibility to obtain multiple steady states in numerical experiments. Here, we use the MIT general circulation model to explore the existence of such alternative steady states around the Permian–Triassic Boundary (PTB). We construct the corresponding bifurcation diagram, taking into account processes on a timescale of thousands of years, in order to identify the stability range of the steady states and tipping points as the atmospheric CO2 content is varied. We find three alternative steady states with a difference in global mean surface air temperature of about 10 °C. We also examine how these climatic steady states are modified when feedbacks operating on comparable or longer time scales are included, namely vegetation dynamics and air-sea carbon exchanges. Our findings on multistability provide a useful framework for explaining the climatic variations observed in the Early Triassic geological record, as well as some discrepancies between numerical simulations in the literature and geological data at PTB and its aftermath. [ABSTRACT FROM AUTHOR]

This article investigates the chaotic analysis and attractive domain of a fractional‐order Φ6$$ {\Phi}&amp;#x0005E;6 $$‐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]

Biometric security has been developed in recent years with the emergence of cancellable biometric concepts. The idea of the cancellable biometric traits is concerned with creating encrypted or distorted traits of the original ones to protect them from hacking techniques. So, encrypted or distorted biometric traits are stored in databases instead of the original ones. This can be accomplished through non-invertible transforms or encryption schemes. In this paper, a cancellable face recognition algorithm is introduced based on face image encryption through a fractional-order multi-scroll chaotic system. The fundamental concept is to create random keys that will be XORed with the three components of color face images (red, green, and blue) to obtain encrypted face images. These random keys are generated from the Least Significant Bits of all state variables of a proposed fractional-order multi-scroll chaotic system. Lastly, the encrypted color components of face images are combined to produce a single cancellable trait for each color face image. The results of encryption with the proposed system are full-encrypted face images that are suitable for cancellable biometric applications. The strength of the proposed system is that it is extremely sensitive to the user's selected initial conditions. The numerical simulation of the proposed chaotic system is done with MATLAB. Phase and bifurcation diagrams are used to analyze the dynamic performance of the proposed fractional-order multi-scroll chaotic system. Furthermore, we realized the hardware circuit of the proposed chaotic system on the PSpice simulator. The proposed chaotic system can be implemented on Field Programmable Gate Arrays (FPGAs). To model our generator, we can use Verilog Hardware Description Language HDL, Xilinx ISE 14.7 and Xilinx FPGA Artix-7 XC7A100T based on Grunwald-Letnikov algorithms for mathematical analysis. The numerical simulation, the circuit simulation and the hardware experimental results confirm each other. Cancellable face recognition based on the proposed fractional-order chaotic system has been implemented on FERET, LFW, and ORL datasets, and the results are compared with those of other schemes. Some evaluation metrics containing Equal Error Rate (EER), and Area under the Receiver Operating Characteristic (AROC) curve are used to assess the cancellable biometric system. The numerical results of these metrics show EER levels close to zero and AROC values of 100%. In addition, the encryption scheme is highly efficient. [ABSTRACT FROM AUTHOR]

With the rapid development of wind power generation in recent years, the demand for detecting weak signals of wind turbine faults has become more urgent. This paper introduces a novel memristor chaotic circuit constructed based on third-order magnetically memristors. The Melnikov chaotic condition of this circuit is analyzed, and its dynamical characteristics are studied through phase trajectory diagrams, bifurcation diagrams, Lyapunov exponent spectra, and Poincaré maps. Leveraging the initial value sensitivity and noise immunity of chaotic systems, the memristor chaotic circuit is employed for the detection of weak signals in wind turbine faults. Using the chaotic system state transition method, we find the threshold for the circuit state to transition from chaotic state to large-scale periodic state, adjust the parameters to make the system in a critical state, input the wind turbine fault vibration signal, and detect the fault signal based on its state transition. Subsequently, the chaotic resonance method is employed, introducing the signal under test, which contains high-intensity chaotic noise, into this novel memristive circuit. This results in chaotic resonance, causing the noise components to be concentrated toward the frequency region where the weak signal under test is located, thereby enhancing the fault signal and facilitating fault identification. The results indicate that this novel memristor chaotic circuit possesses advantages such as high accuracy, strong noise immunity, straightforward operation, and clear judgment in the field of weak signal detection. This circuit shows promising applications in the field of weak signal detection. [ABSTRACT FROM AUTHOR]

