International Journal of Bifurcation and Chaos

Employing bifurcation theory in dynamical systems, we delve into analyzing the nonlinear wave dynamics and asymptotic behaviors of the generalized KdV–mKdV-like system. Our exhaustive analysis across varied parameter regimes uncovers a wide range of wave phenomena tied intricately to the system parameters. Our work elucidates the stability and complex dynamics in related biological systems, paving a transformative way for medical applications. The intricate dynamics revealed by the bifurcation phenomena in the GKdV–mKdV-like system are promising for advancing medical research and clinical practice, particularly through AI-driven optimization. This interdisciplinary approach bridges mathematical theory and medical innovation, opening a door for predictive modeling of health conditions and proactive healthcare strategies.

In this paper, we consider the bifurcation of limit cycles from the Bogdanov–Takens system with an algebraic switching curve under perturbations of piecewise smooth polynomials of degree n. By using Picard–Fuchs equations and the Chebyshev criterion, we give the upper bound of the number of limit cycles that bifurcate from the period annulus for the piecewise smooth Bogdanov–Takens system if the first-order Melnikov function is not identically 0.

A general type of piecewise-smooth dynamic system with two thresholds is analyzed in this work. We define some fundamental notions for the proposed system such as oscillating space and real/virtual equilibrium, which generalizes the counterparts for the planar Filippov system. Moreover, we show that under certain conditions the planar switched system with two thresholds generates a novel limit cycle, and analyze the properties of this periodic solution such as existence, stability, amplitude and period. Interestingly, the existence and stability of this periodic solution in the oscillating space are consistent with the pseudo-equilibrium of the corresponding planar Filippov system. Hence, we establish the connection between the planar switched system with two thresholds and the planar Filippov system. Finally, we apply the modeling and analytical approaches to a piecewise-smooth epidemic model with density-dependent interventions, describing the control measure that is triggered when the number of infected individuals increases and reaches a critical level while being suspended when it decreases down to another level. We prove that the epidemic model stabilizes at either the endemic equilibrium of the free system (the one not under control) or the new periodic solution induced by the two thresholds, depending on the threshold levels. The two-threshold measure is able to suppress the number of infected individuals during the evolution of an infectious disease.

This paper establishes a 16-dimensional multiscale COVID-19 model with three geographical regions and interval parameters, and the model can be investigated by a slow–fast system. In the fast subsystem, there is a globally asymptotically stable equilibrium. The fast–slow theory is applied to investigate the stability and bifurcations of Disease-Free Equilibrium (DFE) and Endemic Equilibrium (EE) of the slow subsystem. It is found that transmission rates in three geographical regions, proportion and average time of individuals spend in public places can result in forward and backward bifurcations. Some criteria for verifying whether the system has forward and backward bifurcations are presented. When there is backward bifurcation, stable DFE coexists with stable EE for Rc < R0 < 1; slow subsystem has saddle-node bifurcation when R0 = Rc; with respect to EE, the subsystem has pitchfork bifurcation when R0 = 1. Interestingly, a critical value Tpc of the average time Tp is obtained from individuals in public places. The dynamics of slow subsystem are appropriate to that of the multiscale system. Chaotic dynamics are found in the multiscale system due to seasonal and stochastic infection. Moreover, several numerical analyses of COVID-19 for Hong Kong and South Africa agree with the theoretical results.

In this paper, we consider the dynamics of a system consisting of two inertial neurons whose activation function is of the hyperbolic tangent type. We distinguish two cases: a homogeneous coupling where each of the two neurons has an activation function with a variable gradient; then, a heterogeneous coupling where one of the two neurons has an activation function with a variable gradient and the other an activation function with a fixed gradient. In each case, the equilibrium points of the coupled system are determined and their stability is examined according to Routh’s criterion combined with Hopf’s bifurcation theorem. Next, we examine the path that leads the Hopf bifurcation system toward grid-scroll chaos when the coupling coefficients are gradually varied. To investigate this, we mainly use the drawings of bifurcation diagrams, basins of attraction and phase portraits. For homogeneous coupling, the system presents a chaotic 3 × 3 scroll chaotic attractor, while the attractor is of the 3 × 2 grid-scroll type for heterogeneous coupling. The route to chaos in each case is also marked by the presence of several parameter zones where the system exhibits multistable behavior. To verify these theoretical analysis results, an experimental study is carried out using an Arduino card.

