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

In a recent paper [Belykh et al., 2024], we proved that the bifurcation structure of a quadratic noninvertible map persists when the parameter increases from zero and the map turns into an invertible multidimensional Henon map. In this paper, we consider a similar problem for a generalized map which combines the Henon-type maps, the Poincaré return map for Shilnikov bifurcation of saddle-focus homoclinic orbit, the Lurie discrete time system, etc. We have obtained the expected result about the persistence of periodic orbits and their bifurcations when passing from a One-Dimensional (1D) endomorphism to the generalized map as a small parameter becomes nonzero. We have revealed the precise mechanism of change of homoclinic orbits and splitting of unstable manifolds as a result of the transition of 1D endomorphism to multidimensional map. Thereby we have derived the reconstruction rules of nonwandering set of orbits and bifurcations of the generalized map from those of 1D endomorphism. [ABSTRACT FROM AUTHOR]

In this paper, a novel locally active memristor is constructed. Based on the 3D Hindmarsh–Rose (HR) and 2D FitzHugh–Nagumo (FHN) neuron, a heterogeneous neuronal system is constructed by connecting the two neurons with the locally active memristor. The equilibrium point and stability of the system are investigated. The dynamic behavior of the system is numerically and experimentally revealed by utilizing dynamic analyses in terms of interspike interval bifurcation diagram, two-parameter bifurcation diagram and so on. The unique and abundant dynamic behavior is found in the proposed neuronal system by varying the coupling strength, external stimulus current, memristive parameter and other system parameters. Multiple bursting firing groups in the tongue-shaped domains have been discovered for the first time. Finally, in order to validate the numerical simulation, an analog equivalent circuit of the heterogeneous neuronal system is devised, which demonstrates that the system is physically realizable. [ABSTRACT FROM AUTHOR]

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

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

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]

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]

Bifurcation analysis is an essential tool in nonlinear dynamics. Bifurcation diagrams help to discover subtle features of investigated dynamics such as chaotic and periodic regimes, hidden attractors and fixed points. However, plotting high-resolution bifurcation diagrams can be a computationally challenging task, especially in multiparametric evaluation. It should be noted that the bifurcation analysis is a task with natural parallelism and thus can be efficiently solved using hybrid central-graphics processing architectures. In this paper, we propose an advanced algorithm and special software for plotting bifurcation diagrams using calculations accelerated by graphics processing unit in combination with a highly efficient semi-implicit ordinary differential equation solver. Time series processing is based on the extraction of amplitude and phase features for the density-based spatial clustering of applications with noise to determine oscillation periodicity. We showcase the features of the application of proposed solutions on a set of test chaotic systems. The performance of the analysis algorithms is investigated in comparison with conventional solutions based on central processing unit and several approaches known from the literature. We explicitly show that the proposed algorithm outperforms known solutions in both calculation speed and precision. [ABSTRACT FROM AUTHOR]

One way to characterize an orbit of a dynamical system is through its frequency content. Using a "spectral bifurcation diagram", it has been shown earlier how the frequency content changes when a system undergoes a period-doubling cascade. In this paper, we extend the scope of this technique to obtain newer insights into various bifurcations like pitchfork bifurcation, border-collision bifurcation, period-adding cascade, and various torus bifurcations. We show that applying this method can enrich our understanding of bifurcations by providing vital information about generating or annihilating frequency components in a bifurcation. [ABSTRACT FROM AUTHOR]

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

This paper presents a global study of the class Q E S ̂ of all real quadratic polynomial differential systems possessing exactly one elemental infinite singular point and one triple infinite singular point, which is either an infinite nilpotent elliptic-saddle or a nilpotent saddle. This class can be divided into three different families, namely, Q E S ̂ (A) of phase portraits possessing three real finite singular points, Q E S ̂ (B) of phase portraits possessing one real and two complex finite singular points, and Q E S ̂ (C) of phase portraits possessing one real triple finite singular point. Here, we provide a comprehensive study of the geometry of these three families. Modulo the action of the affine group and time homotheties, families Q E S ̂ (A) and Q E S ̂ (B) are three-dimensional and family Q E S ̂ (C) is two-dimensional. We study the respective bifurcation diagrams of their closures with respect to specific normal forms, in sub-sets of real Euclidean spaces. The bifurcation diagram of family Q E S ̂ (A) (resp., Q E S ̂ (B) and Q E S ̂ (C)) yields 1274 (resp., 89 and 14) sub-sets with 91 (resp., 27 and 12) topologically distinct phase portraits for systems in the closure Q E S ̂ (A) ¯ (resp., Q E S ̂ (B) ¯ and Q E S ̂ (C) ¯) within the representatives of Q E S ̂ (A) (resp., Q E S ̂ (B) and Q E S ̂ (C)) given by a specific normal form. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to analyze the dynamic characteristics and the meshing state transition characteristics of spalling defect gears under various lubrication conditions. The time-varying geometric parameters, motion parameters and load distribution of the gears in the state of positive meshing and reverse impact are analyzed. The time-varying meshing stiffness considering the time-varying friction coefficient under elastohydrodynamic lubrication (EHL), mixed lubrication and boundary lubrication are calculated, and the spalling defect is introduced in the stiffness calculation process. The equivalent lubrication model of the gear is established, the switching condition of the lubrication state is derived, and the distribution of the lubrication state in the torque-speed plane is discussed. The multimeshing state dynamic model of the spalling defect gear is established, and the evolution characteristics of the coexistence response and the transition of the meshing state under different lubrication conditions are simulated by using the attraction domain, phase trajectory, meshing force and bifurcation diagram. The results show that the phase trajectory of the gear with the spalling defect is deformed compared with that of the healthy gear. The change of lubrication state has little effect on the phase trajectory and attraction domain of the gear with spalling defect, but has a great influence on the meshing force. At the spalling boundary, the meshing force is changed abruptly, and the worse the lubrication state, the more obvious the mutation. Due to the influence of spalling on tooth stiffness, it will also affect the meshing force in the nonspalling area. [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]