We study the global bifurcation diagram of the positive solutions to the problem { Δ u + λ f (u) = 0 in B , u = 0 on ∂ B , where B is the unit ball in R N with N ≥ 3. Under general supercritical growth conditions on f (u) , we show that an unbounded bifurcation curve has no turning point, which indicates the existence of the singular extremal solution. In particular, our theory can be applied to the super-exponential cases of f (u) , and we exhibit that a bifurcation curve for Δ u + λ f (u) = 0 has the same qualitative property as a classical Gel'fand problem Δ u + λ e u = 0 for N ≥ 3 except N = 10. Main technical tools are intrinsic transformations for semilinear elliptic equations and ODE techniques. [ABSTRACT FROM AUTHOR]

We show that delay-differential equations (DDE) exhibit universal bifurcation scenarios, which are observed in large classes of DDEs with a single delay. Each such universality class has the same sequence of stabilizing or destabilizing Hopf bifurcations. These bifurcation sequences and universality classes can be explicitly described by using the asymptotic continuous spectrum for DDEs with large delays. Here, we mainly study linear DDEs, provide a general transversality result for the delay-induced bifurcations, and consider three most common universality classes. For each of them, we explicitly describe the sequence of stabilizing and destabilizing bifurcations. We also illustrate the implications for a nonlinear Stuart–Landau oscillator with time-delayed feedback. [ABSTRACT FROM AUTHOR]

Complex dynamical behaviors such as bifurcation and chaos exist in H-bridge inverter with RLC load, and these nonlinear behaviors will greatly increase the harmonic content of the output current and reduce the stability and reliability of the system. In this paper, a PI controller is added to widen the stable operation domain of the system. The stroboscopic mapping theory is used to model the system, the nonlinear dynamic behavior of the inverter is investigated by the bifurcation diagram, the folding diagram and the phase trajectory diagram are used for comparative verification, and the TDFC method is introduced to inhibit the chaotic behavior of the inverter, which further improves the stable range of the system operation. The fast-change stability theorem is used to analyze the stability of the system theoretically and verify the correctness of the numerical simulation. Therefore, the conclusions of the study provide a reliable theoretical basis for the design of the inverter system, which has important theoretical significance and practical value. [ABSTRACT FROM AUTHOR]

With the rise of the sharing economy, shared bicycles have become an important component of urban transportation. This paper explores the nonlinear dual oligopoly system for the Cournot model in the bike-sharing market; both operators have maximized profits as their competitive goals. The analysis of pivotal factors influencing passenger preferences, including pricing discounts and comfort levels, is meticulously depicted by a bifurcation diagram. A new chaotic attractor—the shared bicycle attractor—is discovered. The research results indicate that larger discounts and adjustment speeds can cause the system to be in a chaotic state, which is not conducive to the long-term development of operators, although discounts can indeed attract more passengers to a certain extent. On the other hand, the increase in the marginal cost of comfort loss can also make it difficult for enterprises to operate, which requires continuous technological innovation to improve the comfort of cycling. [ABSTRACT FROM AUTHOR]

Transient chaos is a characteristic behavior observed in nonlinear systems, where trajectories within a specific region of phase space exhibit chaotic dynamics for a finite duration before transitioning to an external attractor. In many nonlinear systems, transient and chaotic behaviors are closely associated with variations in the system's energy. In this paper, we introduce the concept of an energy variable and establish its connection with the Melnikov integral. We explore the influence of energy variation in an amplitude‐modulated (AM) force‐driven Ueda oscillator through numerical simulations. Our investigation reveals the emergence of transient chaos, chaotic dynamics, and regular behaviors, underscoring the significant role played by the energy variable, denoted as ϵ$$ \epsilon $$, in the system. We employ various analytical tools, including phase diagrams, Poincaré maps, time series plots, and bifurcation diagrams, to characterize and visualize transient chaos, regular, and chaotic motions within the system. [ABSTRACT FROM AUTHOR]

