Bifurcation Parameter Research Articles

An analysis of the stability and bifurcation of a discrete-time predator-prey model with the slow-fast effect on the predator.

In many environments, predators have significantly longer lives and meet several generations of prey, or the prey population reproduces rapidly. The slow-fast effect can best describe such predator-prey interactions. The slow-fast effect ε can be considered as the ratio between the predator's linear death rate and the prey's linear growth rate. This paper examines a slow-fast, discrete predator-prey interaction with prey refuge and herd behavior to reveal its complex dynamics. Our methodology employs the eigenvalues of the Jacobian matrix to examine the existence and local stability of fixed points in the model. Through the utilization of bifurcation theory and center manifold theory, it is demonstrated that the system undergoes period-doubling bifurcation and Neimark-Sacker bifurcation at the positive fixed point. The hybrid control method is utilized as a means of controlling the chaotic behavior that arises from these bifurcations. Moreover, numerical simulations are performed to demonstrate that they are consistent with analytical conclusions and to display the complexity of the model. At the interior fixed point, it is shown that the model undergoes a Neimark-Sacker bifurcation for larger values of the slow-fast effect parameter by using the slow-fast effect parameter ε as the bifurcation parameter. This is reasonable since a large ε implies an approximate equality in the predator's death rate and the prey's growth rate, automatically leading to the instability of the positive fixed point due to the slow-fast impact on the predator and the presence of prey refuge.

Optically injected quantum dot lasers and its complex dynamics

Abstract Lasers are paradigmatic examples of nonlinear systems and have played a crucial role in developing nonlinear dynamics into an interdisciplinary topic. Indeed, a freely operating laser represents a nontrivial system, but the phenomena that occur are much more interesting when the laser is exposed to external factors such as optical feedback (OFB) and optical injection (OI). This paper deals with the analysis of the dynamic behavior of a quantum dot semiconductor laser (QDSL) model under the influence of optical injection from another laser. The proposed model was studied numerically with the help of MATLAB. A QDSL system with optical injection was proposed and studied numerically. The quantum dot (QD) laser was used, which is a quantum semiconductor laser and has strong stability without external perturbation. When it is subjected to feedback, it has a critical effect on the follower laser and changes its stable behavior before injection to a set of nonlinear dynamics. The effect of both optical feedback strength and the delay time on the performance and behavior of the bifurcation patterns in the follower laser output was studied, taking into account changing the bifurcation parameters, the injection force k, the slave laser frequency, and the slave laser delay time. The behavior of the slave was studied each time and compared with the behavior of the master. We observed the chaotic paths (essentially stable and continuous pulsating oscillations evolving into periodic and semi-periodic oscillations and then chaotic ones). The time series corresponding to the bifurcation diagrams and the probability of the time interval between spikes and attractors were studied.

On the stochastic bifurcations regarding random iterations of polynomials of the form

Abstract In this paper, we consider random iterations of polynomial maps $z^{2} + c_{n}$ , where $c_{n}$ are complex-valued independent random variables following the uniform distribution on the closed disk with center c and radius r. The aim of this paper is twofold. First, we study the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness of random Julia sets. Second, we investigate the bifurcation of our random iterations and give quantitative estimates of bifurcation parameters. In particular, we prove that for the central parameter $c = -1$ , almost every random Julia set is totally disconnected with much smaller radial parameters r than expected. We also introduce several open questions worth discussing.

Hopf bifurcation analytical expression and control strategy in direct‐drive permanent magnet synchronous generator

SummaryThe unstable operation of a direct‐drive permanent magnet synchronous generator (DPMSG) is exacerbated by periodic oscillation, having a significant impact on the safe and stable functioning of wind energy generation systems. This paper proposes an approximate solution method for analyzing the periodic oscillation of the Hopf bifurcation and a method for suppressing its bifurcation through H∞ robust control. Firstly, a three‐dimensional nonlinear dynamic model of the DPMSG is constructed. Secondly, the Hopf bifurcation of the system with changes in internal parameters is solved using the time domain and frequency domain methods, and the harmonic balancing method is used to approximate the periodic solution at the point of Hopf bifurcation. The order of the Hopf bifurcation neighborhood is then decreased using the central manifold theorem. Finally, this study suggests a H∞ output feedback control employing bifurcation parameters. The Nyquist criteria are used for evaluating small signal stability and convergence speed of the response system in DPMSG before and after dimensionality reduction. The simulation results suggest that the proposed strategy helps to tackle the periodic solution expression and instability difficulties caused by Hopf bifurcation. This paper provides theoretical suggestions for the future reliable operation of new energy generation systems.