Drug-tolerant persister cells, which are a subpopulation in a transient pseudo-dormant state following cancer therapy, play a crucial role in disease recurrence. This paper primarily focuses on the bifurcations of a model that describes the relationships among sensitive cells, persister cells, and resistant cells in cancer, such as Hopf bifurcation, Bautin bifurcation, and Bogdanov–Takens bifurcations of codimension-2 and codimension-3. The transformation rates among these three types of cells are regarded as the bifurcation parameters. The research discovers that the dynamics on the center manifold are locally topologically equivalent to specific models, and the local representations of the bifurcation curves are presented. Numerical simulations also confirm the theoretical results. This study offers insights into the influence of transformation rates on cancer therapy and treatment strategies for different bifurcations. Additionally, different types of bistable states are revealed, specifically the bistable states of two equilibria or one limit cycle and one equilibrium, along with different paths to bistability. This paper may offer new perspectives on the role of persister cells in cancer recurrence and provide a scientific basis for cancer treatment and prevention strategies.

This paper proposes a novel concept in memristive devices termed “Coupling Memristor (CM)”, aiming to unveil its potential application for modeling complicated neuron circuits and generating complex patterns of oscillations such as bursting in conjunction with other linear, nonlinear or memristive devices. It is well known that traditional Uncoupled Memristor (UM) functions solely based on its inherent properties, while the concept of CM proposed in this paper shares the characteristics of other devices. This study investigates the dynamics of CM and its potential applications in advanced modeling of neuromorphic systems. This work also provides evidence that the small-signal model of a memristor conventionally modeled with the series connection of inductors-resistors and paralleled by another resistor without depending on the interconnected coefficients parameters is only valid when all the states are directly related to the state-dependent Ohm’s law. However, missing one or more states in state-dependent Ohm’s law introduces additional dependencies and feedback effects that require complex modeling techniques for the analysis of the memristor’s small-signal model. Additionally, this work demonstrates from the Chay–Keizer pancreatic β-cell, Phantom Bursting Model (PBM)-II and Chay neuron model of an excitable cell, that the slow dynamics, which communicate with fast dynamics, are in fact a slow modulation CM. It plays a significant role in changing the cell membrane of pancreatic β-cells and excitable cells, thereby facilitating the emergence of bursting patterns, which is an essential mechanism for insulin secretion in pancreatic β-cell. This fundamental principle has the potential to revolutionize the design of new memristive devices and enable precise modeling of neurons, thus fostering comprehensive investigations in neuromorphic systems. The concepts of CM explored in the applications in pancreatic β-cells, PBM-II and excitable cells provide an accurate modeling and a deeper understanding of neuron dynamics. This study could contribute to the development of novel tools for the treatment of neurons, metabolism, and insulin-related disorders.

This study presents an energy-efficient unidirectional coupling strategy for achieving Generalized Synchronization (GS) and bistability in chaotic systems. The method generates decaying coupling signals activated by Poincaré plane crossings of the drive system’s trajectory. Unlike traditional constant coupling, the signal amplitude decays exponentially between crossings, reducing energy consumption while maintaining synchronization. The coupling combines (i) decaying signals activated by Poincaré crossings and (ii) continuous negative feedback from the response/auxiliary systems. Applied to identical pairs of Rössler and Lorenz systems, the method achieves GS and induces identical or indirect synchronization between response and auxiliary systems. The results demonstrate advantages over classical schemes, including lower energy requirements and complex dynamical behaviors. This work advances energy-efficient coupling strategies for chaotic systems, with potential applications in low-power electronics and biological-inspired systems.

For three cubic–quartic optical soliton models of nonlinear refractive index together with nonlinear chromatic dispersion, the corresponding differential systems of the amplitude component are planar dynamical systems with a singular straight line. In this paper, by using the techniques from dynamical systems developed by Li & Chen, 2007 to analyze the parameter conditions of systems and construct the corresponding phase portraits, the dynamical behavior of the amplitude component can be analyzed. Under different parameter conditions, exact explicit envelope solitary wave solutions, periodic wave solutions, periodic peakons as well as the peakon solution, can all be found.