The simultaneous resonance patterns and dynamic attributes of cylindrical bubbles subjected to dual-frequency acoustic excitation are explored in this article. Specifically, analytical models and local stability analysis are constructed for different types of simultaneous resonance under dual-frequency acoustic excitation. Based on the frequency response curve, the mechanisms whereby the core parameters in dual-frequency excitation influence the resonant dynamic characteristics are then explored. Combined with the Lyapunov exponent and amplitude, frequency, and phase, bifurcation diagrams are illustrated for determining the global stability of the cylindrical bubble dual-frequency resonance system. The conclusions from this research are given as follows: (1) Under dual-frequency acoustic excitation, the simultaneous resonance type can be subdivided into three categories, namely, primary–superharmonic, primary–subharmonic, and superharmonic–subharmonic simultaneous resonances. These dual-frequency resonance types exhibit unique and significant dynamic characteristics. (2) Increasing the total amplitude of dual-frequency acoustic excitation significantly enhances the maximum value of the dual-frequency resonance and the vulnerability to instabilities. The effect of the bubble balance radius is similar to that of the total amplitude. Higher values of the nonlinear coefficient reduce the maximum value of the resonance and increase the likelihood of instability. (3) The total amplitude of dual-frequency acoustic excitation is a key factor affecting the stability of bubbles. As the total amplitude increases, the bubble oscillation gradually transforms from periodic to chaotic. [ABSTRACT FROM AUTHOR]

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

In this paper, we consider the grazing–sliding bifurcations in a dry-friction oscillator on a moving belt under periodic excitation. The system is a nonlinear piecewise smooth system defined in two zones whose analytical expressions of the solutions are not available. Thus, we obtain conditions of the existence of grazing–sliding orbits numerically by the shooting method. Then, we compute the lower and higher order approximations of the stroboscopic Poincaré map, respectively, near the grazing–sliding bifurcation point by the method of local zero-time discontinuity mapping. The results of computing the bifurcation diagrams obtained by the lower and higher order maps, respectively, are compared with those from direct simulations of the original system. We find that there are big differences between the lower order map and the original system, while the higher order map can effectively reduce such disagreements. By using the higher order map and numerical simulations, we find that the system undergoes very complicated dynamical behaviors near the grazing–sliding bifurcation point, such as period-adding cascades and chaos. [ABSTRACT FROM AUTHOR]

It is of practical significance to realize a stable and controllable financial system by using chaotic synchronization theory. In this paper, the dynamics and synchronization are studied for a class of fractional-order chaotic financial systems. First, the stability and dynamics of the fractional-order chaotic financial system are analyzed by using the phase trajectory diagram, time series diagram, bifurcation diagram, and Lyapunov exponential diagram. Meanwhile, we obtain the range of each parameter that puts the system in a periodic state, and we also reveal the relationship of the derivative order and the chaotic behaviors. Then, the adaptive control strategy is designed to achieve synchronization of the chaotic financial system. Finally, the theoretical results and control method are verified by numerical simulations. [ABSTRACT FROM AUTHOR]

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

This research article introduces a novel chaotic satellite system based on fractional derivatives. The study explores the characteristics of various fractional derivative satellite systems through detailed phase portrait analysis and computational simulations, employing fractional calculus. We provide illustrations and tabulate the phase portraits of these satellite systems, highlighting the influence of different fractional derivative orders and parameter values. Notably, our findings reveal that chaos can occur even in systems with fewer than three dimensions. To validate our results, we utilize a range of analytical tools, including equilibrium point analysis, dissipative measures, Lyapunov exponents, and bifurcation diagrams. These methods confirm the presence of chaos and offer insights into the system's dynamic behavior. Additionally, we demonstrate effective control of chaotic dynamics using feedback active control techniques, providing practical solutions for managing chaos in satellite systems. [ABSTRACT FROM AUTHOR]