Bistability and Bifurcations of Tumor Dynamics with Immune Escape and the Chimeric Antigen Receptor T-Cell Therapy

Tumor immune escape refers to the inability of the immune system to clear tumor cells, which is one of the major obstacles in designing effective treatment schemes for cancer diseases. Although clinical studies have led to promising treatment outcomes, it is imperative to design theoretical models to investigate the long-term treatment effects. In this paper, we develop a mathematical model to study the interactions among tumor cells, immune escape tumor cells, and T lymphocyte. The chimeric antigen receptor (CAR) T-cell therapy is also described by the mathematical model. Bifurcation analysis shows that there exists backward bifurcation and saddle-node bifurcation when the immune intensity is used as the bifurcation parameter. The proposed model also exhibits bistability when its parameters are located between the saddle-node threshold and backward bifurcation threshold. Sensitivity analysis is performed to illustrate the effects of different mechanisms on the backward bifurcation threshold and basic immune reproduction number. Simulation studies confirm the bifurcation analysis results and predict various types of treatment outcomes using different CAR T-cell therapy strengths. Analysis and simulation results show that the immune intensity can be used to control the tumor size, but it has no effect on the control of the immune escape tumor size. The introduction of the CAR T-cell therapy will reduce the immune escape tumor size and the treatment effect depends on the CAR T-cell therapy strength.

A Novel Method for the Measurement of Retinal Arteriolar Bifurcation.

The purpose of this research was to develop protocols for evaluating the bifurcation parameters of retinal arteriole and establish a reference range of normal values. In this retrospective study, we measured a total of 1314 retinal arteriolar bifurcations from 100 fundus photographs. We selected 200 from these bifurcations for testing inter-measurerand inter-method agreement. Additionally, we calculated the normal reference range for retinal arteriolar bifurcation parameters and analyzed the effects of gender, age, and anatomical features on retinal arteriolar bifurcation. The measurement method proposed in this study has demonstrated nearly perfect consistency among different measurers, with interclass correlation coefficient (ICC) for all bifurcation parameters of retinal arteriole exceeding 0.95. Among healthy individuals, the retinal arteriolar caliber was narrowest in young adults and increased in children, teenagers, and the elderly; retinal arteriolar caliber was greater in females than in males; and the diameter of the inferior temporal branch exceeded that of the superior temporal branch. The angle between the two branches of retinal arteriolar bifurcation was also greater in females than in males. When using the center of the optic disc as a reference point, the angle between the two branches of the retinal arteriole at the proximal or distal ends increased. In contrast, the estimated optimum theoretical values of retinal arteriolar bifurcation were not affected by these factors. The method for the measurement of retinal arteriolar bifurcation in this study was highly accurate and reproducible. The diameter and branching angle of the retinal arteriolar bifurcation were more susceptible to the influence of gender, age, and anatomical features. In comparison, the estimated optimum theoretical values of retinal arteriolar bifurcation were relatively stable. Video available for this article.

Bifurcation analysis of a tristable system with fractional derivative under colored noise excitation

Stochastic bifurcation has received much attention recently and is of great significance in the research on the dynamics of nonlinear systems. In this paper, we study the bifurcation of a self-sustained tristable system containing fractional derivative under the excitation of two colored noises, where the self-sustained tristable system consists of stable limit cycles and a stable state. The multiple scale method and generalized harmonic function are used to convert the original system into an equal system without obvious time delay and fractional derivative. Then the stationary probability density function (SPDF) of the system is obtained by using the stochastic averaging method to discuss stochastic bifurcation. Based on singularity theory, the equation satisfying amplitude is obtained to derive the corresponding bifurcation diagram. From the bifurcation analysis of the system, it is revealed that fractional order, fractional coefficient, and the intensity and correlation time of colored noise can be utilized as bifurcation parameters, causing peculiar stochastic bifurcation phenomena and regulating the output of the system, as well as the relatively big colored noise intensity and relatively small correlation time facilitate the realization of large magnitude limit cycle. Monte Carlo numerical results verify the validity of the theoretical approach. In addition, appropriately changing the delayed feedback parameter helps to govern the bifurcation range and causes richer bifurcation phenomena. The results of this paper contribute to a better understanding of self-sustained tristable systems with fractional derivative and may provide reference value in resolving the problems of chattering in high-speed aircraft and applications to viscoelastic materials.