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

This paper is concerned with a spatial S − I epidemic model with nonlinear incidence rate. First, the existence of the equilibrium is discussed in different conditions. Then the main criteria for the stability and instability of the constant steady-state solutions are presented. In addition, the effect of diffusion coefficients on Turing instability is described. Next, by applying the normal form theory and the center manifold theorem, the existence and direction of Hopf bifurcation for the ordinary differential equations system and the partial differential equations system are given, respectively. The bifurcation diagrams of Hopf and Turing bifurcations are shown. Moreover, a priori estimates and local steady-state bifurcation are investigated. Furthermore, our analysis focuses on providing specific conditions that can determine the local bifurcation direction and extend the local bifurcation to the global one. Finally, the numerical results demonstrate that the intrinsic growth rate, denoted as r , has significant influence on the spatial pattern. Specifically, different patterns appear, with the increase of r. The obtained results greatly expand on the discovery of pattern formation in the epidemic model. [ABSTRACT FROM AUTHOR]

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

In this study, a Cournot duopoly model describing Caputo fractional-order differential equations with piecewise constant arguments is discussed. We have obtained two-dimensional discrete dynamical system as a result of applying the discretization process to the model. By using the center manifold theory and the bifurcation theory, it is shown that the discrete dynamical system undergoes flip bifurcation about the Nash equilibrium point. Phase portraits, bifurcation diagrams, and Lyapunov exponents show the existence of many complex dynamical behaviors in the model such as the stable equilibrium point, period-2 orbit, period-4 orbit, period-8 orbit, period-16 orbit, and chaos according to changing the speed of the adjustment parameter v 1 . The discrete Cournot duopoly game model is also considered on two scale-free networks with different numbers of nodes. It is observed that the complex dynamical networks exhibit similar dynamical behaviors such as the stable equilibrium point, flip bifurcation, and chaos depending on changing the coupling strength parameter c s . Moreover, flip bifurcation and transition chaos take place earlier in more heterogeneous networks. Calculating the largest Lyapunov exponents guarantees the transition from nonchaotic to chaotic states in complex dynamical networks. [ABSTRACT FROM AUTHOR]

Cascaded H-bridge inverters, which belong to strongly nonlinear time-varying system with complex dynamics, have been widely used in high-voltage and high-power engineering fields. In this paper, a carrier phase-shifted sinusoidal pulse-width modulation cascaded H-bridge inverter is taken as an example with piecewise analysis to establish the discrete iterative mathematical model. Furthermore, the nonlinear dynamic behavior of this inverter system is investigated using folding diagrams, spectrum diagrams, bifurcation diagrams, Lyapunov exponents, etc. We find that this inverter can exhibit multiperiod bifurcation, tangential bifurcation and paroxysmal chaos within certain ranges of control coefficients and other circuit parameters, which will influence the system's stability seriously. On the basis of this discovery, the sinusoidal time-delay feedback control method is proposed. The difference between the load current of the controlled object and its own delay is used to form a feedback term, which is then fed into the sine function and proportional link to obtain the control term. Moreover, the Jacobian matrix of this feedback inverter system is derived, and the constraints on the feedback control coefficient are obtained based on the Jury criterion. Simulation results have verified the effectiveness of such a sinusoidal time-delay feedback control method, which can improve the stability and anti-interference of the system. The research can be used to provide some reference and guidance for the circuit design, debugging and control of cascaded H-bridge inverters. [ABSTRACT FROM AUTHOR]

We study a family of one-dimensional quasi-periodically forced maps F a , b (x ,) = (f a , b (x ,) , + ω) , where x is real, is an angle, and ω is an irrational frequency, such that f a , b (x ,) is a real piecewise-linear map with respect to x of certain kind. The family depends on two real parameters, a > 0 and b > 0. For this family, we prove the existence of nonsmooth pitchfork bifurcations. For a < 1 and any b , there is only one continuous invariant curve. For a > 1 , there exists a smooth map b = b 0 (a) such that: (a) For b < b 0 (a) , f a , b has two continuous attracting invariant curves and one continuous repelling curve; (b) For b = b 0 (a) , it has one continuous repelling invariant curve and two semi-continuous (noncontinuous) attracting invariant curves that intersect the unstable one in a zero-Lebesgue measure set of angles; (c) For b > b 0 (a) , it has one continuous attracting invariant curve. The case a = 1 is a degenerate case that is also discussed in the paper. It is interesting to note that this family is a simplified version of the smooth family G a , b (x ,) = (arctan (a x) + b sin () , + ω) for which there is numerical evidence of a nonsmooth pitchfork bifurcation. Finally, we also discuss the limit case when a → ∞. [ABSTRACT FROM AUTHOR]