Pattern formation in biological aggregations is a topic of great interest, due to the complex spatial structure of various aggregations of cells/bacteria/animals that can be observed in nature. While many such aggregations look similar at the macroscopic level, they might differ in their microscopic spatial structure. However, the complexity of the non-linear and sometimes non-local interactions among individuals inside these aggregations makes it difficult to investigate these spatial structures. In this study, we investigate numerically the transitions between different spatial patterns of animal aggregations with various symmetries (even, odd or no symmetry) that characterise the microscopic distribution of individuals inside these aggregations. To this end, we construct a bifurcation diagram starting with perturbations of spatially homogeneous solutions with low, medium, and high amplitudes. For perturbations with low amplitudes, the bifurcating structures show transitions among even-symmetric, odd-symmetric, and non-symmetric solutions. For perturbations with large amplitudes, there are wide parameter regions with non-convergent solutions, characterised by oscillatory transitions between different relatively similar solutions. These numerical results emphasize: (i) the effect of nonlinear and non-local interactions on the microscopically different symmetric/non-symmetric structures of macroscopically similar ecological aggregations; (ii) the difficulty of developing continuation algorithms for this class of non-local models. [ABSTRACT FROM AUTHOR]

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

Unbalance in rotating machines, caused by manufacturing defects and wear, can induce severe vibrations in base structures. As the rotor accelerates from static to high speeds, multiple frequencies may be excited. To effectively suppress vibrations across all speeds, a bi-stable nonlinear energy sink (BNES) is attached to a cantilever beam with an unbalanced rotor. The dynamic equation of the beam-BNES system is derived using Euler-Bernoulli beam theory and the Euler-Lagrange equation, followed by Galerkin discretization. A Multi-Objective Genetic Algorithm optimizes the BNES parameters. Results indicate that the BNES reduces multiple resonant amplitudes by 80% to 95%. Energy dissipation for the first and second beam modes was 85% and 78%, respectively. The system's bifurcation diagram reveals a stable multi-periodic response across a wide range of unbalance levels. Performance measures show that the NES with an additional linear stiffness element or the BNES surpasses traditional NES. [ABSTRACT FROM AUTHOR]

Recently, the research on the modified Leslie–Gower model has become an appealing topic. Due to economic efficiency and the complexity of discrete models, we investigate a discrete modified Leslie–Gower predator–prey model with prey harvesting in this paper. The stability of fixed points and bifurcations of the interior fixed points are studied. According to bifurcation theory and normal forms, we derived the conditions of codimension 2 bifurcations occurred, including 1:1 strong resonance bifurcation and fold-flip bifurcation. These two bifurcations are unusual in bifurcation analysis on discrete systems. In addition, the continuation curves, bifurcation diagrams, and phase diagrams are used to demonstrate theoretical results. Our study shows the interesting dynamics of this model that are very different from the continuous one. [ABSTRACT FROM AUTHOR]

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

• An impact-damping model with multi-point contact and external load is proposed. • The effect of multi-point contact and external load on system impact is studied. • This impact-damping model is applied to the dynamics model of a spur gear pair. • The connection between gear system resonance and teeth impact is revealed. The damping term is an important component of the calculation model of the contact force of collision bodies because it simulates the energy dissipation of collision bodies during the contact-impact process. Therefore, an impact-damping model of a collision body with multi-point contact and external load is established by considering the restitution coefficient. The algorithm for solving this model is proposed based on iterative methods. Then, the accuracy of the established model is verified by the consistency between the calculated and tested results. The effect of multi-point contact and external load on the contact-impact process of the collision body is studied. The results show that the hysteresis damping factor increases from small to large when the collision body enters a stable state through repeated impacts. Further, the established damping model is applied to the dynamics model of a spur gear pair with backlash, extended teeth contact, and structure coupling effect of the gear body. The nonlinear dynamic characteristics, multi-state mesh, and teeth impact of the system are studied through the root-mean-square of transmission error, bifurcation diagrams with multi-mapping sections, and contact forces. The calculation results of the gear dynamics model with the impact-damping model are more consistent with the tested results than the calculation results of the gear dynamics model with the traditional damping model. Meanwhile, the connection between system resonance, response coexistence, teeth disengagement, and teeth impact is revealed. [ABSTRACT FROM AUTHOR]