Emergence of spiral and antispiral patterns and its CGLE analysis in leech-heart interneuron model with electromagnetic induction

Neurons can exhibit various rhythmic activities such as bursting, spiking, and quiescent states when exposed to external input current stimulus. In this paper, a model of medicinal leech's heart (LH) interneuron is considered to describe the dynamics of neurons with a varied range of electrical activities. The crucial insights into the model's dynamics are explored in three different parameter regimes: phasic spiking, regular spiking, and bursting, based upon the codimension-one bifurcation of the model by considering VK2shift as a bifurcation parameter. The spatiotemporal dynamics of the model are explored by allowing 1D and 2D diffusion in the membrane voltage. The 1D diffusive system produces irregular bursting dynamics for the intermediate value of diffusion coefficients, whereas, at higher values, it shows synchronized oscillations. In the presence of 2D diffusion, the emergence of different types of spiral patterns is observed in the system. Furthermore, the system is extended by incorporating electromagnetic induction in the membrane voltage to explore the effect of induction on the various dynamics of neural model. By varying its intensities, the membrane voltage in the extended model produces a variety of discharge modes, such as periodic spiking, fast-spiking, resting, and spike-adding phenomena. In addition, the emergence of anti-spiral patterns in the extended model near subcritical Hopf bifurcation is analytically verified using the complex Ginzburg-Landau equation (CGLE). These findings demonstrate that the firing patterns vary based on the control parameters, and these variations contribute to our understanding of how the brain system transmits and processes the signals.

Cross-diffusion mediated Spatiotemporal patterns in a predator–prey system with hunting cooperation and fear effect

Cooperation during hunting is a common predation plan of action among several large predators to increase their biomass raising capturing potential which can also produce fear in the prey and dispersal-influenced pattern creation is also relevant from both a fundamental and an applied standpoint. Considering these, we present a modified Leslie–Gower predator–prey model incorporating fear, intra-specific competition and hunting cooperation of predators following Berec’s encounter-driven functional response in presence of cross-diffusion. In the non-spatial system, biologically feasible equilibria and their stability conditions are derived. Taking the level of fear as bifurcation parameter, occurrence of limit cycle via Hopf bifurcation and the stability of limit cycle computing Lyapunov coefficient are discussed. In diffusive model, spacial focus on Turing instability conditions and to determine the relevant intervals at which various patterns will begin to appear. In the two dimensional spatial domain, under Neumann boundary condition, various Turing patterns such as spot, stripe, mixture of spot and stripe, labyrinth patterns are emerged varying fear, hunting cooperation and cross-diffusion coefficient. Hopf–Turing region is also a centre for several complex patterns. The numerical simulations indicate that the patterns are in same or opposite phase according to the sign of cross-diffusion coefficient. Intra-specific competition among predators tend the system towards stability. Cross-diffusion causes a paradox to population density with raising fear and hunting cooperation.

A nonlinear transient-dynamics approach to atopic dermatitis: Role of spontaneous remission

Atopic dermatitis (AD) is a common skin disease that can occur in all age groups. An intriguing phenomenon associated with AD is spontaneous remission, in which the symptoms can disappear even without any treatment, especially for patients at a young age. From the point of view of dynamical evolution, spontaneous remission in AD is a transient phenomenon. A clinic implication is that, if the transient time is short, then aggressive treatment may not be necessary. A key question is thus, statistically, how long the transient time can be? Due to the lack of clinic data, mathematical modeling is a viable approach to addressing this question. Modeling AD as a nonsmooth dynamical system and regarding the disease as a transient phenomenon with spontaneous remission marking the end of the transient, we obtain a quantitative understanding of the statistical characteristics of AD. In particular, we find that, depending on the immune state, two different types of transient behaviors can arise. For type-I spontaneous remission characterized by a healthy immune level with its skin state exhibiting mild oscillations, the transient time is short, which typically occurs in patients between infancy and childhood. In contrast, type-II spontaneous remission is characterized by a low immune level and its skin state exhibits severe oscillations with a long recovering time. Quantitatively, a scaling relation exists between the average transient time (or recovery time) and some key physiological parameters, revealing that the transient is superpersistent in the sense that its average lifetime can diverge in a drastic way: e∞ as a bifurcation parameter approaches a critical value. In this case, the disease is essentially permanent, thereby requiring and justifying active treatment.