In this paper, we present a concise four-dimensional (4D) conservative chaotic system with a wide parameter range. Since there are no terms higher than first order, the circuit does not contain multipliers, resulting in a simple circuit implementation. The nonlinear dynamic characteristics, such as phase diagrams, equilibrium points, divergence, Poincaré cross-sections, Lyapunov exponents, bifurcation diagrams, and Lyapunov dimension, are analyzed in detail, which illustrates the conservativity. Besides, the system exhibits different offset boosting behaviors. Through offset boosting, the system can propagate along a line, convert signal polarity, control variable amplitude, generate coexisting attractors, and even induce changes in its state. Specially, we realize the transition from a single-vortex attractor to a multivortex one by some changes in the initial values. Furthermore, the Pearson correlation coefficient is used to demonstrate the higher initial value sensitivity of the system. Finally, the system is implemented through Multisim simulation and analog circuit separately, and their consistency validates the system effectively. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a Holling-III predator–prey model considering the perturbation of slow-varying, carrying capacity parameters. The study aims to address how the slow changes in carrying capacity influence the dynamics of the model. Based on the bifurcation theory and the slow–fast analysis method, the existence and the equilibrium of the autonomous system are explored, and then, the critical condition of Hopf bifurcation and transcritical bifurcation is established for the autonomous system. The slow–fast coupled nonautonomous system has quasiperiodic oscillations, single Hopf bursting oscillations, and transcritical–Hopf bursting oscillations within a certain range of perturbation amplitude variation if the carrying capacity perturbation amplitude crosses some critical values, such that the predator–prey management is challenging for the extinction of predator populations under the critical value. The motion pattern of the nonautonomous system is closely related to the transcritical bifurcation, Hopf bifurcation and attractor type of the autonomous system. Finally, the effects of changes in parameters related to predator aggressiveness on system behavior are investigated. These results show how crucial the predator–prey control is for varying carrying capacities. [ABSTRACT FROM AUTHOR]

This contribution investigates a photosensitive circuit based on a mutator for emulating memristor, memcapacitor, and meminductor. The circuit contains a phototube, a mutator, and several basic electronic components. The phototube is employed within the circuit to capture the external illumination for energy injection in order to obtain a continuous signal source. By varying the light intensity on the phototube, chaotic dynamics such as attractors, bifurcation diagrams, and Lyapunov exponents spectrum are analyzed, while the mutator is switched to different mem-elements without changing the structure of the circuit. Especially, the special phenomenon of the coexistence of attractors is discovered. Ultimately, the circuit is realized using DSP. This circuit holds promise in offering potential insights into the dynamics of neural networks, and it could find applications in the development of highly sensitive sensors. [ABSTRACT FROM AUTHOR]

We investigate how the phase space structure of a Three-Dimensional (3D) autonomous Hamiltonian system evolves across a series of successive Two-Dimensional (2D) and 3D pitchfork and period-doubling bifurcations, as the transition of the parent families of Periodic Orbits (POs) from stability to simple instability leads to the creation of new stable POs. Our research illustrates the consecutive alterations in the phase space structure near POs as the stability of the main family of POs changes. This process gives rise to new families of POs within the system, either maintaining the same or exhibiting higher multiplicity compared to their parent families. Tracking such a phase space transformation is challenging in a 3D system. By utilizing the color and rotation technique to visualize the Four-Dimensional (4D) Poincaré surfaces of section of the system, i.e. projecting them onto a 3D subspace and employing color to represent the fourth dimension, we can identify distinct structural patterns. Perturbations of parent and bifurcating stable POs result in the creation of tori characterized by a smooth color variation on their surface. Furthermore, perturbations of simple unstable parent POs beyond the bifurcation point, which lead to the birth of new stable families of POs, result in the formation of figure-8 structures of smooth color variations. These figure-8 formations surround well-shaped tori around the bifurcated stable POs, losing their well-defined forms for energies further away from the bifurcation point. We also observe that even slight perturbations of highly unstable POs create a cloud of mixed color points, which rapidly move away from the location of the PO. Our study introduces, for the first time, a systematic visualization of 4D surfaces of section within the vicinity of higher multiplicity POs. It elucidates how, in these cases, the coexistence of regular and chaotic orbits contributes to shaping the phase space landscape. [ABSTRACT FROM AUTHOR]

Benefiting from trigonometric and hyperbolic functions, a nonlinear megastable chaotic system is reported in this paper. Its nonlinear equations without linear terms make the system dynamics much more complex. Its coexisting attractors' shape is diamond-like; thus, this system is said to have diamond-shaped oscillators. State space and time series plots show the existence of coexisting chaotic attractors. The autonomous version of this system was studied previously. Inspired by the former work and applying a forcing term to this system, its dynamics are studied. All forcing term parameters' impacts are investigated alongside the initial condition-dependent behaviors to confirm the system's megastability. The dynamical analysis utilizes one-dimensional and two-dimensional bifurcation diagrams, Lyapunov exponents, Kaplan–Yorke dimension, and attraction basin. Because of this system's megastability, the one-dimensional bifurcation diagrams and Kaplan–Yorke dimension are plotted with three distinct initial conditions. Its analog circuit is simulated in the OrCAD environment to confirm the numerical simulations' correctness. [ABSTRACT FROM AUTHOR]