When a person feels some emotion such as happiness, sadness, or love, the person consciously knows that such feelings appear, or it is done unconsciously, and sometimes it appears as a compound result of the two. In this paper, these two are defined as cognition and noncognition. By applying these two to the fuzzy love model using complex fuzzy numbers, we observe the chaotic behavior that appears in this model. We verify chaotic behaviors in the love model with fuzzy triangular and trapezoidal external forces using phase portrait and bifurcation diagram. The love model is known as the differential equation that can represent how a person feels love with respect to the time when the positive or negative external force is changed. Because love is a person's feeling that includes vagueness and ambiguity of human emotion. Even more the external force also can express some external influence that is also can be human's response. Because a person's feeling is vague and ambiguous, fuzzy valued sinusoidal functions and Gaussian have been used to express those feelings and external forces. [ABSTRACT FROM AUTHOR]

Sealing aerodynamic characteristics are affected by the seal structure, and thus the stability of the rotor system is affected too. A 1.5-stage, three-dimensional, full-cycle model of the high-pressure cylinder of a 1000 MW steam turbine was established. The high eccentricity whirl of the rotor was realized using mesh deformation technology and the multi-frequency whirl model. The nonlinear steam-flow-exciting force of different sealing structures was obtained using CFD/FLUENT, and the motion equations with a nonlinear steam-exciting force were solved using the Runge–Kutta method. The motion characteristics and stability of the rotor system with different sealing structures were obtained. The results show that there are "inverted bifurcation" and "double bifurcation" phenomena in the bifurcation diagrams of different tooth numbers, boss numbers, and tooth lengths, and a 1/2 power frequency of different sealing structures goes through the process of weakening, disappearing, reproducing, and evolving into a 1/3 power frequency and a 2/3 power frequency. With the increasing load, the steam-flow-exciting force becomes stronger, and the multi-frequency vibration and dense frequency phenomena are significant. Under some load conditions, the change curves of three kinds of teeth in 1/3 and 2/3 power frequency vibrations are highly similar, and the tooth number has little influence on the system stability. Under the high load condition, with the boss number increasing, the chaos phenomenon is weakened. Increasing the tooth length is beneficial to the stability of the rotor. [ABSTRACT FROM AUTHOR]

In this work, numerical and experimental implementation of a state observer applied to an erbium-doped fiber laser (EDFL) has been developed. The state observer is designed through the mathematical model of the EDFL to estimate the non-measurable variable; however, in numerical estimation, the state variables can be measurable given the mathematical model. Only the laser intensity variable was experimentally measured. The state observer estimated the population inversion through the obtained experimental laser intensity time series fitted with their numerical laser intensity using the mean square error (MSE) tool. A bifurcation diagram of the population inversion time series local maximum was built from the state observer. The state space of the experimental laser intensity versus observed population inversion was built. [ABSTRACT FROM AUTHOR]

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

Main objective of this research to eliminate the resonant vibrations and stabilize the unstable motion of a self-excited structure through the implementation of an innovative active control strategy. The control strategy coupling the self-excited structure with a second-order filter, which feedback gain λ and control gain β , as well as a first-order filter, which feedback gain δ and control gain γ. The coupling of the second-order filter to establish an energy bridge between the structure and the filter to pump out the structure's vibration energy to the filter. In contrast, the primary purpose of coupling the first-order filter to stabilize the closed loop by adjusting the damping of the system using the control keys δ and γ. Accordingly, the mathematical model of the proposed control system formulated, incorporating the closed-loop time delay τ. An analytical solution for the system model obtained, and a nonlinear algebraic system for the steady-state dynamics of the controlled structure extracted. The system's bifurcation characteristics analyzed in the form of stability charts and response curves. Additionally, the system's full response simulated numerically. Findings the high performance of the introduced controller in eliminating the structure's resonant vibrations and stabilizing non-resonant unstable motion. In addition, analytical and numerical investigations revealed that the frequency band within which the second-order filter can absorb the structure's resonant oscillation relies on the algebraic product of β and λ. Furthermore, it was found that the equivalent damping of the system depends on the algebraic product of γ and δ , which can be employed to stabilize the negatively damped self-excited systems. Finally, it reported that although the loop delay can potentially degrade vibration control performance, the time-delay stability margin is nonlinearly proportional to the product of γ and δ. This finding that increasing the value of γ × δ can compensate for the adverse effects of loop delay on both system stability and vibration suppression efficiency. [ABSTRACT FROM AUTHOR]