We introduce a novel chaotic dynamical system formulated as a second-order difference equation. The model includes one time-delay linear term and one nonlinear term composed of the product of two different time-delay terms. As a result, the nonlinear term gives rise to strange periods. The dynamical behaviors of the model are investigated by the Lyapunov spectrum and bifurcation diagram. We found that the model can have a positive Lyapunov exponent. We also found that while the bifurcation diagram has a typical Arnol’d tongue structure, there are unique shapes that show the progression of overlapping fish hooks in the synchronization region.

Steady-state bifurcation analysis of real-world networks is challenging due to their sizes and complexity. It turns out that, small networks, known as motifs, often serve as fundamental building blocks of larger networks. This raises the question of whether knowledge of steady-state bifurcations in motif networks can provide insights into the steady-state bifurcations in larger networks. In this paper, we explore this question by investigating a particular type of networks obtained by the combination of two motif networks through the coalescence operation, where the two motifs share a common node. We conclude that, in general, bifurcation conditions in a coalescence network are not necessarily bifurcation conditions in its component networks. Consequently, in general, bifurcations in the component networks do not offer clear insights into the bifurcations of the coalescence network. However, for a specific class of networks — feedforward coalescence networks — we prove that the codimension-1 steady-state bifurcations of the coalescence network relate to those in their component networks. In particular, we show that it is possible to infer the bifurcation branches in the coalescence network from those in the component networks. Notably, we conclude that the growth rate of bifurcation branches in the coalescence network depends on the interconnections between the component networks.

Fault Diagnosis (FD) is vital for industrial systems’ safety and reliability. While traditional approaches — Model-Based (MB), Signal-Based (SB), and Data-Driven (DD) methods — have made significant progress, they face inherent limitations with increasingly complex systems. MB methods struggle with precise mathematical modeling of nonlinear systems, SB techniques falter when fault signatures are obscured by noise, and DD methods face challenges in data quality, model generalization, and real-time constraints. Chaos theory, which addresses these limitations utilizing its special features of deterministic nonlinearity, sensitiveness to initial conditions, and pseudo-randomness, has emerged as a transformative paradigm in FD. Chaotic systems enhance FD methods by enabling complex system modeling, sensitivity to initial conditions improves SB methods by enhancing signal-to-noise ratios, and pseudo-random trajectories boost global optimization capabilities in DD techniques. Each approach is evaluated through theoretical analysis, implementation frameworks, and industrial applications, supported by representative case studies across various equipment domains.

In this paper, we consider a Holling type-II prey–predator model with cross-diffusion under homogeneous Neumann boundary conditions. The existing literature shows that the corresponding ODE model admits a unique globally asymptotically stable periodic solutions and these periodic solutions become locally asymptotically stable homogeneous periodic solutions in the diffusive model, which means that passive diffusion does not create Turing instability of periodic solutions. When cross-diffusions are introduced in the model, the Turing instability of periodic solutions and the existence of nonconstant positive stationary solutions are studied. We derive exact conditions on the cross-diffusion coefficients so that under these conditions, the periodic solutions can undergo Turing instability. Once Turing instability of periodic solutions occurs, numerical simulations show that new irregular patterns emerge. In addition, by using the fixed point index theory and analytical techniques, we obtain sufficient conditions for the existence of nonconstant positive stationary solutions. Our results show that cross-diffusion plays a crucial role in the formation of spatiotemporal patterns, that is, it can create new irregular patterns, which is a sharp contrast to the case without cross-diffusion.

In this paper, we consider dynamics and exact solutions for the generalized nonlinear Schrödinger equation with nonlinear chromatic dispersion and quintuple power law of refractive index in optical fibers. By investigating the bifurcations of phase portraits of the corresponding stationary solution system of the PDE model, we reveal the dynamical behavior of stationary envelope solutions. For the level curves having zero energy, we derive possible explicit exact parametric representations of the envelope soliton solutions, periodic solutions, peakon, periodic peakon as well as compacton solutions under given parameter conditions.

With the popularity of new energy vehicles, how to construct the recycling supply chain of new energy vehicle batteries has become a research hotspot. Considering the uncertainty of remanufactured products after recycling and the reward–penalty strategy, we seek to construct four recycling supply chain models with different structures: single-channel manufacturer recycling, single-channel third-party recycling, decentralized dual-channel recycling, and centralized dual-channel recycling, and investigate two problems: which is the best model in terms of profit and recycling amount; and how to coordinate a centralized dual-channel recycling supply chain? Subsequently, the complexity of the recycling supply chain model is analyzed in depth, and the steady-state parameter domain of the optimal solution is clarified to fill the gap in the existing literature. The results show that, first, the dual-channel model is most beneficial to manufacturers. Second, the revenue-sharing contract constructed in this paper successfully coordinates a centralized dual-channel recycling supply chain system. Finally, manufacturers and third parties provide more aggressive battery recycling services to consumers, stimulated by the reward and punishment strategies.