This paper investigates the nonlinear bifurcation and chaos characteristics of piezoelectric composite lattice sandwich plates at 1:3 internal resonance. The nonlinear vibration partial differential equation is first discretized to become an ordinary differential equation by applying the Galerkin method. The main resonance modulation equations and the parametric resonance for the external excitation frequency that is close to the system's second-order modal frequency are then obtained by using the multiscale method. The Newton–Raphson method is subsequently applied to obtain the bifurcation diagram of the steady-state equilibrium of modulation equation with varying system parameters. The equilibrium stability is finally analyzed. We confirm the existence of static bifurcation such as saddle-node bifurcation, pitchfork bifurcation, and Hopf dynamic bifurcation. A detailed analysis is also conducted on the complex nonlinear jump phenomenon caused by the presence of multiple nonlinear steady-state solution regions. In the dynamic Hopf bifurcation interval, the fourth-order Runge–Kutta method is used to continue to track the dynamic periodic solution of the modulation equation in Cartesian coordinates. It is found that there are multiple boundary crises and attractor merging crises in the vibration system for piezoelectric composite lattice sandwich plates. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the impact of functional shifts in a time-discrete cross-catalytic system. We use the hypercycle model considering that one of the species shifts from a cooperator to a degrader. At the bifurcation caused by this functional shift, an invariant curve collapses to a point P while, simultaneously, two fixed points collide with P in a transcritical bifurcation. Moreover, all points of a line containing P become fixed points at the bifurcation and only at the bifurcation in a degenerate scenario. We provide a complete analytical description of this degenerate bifurcation. As a result of our study, we prove the existence of the invariant curve arising from the transition to cooperation. [ABSTRACT FROM AUTHOR]

Global bifurcations with sliding have rarely been studied in three-dimensional piecewise smooth systems. In this paper, codimension-2 bifurcations of nondegenerate sliding heteroclinic cycle Γ are investigated in three-dimensional Filippov systems. Two cases of sliding heteroclinic cycle are discussed: (C 1) connecting two saddle-foci, (C 2) connecting one saddle-focus and one saddle. It is proved that at most one sliding homoclinic or one sliding periodic orbit can bifurcate from Γ under certain conditions at the eigenvalues of the equilibria, but they cannot coexist. The asymptotic stability of the sliding periodic orbit and the structural feature of the bifurcation curves of homoclinic orbits are further studied. Finally, two numerical examples corresponding to cases (C 1) and (C 2) , respectively, are simulated to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The renowned 2D invertible Hénon map turns into 1D noninvertible quadratic map when its leading parameter b becomes zero. This well-known link was studied by Mira who demonstrated that the bifurcation set of Hénon diffeomorphism is similar to his "box-within-a-box" bifurcation structure of 1D endomorphism. In general, such similarity has not been strictly established, especially in multidimensional cases. In this paper, we proved that the Mira bifurcation structure of a quadratic noninvertible map persists when the parameter increases from zero and the map turns into an invertible multidimensional generalized Hénon map. The changes of periodic and homoclinic orbits and chaotic attractors at this transition are described. We proved the existence of Newhouse regions is different from those Mira boxes that accumulate to the homoclinic bifurcations. [ABSTRACT FROM AUTHOR]

In this paper, we study the family of quadratic Riccati differential systems. Our goal is to obtain the complete topological classification of this family on the Poincaré disk compactification of the plane. The family was partially studied before but never from a truly global viewpoint. Our approach is global and we use geometry to achieve our goal. The geometric analysis we perform is via the presence of two invariant parallel straight lines in any generic Riccati system. We obtain a total of 119 topologically distinct phase portraits for this family. Furthermore, we give the complete bifurcation diagram in the 12-dimensional space of parameters of this family in terms of invariant polynomials, meaning that it is independent of the normal forms in which the systems may be presented. This bifurcation diagram provides an algorithm to decide for any given quadratic system in any form it may be presented, whether it is a Riccati system or not, and in case it is to provide its phase portrait. [ABSTRACT FROM AUTHOR]

This paper investigates a spatiotemporal SI epidemiological model under homogeneous Neumann boundary conditions. First, the long-time behavior of the solutions is described, a priori estimates of nonconstant positive solutions are given, and the nonexistence of nonconstant positive steady states is proved by the energy method. Second, the Turing instability of the positive constant steady-state is discussed, and the existence of nonconstant positive steady states is shown by using the degree theory. Moreover, applying the bifurcation theory, we establish the local and global structures of the steady-state bifurcation from simple eigenvalues, and describe some conditions for determining the direction of bifurcation, where the techniques of space decomposition and implicit function theorem are adopted to deal with the local structure of the steady-state bifurcation from double eigenvalues. Finally, some analysis results are supplemented by numerical simulations. [ABSTRACT FROM AUTHOR]

The two-degree-of-freedom (2-DOF) aerial manipulator is composed of a quadrotor aircraft and a 2-DOF manipulator, which significantly expands the scope of grabbing and transporting objects. After the manipulator is installed on the quadrotor, the manipulator and the load will cause serious interference to the quadrotor, resulting in difficulty of system control and even instability. This paper presents a mathematical model of the angular velocity system of the 2-DOF aerial manipulator. The model considers the influence of the manipulator and the load on the quadrotor. Based on this model, the nonlinear dynamics of the angular velocity system of the 2-DOF aerial manipulator are analyzed by solving the equilibrium points, calculating the Lyapunov exponents, analyzing the dynamic bifurcation diagram, and drawing the dynamic region distribution map. It is found that angular velocity can produce the dynamic behaviors of sink, period-doubling, and chaos under certain circumstances. By analyzing the nonlinear dynamic behaviors of the angular velocity system under different manipulator postures, different manipulator configurations, different load masses, and different load resistances, the stability of the angular velocity system is analyzed to guide the use of the aerial manipulator more safely and efficiently. [ABSTRACT FROM AUTHOR]