In recent years, conventional Ricker maps have enjoyed widespread applications across crucial domains such as modeling and security. However, their limitation to a single changeable parameter poses constraints on their adaptability. This paper introduces a generalized form of the Ricker map, incorporating arbitrary powers, thus offering enhanced versatility compared to the traditional Ricker map. By introducing an additional parameter (arbitrary power), the map gains increased degrees of freedom, thereby accommodating a broader spectrum of applications. Consequently, the conventional Ricker map emerges as merely a special case within each proposed framework. This newfound parameter enhances system flexibility and elucidates the conventional system's performance across diverse contexts. Through numerous illustrations, we meticulously investigate the impact of the arbitrary power and equation parameters on equilibrium points, their positions, basin of attraction, stability conditions, and bifurcation diagrams, including the emergence of chaotic behavior. [ABSTRACT FROM AUTHOR]

This paper investigates two new Filippov-Gause predator-prey models that show the transition between individual and cooperative hunting dynamics among predators with respect to the critical prey population size. By using Filippov systems, this research offers a more realistic and complete representation of the nonlinear dynamics and discontinuities inherent in this complex ecological system so that the proposed models provide a more complete view of predator-prey interactions in different realistic environmental contexts. For the case where predators only cooperate in hunting when the prey population size is larger than its critical value, we could have at least one asymptotically stable limit cycle around a single positive inner equilibrium, or two limit cycles around the pseudo-stable equilibrium. However, in the case where predators only cooperate in hunting when the prey population size is less than its critical value, we could have one limit cycle around two inner equilibria and one pseudo-equilibrium that is unstable. In general, as the two proposed models differ in the presence of a sliding or escaping segment, the overall dynamics and bifurcation diagrams of both models change significantly. [ABSTRACT FROM AUTHOR]

The intricate relationship between temporal and spatiotemporal dynamics in a Crowley–Martin predator–prey model, enriched with fear effect and anti-predator behavior, is investigated in this study. A careful mathematical analysis is conducted to explore the feasible equilibria of the model system, followed by an examination of their stability, instability, and all possible bifurcation scenarios. Asymptotic stability, bistability, and various codimension-1 and codimension-2 bifurcations, including transcritical, saddle–node, Hopf, cusp, and Bogdanov–Takens bifurcations, are demonstrated by the model. The analytical findings and the model's applications are validated through numerical simulations, employing a biparameter bifurcation diagram for quantitative analysis. The study also observes bubbling phenomena and a scenario leading to the density-dependent hydra effect. The diffusion effect is investigated with special attention to the system's nonlinearity and chosen parameter values. Numerical simulations reveal the emergence of spatiotemporal patterns both within and beyond the Turing space. The evolution of diffusion-driven pattern generation, including cold spots, stripes, labyrinthines, mixtures of stripes and cold spots, and complex non-Turing patterns, is demonstrated on the plane. These spatial patterns are shown to be influenced biologically by both the fear effect and anti-predator behavior. [ABSTRACT FROM AUTHOR]