One of the most challenging problems in the qualitative theory of piecewise linear differential systems is determining their maximum number of crossing limit cycles. This study addresses this issue by examining the systems in ℝ3 formed by an arbitrary linear differential center and the relay differential systems separated by quadric surfaces. This study aims to find out whether such piecewise linear differential systems possess limit cycles and, if so, how many. We demonstrate that these discontinuous piecewise linear differential systems can exhibit at most two limit cycles when the separating surface is a paraboloid. Conversely, we prove that they can have up to four limit cycles for a hyperbolic separation surface.

An intraguild predation model with Holling II functional response is studied. Previous research has studied the consistent persistence of species, the non-negative boundedness of solutions, and thoroughly analyzed the existence and stability of boundary equilibria, but the existence of interior equilibria and the bifurcation of equilibria of system are unknown. In this paper, we consider the existence conditions for the interior equilibria with parameters in R+9 semialgebraic space. We find that system has at most seven and at least two coexisting equilibria, and give conditions for having the exact number of equilibria. Furthermore, we investigate the local stability and global stability of the interior equilibria. Moreover, we also verify the existence of transcritical bifurcations and Hopf bifurcations. Finally, the simulation results show that the three species of the system can coexist and maintain a relatively stable state under the given parameter conditions.

In this paper, we conduct a mathematical analysis of an age-structured SIRS infectious disease model, which incorporates a general nonlinear infection rate and accounts for the diffusion of the disease. The main objective of this study is to investigate the existence of periodic wave train solutions for the age-structured SIRS epidemic model with diffusion by utilizing the Hopf bifurcation theory for nondensely defined Cauchy problem and the persistence of nondegenerate Hopf bifurcation of singularly perturbed second-order equations. The outcomes indicate that the temporal oscillations derived from the age-structured part would interact with the spatial diffusion to result in the existence of periodic wave train solutions. At the same time, to illustrate our theoretical analysis more vividly, numerical simulations are performed.

The behavior of the Generalized Alignment Index (GALI) method has been extensively studied and successfully applied for the detection of chaotic motion in conservative Hamiltonian systems, yet its application to non-Hamiltonian dissipative systems remains relatively unexplored. In this work, we fill this gap by investigating the GALI’s ability to identify stable fixed points, stable limit cycles, chaotic (strange) and hyperchaotic attractors in dissipative systems generated by both continuous and discrete time dynamics, and compare its performance to the analysis achieved by the computation of the spectrum of Lyapunov exponents. Through a comprehensive study of three classical dissipative models, namely the 3D Lorenz system, a modified Lorenz 4D hyperchaotic system, and the 3D generalized hyperchaotic Hénon map, we examine GALI’s behavior and possible limitations in detecting chaotic motion, as well as the presence of different types of attractors occurring in dissipative dynamical systems. We find that the GALI successfully detects chaotic motion, as well as stable fixed points, but it faces difficulties in distinctly discriminating between stable limit cycles, chaotic attractors, and hyperchaotic motion.

Electronic circuit implementation has been widely used to corroborate the numerically detected occurrence of chaotic dynamics in nonlinear dynamical systems. In this paper, we demonstrate that careful consideration is required when dealing with such behaviors, both in terms of numerical methods and the corresponding electronic circuit designs. Specifically, we investigate the dynamics of a circuit consisting of three elements: a passive linear inductor, a passive linear capacitor, and a locally active generic memristor, which is modeled by a three-dimensional four-parameter differential system, having the plane x = 0 invariant with respect to its flow in the phase space. For a particular set of parameter values, the system exhibits a chaotic attractor, numerically obtained. However, depending on the numerical method used, solutions traverse the invariant plane x = 0, which contradicts the analytical properties of the system. Furthermore, an electronic circuit implementation of the model surprisingly mimics such a numerical error, producing the same erroneous crossing behavior. This peculiarity made us consider other numerical methods, seeking to identify the reasons for the behavior, either via stiffness, or by constant step methods, etc. Going beyond the usual numerical difficulties, the search for a more suitable method was intense and culminated in an implementation specifically made to be used in problems with sharp growth, known as the Cash–Karp method, which produced reasonable solutions. In this context, we provide results by different numerical methods, highlighting potential pitfalls in both numerical simulations and physical implementation in the study of chaotic systems.