Periodic and chaotic spiking behaviors in a simplified memristive Hodgkin-Huxley circuit

The famous Hodgkin-Huxley circuit contains two time-varying resistors to describe the electrophysiological characteristics of sodium and potassium ion channels. But the time-varying resistors are expressed by mixed exponential equations, which are unavailable to directly implement in analog circuit and hinders hardware application of the Hodgkin-Huxley circuit. To hit this issue, a simplified memristive Hodgkin-Huxley (m-HH) circuit is proposed, which only involves two locally active memristors (LAMs) to depict the sodium and potassium ion channels, a capacitor to describe the neuron membrane, a DC current to represent external stimulus, and two DC voltages to express the reversal potentials. MATLAB-based numerical simulations and analog circuit-based hardware measurements display that the simplified m-HH circuit can generate memristor parameter- and DC current-related periodic spiking behaviors with different periodicities and chaotic spiking behavior. This delights that electrophysiological characteristics of ion channels and external stimulus can be employed for regulating these periodic and chaotic spiking behaviors. It is interesting that the simplified m-HH circuit can generate frequency self-adaptation and coexisting firing patterns. The inter-spike interval bifurcation diagram shows that the frequency of the periodic spiking behavior increases with the increase of externally applied DC current. Besides, the hybrid parameter bifurcation diagram displays that the simplified m-HH circuit can generate memristor initial state-related coexisting firing patterns of period-6 with chaos, period-3 with chaos, and period-3 with period-8. These offer us the unique insight into explaining some biological firing patterns. The most significance is that the analog hardware circuit is feasible in reproducing these periodic and chaotic spiking behaviors and benefits for developing spiking-based neuromorphic hardware.

Bionic modeling and dynamics analysis of heterogeneous brain regions connected by memristive synaptic crosstalk

The coupling dynamics between different brain regions has long been a gap in the study of neural dynamics. To overcome this bottleneck, the synaptic crosstalk characterized by coupled memristors is creatively connected to different brain regions characterized by two heterogeneous Hopfield neural networks (HNNs), forming a novel bionic memristive synaptic crosstalk coupled-HNN (MSC-HNN) model for the first time. Rigorous boundedness and stability analyses ensure that coupling dynamics are generated within a bounded region. Utilizing innovative dynamics methods, it is revealed that the plasticity of synaptic crosstalk allows MSC-HNN to exhibit homogeneous dynamics properties globally, and the two internal HNNs preserve heterogeneous bifurcation processes and attractors. Interestingly, when the initial conditions undergo slight variations, the multistability phenomenon is simultaneously represented by the coexisting parameter bifurcation and the local basin of attraction, which enhances the diversity of activities in MSC-HNN. Furthermore, these novel dynamics effects are evaluated and validated through quantitative calculations of complexity measures and spectra. Subsequently, the developed analog and digital circuit implementation paradigms confirm the feasibility and practical value of MSC-HNN.

Nonlinear dynamics of a heaving spar-floater system integrated with inerter pendulum vibration absorber power take-off for wave energy conversion

A power take-off based on the inerter pendulum vibration absorber (called IPVA-PTO) is integrated with a spar-floater system to study its hydrodynamic response suppression and wave energy conversion capabilities in regular waves. The hydrodynamics of the spar-floater system is computed using the boundary element method with linear wave theory. With the wave height and wave frequency as the bifurcation parameters, it is found that the system can undergo two bifurcations: period-doubling bifurcation around the first resonance frequency (spar mode) and secondary Hopf bifurcation around the second resonance frequency (floater mode). The period-doubling bifurcation results in an energy transfer between the spar-floater system and the IPVA-PTO for small electrical damping values. As a result, the IPVA-PTO system simultaneously reduces the maximum response amplitude operator (RAO) of the spar and increases the normalized capture width in comparison with the optimal linear benchmark. Experiments performed on a “dry” single-degree-of-freedom system integrated with the IPVA-PTO where the base excitation is substituted for the wave excitation verify the simultaneous performance enhancement due to the period-doubling bifurcation. The system performance beyond the period-doubling bifurcation is also experimentally investigated. On the other hand, as the wave height approaches and passes the secondary Hopf bifurcation, the pendulum responses transition from primary harmonic responses to quasi-periodic responses to rotations. When the rotations occur, the IPVA-PTO system increases the maximum normalized capture width threefold to fivefold compared with the optimal linear benchmark, yet slightly increases the RAO around the second resonance frequency. Nevertheless, the RAO remains smaller than the global maximum RAO of the optimal linear benchmark. Finally, parametric studies are performed to study the effects of parameters on the bifurcations. It is observed that by varying the electrical damping, the wave height required for achieving the period-doubling bifurcation can be changed significantly, which can be exploited to stabilize the spar.