The use of ideal memristors in a continuous-time (CT) nonlinear circuit is known to greatly enrich the dynamic behavior with respect to the memristorless counterpart, which is a crucial property for applications in future analog electronic circuits. This can be explained via the flux–charge analysis method (FCAM), according to which CT circuits with ideal memristors have for structural reasons first integrals (or invariants of motion, or conserved quantities) and their state space can be foliated in infinitely many invariant manifolds where they can display different dynamics. The paper introduces a new discretization scheme for the memristor which, differently from those adopted in the literature, guarantees that the first integrals of the CT memristor circuits are preserved exactly in the discretization, and that this is true for any step size. This new scheme makes it possible to extend FCAM to discrete-time (DT) memristor circuits and rigorously show the existence of invariant manifolds and infinitely many coexisting attractors (extreme multistability). Moreover, the paper addresses standard bifurcations varying the discretization step size and also bifurcations without parameters, i.e. bifurcations due to varying the initial conditions for fixed step size and circuit parameters. The method is illustrated by analyzing the dynamics and flip bifurcations with and without parameters in a DT memristor–capacitor circuit and the Poincaré–Andronov–Hopf bifurcation in a DT Murali–Lakshmanan–Chua circuit with a memristor. Simulations are also provided to illustrate bifurcations in a higher-order DT memristor Chua's circuit. The results in the paper show that DT memristor circuits obtained with the proposed discretization scheme are able to display even richer dynamics and bifurcations than their CT counterparts, due to the coexistence of infinitely many attractors and the possibility to use the discretization step as a parameter without destroying the foliation in invariant manifolds. [ABSTRACT FROM AUTHOR]

The dynamics of a delay Sel'kov–Schnakenberg reaction–diffusion system are explored. The existence and the occurrence conditions of the Turing and the Hopf bifurcations of the system are found by taking the diffusion coefficient and the time delay as the bifurcation parameters. Based on that, the existence of codimension-2 bifurcations including Turing–Turing, Hopf–Hopf and Turing–Hopf bifurcations are given. Using the center manifold theory and the normal form method, the universal unfolding of the Turing–Hopf bifurcation at the positive constant steady-state is demonstrated. According to the universal unfolding, a Turing–Hopf bifurcation diagram is shown under a set of specific parameters. Furthermore, in different parameter regions, we find the existence of the spatially inhomogeneous steady-state, the spatially homogeneous and inhomogeneous periodic solutions. Discretization of time and space visualizes these spatio-temporal solutions. In particular, near the critical point of Hopf–Hopf bifurcation, the spatially homogeneous periodic and inhomogeneous quasi-periodic solutions are found numerically. [ABSTRACT FROM AUTHOR]

In this paper, a type of modified dual memristive Shinriki oscillator is constructed with a flux-controlled absolute-type memristor and a voltage-controlled generic memristor, and the proposed oscillator with abundant dynamical behaviors, including the multistability and antimonotonicity, is comprehensively studied through dynamical distribution graphs, bifurcation diagrams, Lyapunov exponents and phase portraits. It is found that the passive/active state of memristor, which means different characteristics in the v – i domain with positive and negative parameters of the elements, can affect the state of the oscillator. For example, if the memristor is active, the oscillator will change more frequently in the multistable region. Also, it is noted that, for inherent initial-related symmetry and circuit structures with duality, both phenomena have strong symmetric characteristics and opposite evolution trends modulated by values of corresponding components. Especially, the bubbles, which are symmetric about parameters with duality and own complex evolution laws, have rarely been explored in previous works. In addition, the memristive oscillator is modularized based on field programmable gate array (FPGA) technology, and the multiple coexisting attractors are captured, which verifies the accuracy of the numerical results. [ABSTRACT FROM AUTHOR]

This study investigates a structurally unstable synchronization phenomenon observed in the two-coupled Izhikevich neuron model. As the result of varying the system parameter in the region of parameter space close to where the unstable synchronization is observed, we find significant changes in the stability of its periodic motion. We derive a discrete-time dynamical system that is equivalent to the original model and reveal that the unstable synchronization in the continuous-time dynamical system is equivalent to border-collision bifurcations in the corresponding discrete-time system. Furthermore, we propose an objective function that can be used to obtain the parameter set at which the border-collision bifurcation occurs. The proposed objective function is numerically differentiable and can be solved using Newton's method. We numerically generate a bifurcation diagram in the parameter plane, including the border-collision bifurcation sets. In the diagram, the border-collision bifurcation sets show a novel bifurcation structure that resembles the "strike-slip fault" observed in geology. This structure implies that, before and after the border-collision bifurcation occurs, the stability of the periodic point discontinuously changes in some cases but maintains in other cases. In addition, we demonstrate that a border-collision bifurcation set successively branches at distinct points. This behavior results in a tree-like structure being observed in the border-collision bifurcation diagram; we refer to this structure as a border-collision bifurcation tree. We observe that a periodic point disappears at the border-collision bifurcation in the discrete-time dynamical system and is simultaneously replaced by another periodic point; this phenomenon corresponds to a change in the firing order in the continuous-time dynamical system. [ABSTRACT FROM AUTHOR]