The bifurcation of limit cycles is investigated from asymmetric cubic Hamiltonian systems under the perturbation of five-order polynomials. The approach applies Green’s theorem to the method of detection function: First, the Abelian integrals with irregular regions are computed, improving the accuracy of bifurcation parameter values. Second, it is shown that there exist at least 11 limit cycles with two distinct distributions. Third, precise detection curves are presented to validate all analytical results. Finally, four cases are investigated for the number and distribution of limit cycles by considering Hopf, heteroclinic, and homoclinic bifurcation values to obtain parameter conditions.

Predator–prey interaction is considered a natural phenomenon in the ecological system. Empirical research conducted on vertebrates has demonstrated that the presence of predators can have a significant impact on the survival rates and reproductive capabilities of prey populations. Recently, there has been research on mathematical models of predator–prey systems that include different predator functional responses and fear effects. These studies have overlooked the influence of fear on the death rates of species that are hunted. From the given findings, we present a mathematical model of predator–prey systems that includes fear costs impacting the rates of reproduction and death in prey population. By reducing the discrete model into different normal forms, we prove that there exists a set of codimension-1 and codimension-2 bifurcations, which include transcritical, flip, Neimark–Sacker bifurcations, 1:2 and 1:4 strong resonance bifurcations. These findings indicate that, compared with the system without fear effect, the increase of the fear effect parameter k1 that affects the birth rate of prey and the fear effect parameter k2 that affects the death rate of prey will strengthen the oscillation of prey population and reduce the oscillation of predator population. In addition, the increases of k1 and k2 have no effect on the density of the prey population but reduce the density of the predator population. When the fear effect k1 and other parameter values remain the same, the system generates an expanding limit circle as k2 increases, indicating that the effect of fear effects on the death rate enhances the stability of the system.

In this study, we explore the dynamics of a triopoly game involving firms with bounded rationality. Our analysis examines the existence and stability of equilibrium points, with a specific focus on the Nash equilibrium point. We found that an increase in the parameters k1 (adjustment speed) and a1 (product substitution) destabilizes the system, leading to more chaotic behavior through flip and Neimark–Sacker bifurcations. This chaotic behavior is examined through numerical simulations, where various parameters are adjusted to observe their effects. The basins of attraction are analyzed, revealing that the basin of attraction varies as the parameter k1 increases. Our approach aims to control the chaotic behavior and stabilize the system by employing the OGY method. Importantly, our analysis emphasizes that the product substitution parameter a1 is a key factor in achieving stabilization within the game. To clearly illustrate the effectiveness of our proposed stabilization strategy in this competitive context, we show the numerical simulations.

In this paper, we delve into a predator–prey model that includes fear effect, particularly focusing on the impact of fear on the predation rate of medium-sized predators. First, the existence and type of equilibria of the model are discussed. Then, by using bifurcation theory, we find that the model will experience saddle-node bifurcation, Hopf bifurcation, degenerate Hopf, double limit cycle, Bogdanov–Takens bifurcation, saddle-node bifurcation of limit cycles and saddle-node homoclinic bifurcation. The analysis results show that as the level of fear increases, the oscillation amplitude of the model gradually decreases, which actually enhances the stability of the ecosystem. However, when the level of fear is too high, the dynamic behavior of the model will undergo significant changes, and may even lead to the extinction of medium-sized predator populations, which poses a threat to the stability of the ecosystem. This suggests that fear can effectively control the density of medium predators and prevent the excessive growth of species below the food chain, thus maintaining the balance of the ecosystem.

In this paper, we study the FitzHugh–Nagumo (1,1)-fast–slow system where the vector fields associated to the fast/slow equations come from the reduction of the Hodgkin–Huxley model for the nerve impulse. After deriving the dynamical properties of the singular and regular cases, we perform a bifurcation analysis and investigate how the parameters (of the affine slow equation) impact the dynamics of the system. The study of codimension-1 bifurcations and the numerical locus of canards concludes this case study. All theoretical results are numerically illustrated.