FILIPPOV TYPE PREY–PREDATOR SYSTEM FOR SELECTIVE HARVESTING OF PREY

In this paper, the dynamics of a Filippov system with logistically growing prey and Holling Type II predation is explored. The selective nonlinear harvesting of prey is carried out when the ratio of prey to predator is above a specified threshold value. In order to prevent the extinction of both the species and sustainable economic gains, no harvesting is allowed when the ratio is below a threshold value. The equilibrium states and their stability for the individual smooth are analyzed. Filippov convex method is applied to determine the dynamics on the boundary. The sliding mode dynamics is scrutinized, and it is observed that there is no escaping region. The existence of boundary points, tangent points and pseudo-equilibrium points are established. The conditions are obtained for the visibility of the tangent points. The complex behavior within the sliding region is studied by experimenting with different initial conditions. The increase in harvesting effort may lead to extinction of both the species through the sliding region of the boundary. The change in the stability behavior of both the interior equilibrium states is investigated with respect to model parameters. Border cross bifurcates with threshold value as the bifurcation parameter is shown for both the interior equilibrium states.

On the Asymptotic of Solutions of Odd-Order Two-Term Differential Equations

This work is devoted to the development of methods for constructing asymptotic formulas as x→∞ of a fundamental system of solutions of linear differential equations generated by a symmetric two-term differential expression of odd order. The coefficients of the differential expression belong to classes of functions that allow oscillation (for example, those that do not satisfy the classical Titchmarsh–Levitan regularity conditions). As a model equation, the fifth-order equation i2p(x)y‴″+p(x)y″‴+q(x)y=λy, along with various behaviors of coefficients p(x),q(x), is investigated. New asymptotic formulas are obtained for the case when the function h(x)=−1+p−1/2(x)∉L1[1,∞) significantly influences the asymptotics of solutions to the equation. The case when the equation contains a nontrivial bifurcation parameter is studied.

Domain walls and vector solitons in the coupled nonlinear Schrödinger equation

We outline a program to classify domain walls (DWs) and vector solitons in the 1D two-component coupled nonlinear Schrödinger (CNLS) equation without restricting the signs or magnitudes of any coefficients. The CNLS equation is reduced first to a complex ordinary differential equation (ODE), and then to a real ODE after imposing a restriction. In the real ODE, we identify four possible equilibria including ZZ, ZN, NZ, and NN, with Z(N) denoting a zero (nonzero) value in a component, and analyze their spatial stability. We identify two types of DWs including asymmetric DWs between ZZ and NN and symmetric DWs between ZN and NZ. We identify three codimension-1 mechanisms for generating vector solitons in the real ODE including heteroclinic cycles, local bifurcations, and exact solutions. Heteroclinic cycles are formed by assembling two DWs back-to-back and generate extended bright-bright (BB), dark-dark (DD), and dark-bright (DB) solitons. Local bifurcations include the Turing (Hamiltonian–Hopf) bifurcation that generates Turing solitons with oscillatory tails and the pitchfork bifurcation that generates DB, bright-antidark, DD, and dark-antidark solitons with monotonic tails. Exact solutions include scalar bright and dark solitons with vector amplitudes. Any codimension-1 real vector soliton can be numerically continued into a codimension-0 family. Complex vector solitons have two more parameters: a dark or antidark component can be numerically continued in the wavenumber, while a bright component can be multiplied by a constant phase factor. We introduce a numerical continuation method to find real and complex vector solitons and show that DWs and DB solitons in the immiscible regime can be related by varying bifurcation parameters. We show that collisions between two DB solitons with a nonzero phase difference in their bright components typically feature a mass exchange that changes the frequencies and phases of the two bright components and the two soliton velocities.