This paper presents a discrete predator–prey interaction model involving hibernating vertebrates, with detailed analysis and simulation. Hibernation contributes to the survival and reproduction of organisms and species in the ecosystem as a whole. In addition, it also constitutes a wise sharing of time, space, and resources with others. We have created a new predator–prey model by integrating the two species, Holling-III and Holling-I, which have a bifurcation within a specified parameter range. We discovered that this system possesses the stability of fixed points as well as several bifurcation behaviors. To accomplish this, the center manifold theorem and bifurcation theory are applied to create existence conditions for period-doubling bifurcations and Neimark–Sacker bifurcations, which are depicted in the graph as distinct structures. Examples of numerical simulations include bifurcation diagrams, maximum Lyapunov exponents, and phase portraits, which demonstrate not just the validity of theoretical analysis but also complex dynamical behaviors and biological processes. Finally, the Ott–Grebogi–Yorke (OGY) method and phases of chaos control bifurcation were used to control the chaos of predator–prey model in hibernating vertebrates. [ABSTRACT FROM AUTHOR]

This work presents a numerical method to color the bifurcation diagram of any discrete map, based on the distribution of the map's values in its domain. This density-colored diagram reveals information on the uniformity of the map's value distribution across its domain set, as a bifurcation parameter is increased. This diagram can serve as a complementary visual tool for the analysis of chaotic maps, and can be vital for the application of chaotic dynamics. [ABSTRACT FROM AUTHOR]

This paper thoroughly discusses the electric activities generated by ion communication between cells and their surrounding environment. Specifically, it focuses on the transients of firing activities of a four-dimensional pituitary model that evolves on three disparate timescales. To examine the impact of inward rectifying K + current and calcium concentration on the firing activities, a bifurcation analysis is conducted, categorizing three primary behaviors: resting, tonic spiking, and bursting. Each behavior is validated through their respective time courses, with the pituitary cells showing higher secretion rates of hormones and neurotransmitters during bursting than spiking. The geometric singular perturbed theory is applied to reveal hidden geometric features and the transient mechanisms associated with bursting, particularly mixed-mode oscillations (MMOs). Singular orbit construction performed in two-timescale separation with different viewpoints offers clarity on the underlying dynamic mechanisms. Canard-induced MMOs are observed in the context of 1 fast/3 slow and 2 fast/2 slow separations, facilitated by the presence of folded saddle-node and folded node, respectively. Additionally, the fast–slow analysis of the 3 fast/1 slow subsystem, which treats calcium concentration c as a parameter, in conjunction with the singular orbit constructions, effectively illustrates the system's complex dynamics. Furthermore, the information obtained in 1 fast/3 slow and 3 fast/1 slow discussions is interplayed in the context of three-timescale separation. The singular orbits identified within three-timescale framework offer a supplemental perspective to the delicate firing patterns observed in two-timescale analysis, enriching the overall understanding of the transient and long-term firing behaviors of the pituitary cells. This study presents valuable insights into the firing features in pituitary cells from the perspectives of dynamic systems. The singular perturbation analysis provides useful viewpoints for accessing firing patterns in multi-timescale systems. [ABSTRACT FROM AUTHOR]

We consider a discrete-time model of a population of agents participating in a minority game using a quantum cognition in an approach with binary choices. As the agents make decisions based on both their present and past states, the model is inherently two-dimensional, but can be reduced to a one-dimensional system governed by a bi-valued function. Through this reduction, we prove how the complex bifurcation structure in the model's 2D parameter space can be explained by a few codimension-2 bifurcation points of a type not yet reported in the literature. These points act as organizing centers for period-adding structures that partially overlap, leading to bistability. [ABSTRACT FROM AUTHOR]

The three-dimensional Muthuswamy–Chua–Ginoux (MCG) circuit model is a generalization of the paradigmatic canonical Muthuswamy–Chua circuit, where a physical memristor assumes the role of a thermistor, and it is connected in series with a linear passive capacitor, a linear passive inductor, and a nonlinear resistor. The physical memristor presents an electrical resistance which is a function of temperature. Nowadays, the MCG circuit model has gained considerable attention due to the lack of extensive numerical explorations and their distinct dynamical properties, exemplified by phenomena such as the transition from torus breakdown to chaos, giving rise to a double spiral attractor associated to independent period-doubling cascades. In this contribution, the complex dynamics of the MCG circuit model is studied in terms of the Lyapunov exponents spectra, Kaplan–Yorke (KY) dimension, and the number of local maxima (LM) computed in one period of oscillation, as two parameters are simultaneously varied. Using the Lyapunov spectra to distinguish different dynamical regimes, KY dimension to estimate the attractors' dimension, and the number of LM to characterize different periodic attractors, we construct high-resolution two-dimensional stability diagrams considering specific ranges of the parameter pairs (α ,). These parameters are associated with the inverse of the capacitance in the passive capacitor, and the heat capacitance and dissipation constant of the thermistor, respectively. Unexpectedly, we identify sequences of infinite self-organized generic stable periodic structures (SPSs) and Arnold tongues-like structures (ATSs) merged into chaotic dynamics domains, and the coexistence of different attracting sets (attractors) for the same parameter combinations and different initial conditions (multistability). We explore the multistable dynamics using the stability analysis and computation of Lyapunov coefficients, the inspection of the coexisting attractors, bifurcations diagrams, and basins of attraction. The periods of the ATSs and a particular sequence of shrimp-shaped SPSs obey specific generating and recurrence rules responsible for the bifurcation cascades. As the MCG circuit model has the crucial properties presented by the usual Muthuswamy–Chua circuit model, specific properties explored in our study should be helpful in real problems involving circuits with the presence of physical memristor playing the role of thermistors. [ABSTRACT FROM AUTHOR]