This paper proposes a novel Switching Chaotic Coupled Map Lattices (SCCML) system based on Elementary Cellular Automata (ECA) to address issues found in Coupled Map Lattices (CML) systems such as period windows, poor lattice correlation, and limited chaotic behavior parameters. The system uses a bidirectional coupling method and introduces two ECA with different rules to continuously alter the indexes of the coupled lattices and the underlying chaotic mapping. This approach effectively mitigates correlation issues in CML systems. At the same time, the iterative results of ECA are cascaded with the chaotic mapping as a perturbation of the SCCML system to extend the parameter range of chaotic behavior. The Kolmogorov–Sinai entropy density and universality, bifurcation diagrams, uniformity and correlation are analyzed to investigate the dynamical properties of the proposed system. Theoretical analysis and numerical simulations demonstrate that, compared with other novel spatiotemporal chaotic systems, the proposed system eliminates periodic windows, expands the chaotic behavior parameter range, and decreases the correlation between different lattices. The above experimental results demonstrate the proposed system's excellent potential for cryptosystems and secure communication. Furthermore, an Encrypted Code Index Modulation (E-CIM) scheme based on reversible ECA encryption is proposed to address the problems of limited pseudo-random sequence resources and low spectrum utilization in direct sequence spread spectrum systems. The proposed scheme uses a chaotic sequence with good correlation and randomness generated by the SCCML spatiotemporal chaos system as the spreading code sequence. In the scheme, the information bits are divided into two parts, namely modulation bits and mapping bits. The modulation bits are mapped to points on the modulation constellation diagram. The mapping bits are encrypted using reversible ECA, and the mapped bits before and after encryption are mapped as spreading sequences with different indexes. Simulation and analysis show that under the same spectrum efficiency conditions, the BER performance of E-CIM is superior to that of the Code Index Modulation (CIM) and Generalized Code Index Modulation (GCIM) schemes by about 2–4 dB and superior to that of the Nonorthogonal Code Shift Keying and Code Index Modulation (N-CSK-CIM) scheme by about 0.5 dB in an additive Gaussian white noise channel when the BER is 10 − 5 . [ABSTRACT FROM AUTHOR]

This paper consists of four parts. It begins by using the authors' generalized Schauder formula, [41] , and the algebraic multiplicity, χ , of Esquinas and López-Gómez [15,14,31] to package and sharpening all existing results in local and global bifurcation theory for Fredholm operators through the recent author's axiomatization of the Fitzpatrick–Pejsachowicz–Rabier degree, [42]. This facilitates reformulating and refining all existing results in a compact and unifying way. Then, the local structure of the solution set of analytic nonlinearities F (λ , u) = 0 at a simple degenerate eigenvalue is ascertained by means of some concepts and devices of Algebraic Geometry and Galois Theory, which establishes a bisociation between Bifurcation Theory and Algebraic Geometry. Finally, the unilateral theorems of [31,33] , as well as the refinement of Shi and Wang [53] , are substantially generalized. This paper also analyzes two important examples to illustrate and discuss the relevance of the abstract theory. The second one studies the regular positive solutions of a multidimensional quasilinear boundary value problem of mixed type related to the mean curvature operator. [ABSTRACT FROM AUTHOR]

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

A comprehensive theoretical investigation on the occurrence of secondary resonances in parametrically excited unbalanced spinning composite beams under the stretching effects is conducted numerically and analytically. Based on an optimal stacking sequence and Rayleigh’s beam theory, the governing equations of the system are derived using extended Hamilton’s principle. The system’s partial differential equations are then discretized using the Galerkin method. Numerical (Runge–Kutta technique) and analytical (multiple scales method) approaches are exploited to solve the reduced-order equations, and their results are compared and verified accordingly. Comparison and convergence investigations are performed to guarantee the validity of the outcomes. Stability and bifurcation analyses are accomplished, and resonance effects are thoroughly studied utilizing frequency-response diagrams, phase portraits, Poincaré maps and time-history responses. It is observed that among the various types of secondary resonance, only a combination resonance can be observed in the system dynamics. The outputs reveal that, in this resonance, the gyroscopic coupling results in the steady-state time response consisting of three main frequencies. By examining the effects of damping, eccentricity, and beam length, it is exhibited that this resonance does not occur in the system’s dynamics for any combination of these parameters. Therefore, these parameters can be adjusted in the design of asymmetric beams to prevent this type of resonance. [ABSTRACT FROM AUTHOR]