Extrapolating tipping points and simulating non-stationary dynamics of complex systems using efficient machine learning

Model-free and data-driven prediction of tipping point transitions in nonlinear dynamical systems is a challenging and outstanding task in complex systems science. We propose a novel, fully data-driven machine learning algorithm based on next-generation reservoir computing to extrapolate the bifurcation behavior of nonlinear dynamical systems using stationary training data samples. We show that this method can extrapolate tipping point transitions. Furthermore, it is demonstrated that the trained next-generation reservoir computing architecture can be used to predict non-stationary dynamics with time-varying bifurcation parameters. In doing so, post-tipping point dynamics of unseen parameter regions can be simulated.

Two-parametric unfoldings for planar invisible double-tangency singularities.

We consider the family of planar discontinuous piecewise linear systems in two regions with a straight line as boundary and one tangency point in each region. First, we recall a seven-parametric canonical form for such a family. Next, we select two parameters as bifurcation parameters. One of them represents the distance between the tangency points. It is known that this bifurcation parameter unfolds the called pseudo-Hopf bifurcation. The coefficient that determines the stability of the crossing limit cycle that emerges from this bifurcation mechanism, called the first Lyapunov coefficient, is considered our second bifurcation parameter. Finally, with these two bifurcation parameters, we give a two-parametric unfolding for the invisible fold-fold and focus-fold singularities. The bifurcation diagram for each unfolding consists of two curves of bifurcation points: a curve of pseudo-Hopf bifurcation points and a curve of saddle-node bifurcation points for crossing limit cycles. We call this phenomenon the pseudo-Bautin (pB) bifurcation because of the dynamical behavior that is same as the Bautin bifurcation for smooth dynamical systems.

Combined impact of fear and Allee effect in predator-prey interaction models on their growth

<p>We considered predator-prey models which incorporated both an Allee effect and a new fear factor effect together, and where the predator predated the prey with a Holling type I functional response. We started off with a two-dimensional model where we found possible equilibria and examined their stabilities. By using the predator mortality rate as the bifurcation parameter, the model exhibited Hopf-bifurcation for the coexistence equilibrium. Furthermore, our numerical illustrations demonstrated the effect of fear and the Allee effect on the population densities, and we found that the level of fear had little impact on the long-term prey population level. The population of predators, however, declined as the fear intensity rose, indicating that the fear effect might result in a decline in the predator population. The dynamics of the delayed system were examined and Hopf-bifurcation was discussed. Finally, we looked at an eco-epidemiological model that took into account the same cost of fear and the Allee effect. In this model, the prey was afflicted with a disease. The prey was either susceptible or infected. Numerical simulations were carried out to show that as the Allee threshold rose, the uninfected prey and predator decreased, while the population of infected prey increased. When the Allee threshold hit a certain value, all populations became extinct. As fear intensity increased, the population of uninfected prey decreased, and beyond a certain level of fear, habituation prevented the uninfected prey from changing. After a certain level of fear, the predator population went extinct and, as a result, the only interaction left was between uninfected and infected prey which increased disease transmission, and so the infected prey increased. Hopf-bifurcation was studied by taking the time delay as the bifurcation parameter. We estimated the delay length to preserve stability.</p>

Chaos detection in predator-prey dynamics with delayed interactions and Ivlev-type functional response

<p>Regarding delay-induced predator-prey systems, extensive research has focused on the phenomenon of delayed destabilization. However, the question of whether delays contribute to stabilizing or destabilizing the system remains a subtle one. In this paper, the predator-prey interaction with discrete delay involving Ivlev-type functional response is studied by theoretical analysis and numerical simulations. The positivity and boundedness of the solution for the delayed model have been discussed. When time delay is accounted as a bifurcation parameter, stability analysis for the coexistence equilibrium is given in theoretical aspect. Supercritical Hopf bifurcation is detected by numerical simulation. Interestingly, by choosing suitable groups of parameter values, the chaotic solutions appear via a cascade of period-doubling bifurcations, which is also detected. The theoretical analysis and numerical conclusions demonstrate that the delay mechanism plays a crucial role in the exploration of chaotic solutions.</p>