In this paper, we consider a special kind of geometrical nonlinear oscillator with a mass parameter admitting two different dynamical states leading to a double-valued potential energy. A cylindrical manifold is introduced to formulate the equation of motion to describe the distinguished dynamical behaviors. With the help of Hamiltonian, complex bifurcations are demonstrated with varying parameters including periodic solutions, the steady states and the blowing up phenomenon near = ± π 2 to infinity. A toroidal manifold is introduced to map the infinities into (0 , ± 2 , 0) on the torus exhibiting saddle-node-like behavior, where the uniqueness of solution is lost, for which a special "collision" parameter is introduced to define the possible motion leaving from infinities. Numerical calculation is carried out to generate bifurcation diagrams using Poincaré sections for the perturbed system to exhibit complex dynamics including the coexistence of periodic solutions, chaos from the coexisting periodic doubling and also instant chaos from the coexisting periodic solutions. The results demonstrated herein this paper provide a brand new insight into the understanding of enriched nonlinear dynamics and an essential explanation about "collision" of mechanical system with both geometrical and mass parameters. [ABSTRACT FROM AUTHOR]

We study bifurcation behavior in periodic perturbations of two-dimensional symmetric systems exhibiting codimension-two bifurcations with a double zero eigenvalue when the frequencies of the perturbation terms are small. We transform the periodically perturbed systems to simpler ones which are periodic perturbations of the normal forms for the codimension-two bifurcations, and apply the subharmonic and homoclinic Melnikov methods to analyze bifurcations occurring there. In particular, we show that there exist transverse homoclinic or heteroclinic orbits, which yield chaotic dynamics, in wide parameter regions. These results can be applied to three or higher-dimensional systems and even to infinite-dimensional systems with the assistance of center manifold reduction and the invariant manifold theory. We illustrate our theory for a pendulum subjected to position and velocity feedback control when the desired position is periodic in time. We also give numerical computations by the computer tool AUTO to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

Research on nonautonomous delayed differential equations (DDEs) is crucial and very difficult due to nonautonomy and time delay in many fields. The main work of the present paper is to discuss complex dynamics of nonautonomous DDEs, such as Hopf bifurcation, periodic solutions and multistability. We consider three examples of nonautonomous DDEs with time-varying coefficients: a harmonically forced Duffing oscillator with time delayed state feedback and periodic disturbance, generalized van der Pol oscillator with delayed displacement difference feedback and periodic disturbance, and an electro-mechanical system with delayed and periodic disturbance. Firstly, we obtain the amplitude equations of these three examples by the method of multiple scales (MMS), and then analyze the stability of approximate solutions by the Routh–Hurwitz criterion. The obtained amplitude equations are used to construct the bifurcation diagrams, so that we can observe the occurrence of the Hopf bifurcation and judge its type (super- or subcritical) from the bifurcation diagrams. We discover rich dynamic phenomena of the three systems under consideration, such as Hopf bifurcation, quasi-periodic solutions and the coexistence of multiple stable solutions, and then discuss the impact of some parameter changes on the system dynamics. Finally, we validate the correctness of these theoretical conclusions by software WinPP, and the numerical simulations are consistent with our theoretical findings. Therefore, the MMS we use to analyze the dynamics of nonautonomous DDEs is effective, which is of great significance to the research of nonautonomous DDEs in many fields. [ABSTRACT FROM AUTHOR]

Children frequently experience absence seizures, a form of seizure that is characterized by brief periods of unconsciousness and staring spells. While many studies have been conducted on absence seizures, there is still some uncertainty regarding the precise mechanisms causing absence seizures. The basal ganglia are believed to be essential in regulating thalamocortical network activity responsible for such seizures. Controlling or designing a treatment for this disorder requires an understanding of the contribution of the basal ganglia regions in the absence seizures. In this regard, efforts have been made to propose a mathematical model of brain neuronal substructures and their connections in the basal ganglia. The basal ganglia cortex-thalamus (BGCT) model is one of the most-studied mathematical models investigating absence seizures. However, this model has not been comprehensively studied from the viewpoint of dynamical behavior. Hence, to evaluate the BGCT model, this paper is devoted to studying a detailed and in-depth bifurcation analysis of the basal ganglia regions in the BGCT loop. Moreover, the 0–1 test for chaos is performed to confirm the results shown in the bifurcation diagrams. Our results suggest that the BGCT model can exhibit chaotic behavior in small regions of the coupling parameter, which is consistent with the complex nature of the brain neuronal network. [ABSTRACT FROM AUTHOR]

Dynamic behavior investigations of infectious disease models are central to improve our understanding of emerging characteristics of model states interaction. Here, we consider a Susceptible-Infected (SI) model with a general state-dependent delay, which covers an immuno-epidemiological model of pathogen transmission, developed in our early study, using a threshold delay to examine the effects of multiple exposures to a pathogen. The analysis in the previous work showed the appearance of forward as well as backward bifurcations of endemic equilibria when the basic reproductive ratio R 0 is less than unity. The analysis, in the present work, of the endemically infected equilibrium behavior, through the study of a second order exponential polynomial characteristic equation, concludes the existence of a Hopf bifurcation on the upper branch of the backward bifurcation diagram and gives the criteria for stability switches. Furthermore, the inclusion of state-dependent delays is shown to entirely change the dynamics of the SI model and give rise to rich behaviors including periodic, torus and chaotic dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we report the bifurcations of mode-locked periodic orbits occurring in maps of three or higher dimensions. The "torus" is represented by a closed loop in discrete time, which contains stable and unstable cycles of the same periodicity, and the unstable manifolds of the saddle. We investigate two types of "doubling" of such loops: (a) two disjoint loops are created and the iterates toggle between them, and (b) the length of the closed invariant curve is doubled. Our work supports the conjecture of Gardini and Sushko, which says that the type of bifurcation depends on the sign of the third eigenvalue. We also report the situation arising out of Neimark–Sacker bifurcation of the stable and saddle cycles, which creates cyclic closed invariant curves. We show interesting types of saddle-node connection structures, which emerge for parameter values where the stable fixed point has bifurcated but the saddle has not, and vice versa. [ABSTRACT FROM AUTHOR]

Consolidation of new information in memory occurs through the simultaneous occurrence of sharp-wave ripples (SWR) in the hippocampus network, fast–slow spindles in the thalamus network, and up and down oscillations in the cortex network during sleep. Previous studies have investigated the influential and active role of spindles and sharp-wave ripples in memory consolidation. However, a detailed investigation of the effect of membrane voltage of neurons and synaptic connections between neurons in the cortex, hippocampus, and thalamus networks to create spindle and SWR is required. This paper studies the dynamic behaviors of a hippocampal-thalamic-cortical network as a function of synaptic connection between excitatory neurons, inhibitory neurons (in the hippocampus and cortex), reticular neurons, and thalamocortical neurons (in the thalamic network). The bifurcation diagrams of the hippocampus, cortex, and thalamus networks are obtained by varying the strengths of different synaptic connections. The power diagrams for SWR and sleep spindles are shown accordingly. The results show that variations in synaptic self-connection (and inhibitory synaptic connection) of excitatory neurons in the CA3 region, as well as synaptic connection between excitatory neurons from CA1 region to excitatory neurons (and inhibitory neurons) in the cortex network have the most significant influence on dynamical behavior of the network. Furthermore, comparing diagrams for different synaptic connections shows that SWR is formed by excitatory neurons in CA3 region of the hippocampal network, passes through CA1 region, and enters cortex network. [ABSTRACT FROM AUTHOR]

Employing the Riccati–Bernoulli sub-ODE method (RBSM) and improved Weierstrass elliptic function method, we handle the proposed (2 + 1) -dimensional nonlinear fractional electrical transmission line equation (NFETLE) system in this paper. An infinite sequence of solutions and Weierstrass elliptic function solutions may be obtained through solving the NFETLE. These new exact and solitary wave solutions are derived in the forms of trigonometric function, Weierstrass elliptic function and hyperbolic function. The graphs of soliton solutions of the NFETLE describe the dynamics of the solutions in the figures. We also discuss elaborately the effects of fraction and arbitrary parameters on a part of obtained soliton solutions which are presented graphically. Moreover, we also draw meaningful conclusions via a comparison of our partially explored areas with other different fractional derivatives. From our perspectives, by rewriting the equation as Hamiltonian system, we study the phase portrait and bifurcation of the system about NFETLE and we also for the first time discuss sensitivity of the system and chaotic behaviors. To our best knowledge, we discover a variety of new solutions that have not been reported in existing articles V 1 , 2 ∗ ∗ , ... , V 7 , 8 ∗ ∗ , V 9 , 1 0 , ... , V 1 3 , 1 4 . The most important thing is that there are iterative ideas in finding the solution process, which have not been seen before from relevant articles such as [Tala-Tebue et al., 2014; Fendzi-Donfack et al., 2018; Ashraf et al., 2022; Ndzana et al., 2022; Halidou et al., 2022] in seeking for exact solutions about NFETLE. [ABSTRACT FROM AUTHOR]

Considering the effects of time-varying meshing stiffness, time-varying support stiffness, transmission errors, tooth side clearance and bearing clearance, a nonlinear dynamics model of the coupled gear-rotor-bearing transmission system of a new energy vehicle is constructed. Firstly, the fourth-order Runge–Kutta integral method is used to solve the differential equations of the system dynamics, and the time-varying meshing force diagram, time history diagram, phase diagram, FFT spectrum diagram, Poincaré map and bifurcation diagram of the system are obtained to study the influence of the external load excitation frequency on the dynamics characteristics of the system. In addition, the multiscale method is used to analyze the main resonance characteristics of the system and to determine the main resonance stability conditions of the system. The effect of time lag control parameters and external load excitation frequency on the main resonance of the system is analyzed by numerical methods. The results show that the gear-rotor-bearing coupled transmission system of the new energy vehicle has obviously nonlinear characteristics, avoiding the system instability interval reasonable selection of external load excitation frequency, meshing damping, time lag parameters and load fluctuations, which can be used to improve the stability of the transmission system of the new energy vehicle. [ABSTRACT FROM AUTHOR]