International Journal of Bifurcation and Chaos

In this article, we construct an excitable memristive diffusive neuron model by considering a biophysical slow–fast bursting oscillator and study the effects of electromagnetic induction on the dynamics of the single model as well as the coupled systems. We explore various firing regimes such as tonic spiking, bursting, and mixed-mode oscillations depending on the bifurcation structure with different injected current stimuli, then perform a comparative analysis on the synchronization of the coupled oscillators by setting the model into two different network architectures. First, a diffusively coupled network is considered, and later a global network is constructed. The results suggest that the diffusively connected neurons show complete synchronization at higher couplings for bursting and tonic spiking regimes. Furthermore, we show that the extended spatial system can generate spiral-like patterns in the vicinity of a Hopf bifurcation point and observe the impact of Gaussian white noise to study its effects on pattern formation. These types of patterns are robust in the excitable model. Our results might contribute significantly to the dynamical studies of irregular neural computation.

In this paper, we investigate the stability of equilibrium points for the planar restricted equilateral four-body problem in the case that one particle of negligible mass is moving under the Newtonian gravitational attraction of three positive masses $m_1$, $m_2$ and $m_3$ (called primaries). These always lie at the vertices of an equilateral triangle (Lagrangian configuration) and move with constant angular velocity in circular orbits around their center of masses. We consider the case where all the primaries have unequal masses, and investigate the nonlinear stability (in the sense of Lyapunov) of the elliptic equilibrium for the specific values of the mass $m_1=0.01$ and $0.99$ of the primary, fixed on the horizontal axis. Moreover, the $2$:$-1$ four-order resonant cases are determined and the stability is investigated. In this study, Markeev’s theorem and Arnold’s theorem become key ingredients.

In this paper, we study equilibria of differential equation models for networks. When interactions between nodes are taken to be piecewise constant, an efficient combinatorial analysis can be used to characterize the equilibria. When the piecewise constant functions are replaced with piecewise linear functions, the equilibria are preserved as long as the piecewise linear functions are sufficiently steep. Therefore the combinatorial analysis can be leveraged to understand a broader class of interactions. To better understand how broad this class is, we explicitly characterize how steep the piecewise linear functions must be for the correspondence between equilibria to hold. To do so, we analyze the steady state and Hopf bifurcations which cause a change in the number or stability of equilibria as slopes are decreased. Additionally, we show how to choose a subset of parameters so that the correspondence between equilibria holds for the smallest possible slopes when the remaining parameters are fixed.

This paper aims to study the nonlinear dynamics and bifurcation structures of a new mathematical model of the $\gamma$-Ricker population model with a Holling type II per-capita birth function, where the Allee effect parameter is $\gamma =2$. A generalized $r$-Lambert function is defined on the 3D parameters space to determine the existence and variation of the number of nonzero fixed points of the homographic $2$-Ricker maps considered. The singularity points of the generalized $r$-Lambert function are identified with the cusp points on a fold bifurcation of the homographic $2$-Ricker maps. In this approach, the application of the transcendental generalized $r$-Lambert function is demonstrated based on the analysis of local and global bifurcation structures of this three-parameter family of homographic maps. Some numerical studies are included to illustrate the theoretical results.

In this work, we continue the study of the bifurcations of the critical points in a symmetric Caldera potential energy surface. In particular, we study the influence of the depth of the potential on the trajectory behavior before and after the bifurcation of the critical points. We observe two different types of trajectory behavior: dynamical matching and the nonexistence of dynamical matching. Dynamical matching is a phenomenon that limits the way in which a trajectory can exit the Caldera based solely on how it enters the Caldera. Furthermore, we discuss two different types of symmetric Caldera potential energy surface and the transition from the one type to the other through the bifurcations of the critical points.

The study lies in the field of physical reservoir computing and aims to develop a classifier using Fisher Iris dataset for benchmark tasks. Single Chua chaotic oscillator acts as a physical reservoir. The study was performed using computer simulation. The features of Iris flowers are represented as the consequence of short pulses at a constant level of a control parameter, which is fed to the oscillator, changing its dynamics. During the classification of flowers, the oscillator works without being reset, so each pulse on the input changes the phase trajectory and makes it unique for each Iris flower. Finally, the estimation of the symmetry of an attractor makes it possible to connect each species of Iris with the properties of the attractor.

The resulting architecture of the classifier includes a single-node externally-driven Chua oscillator with time-delayed input. The classifier shows two mistakes in classifying the dataset with 75 samples working in chaotic mode.

Formation and study of periodic orbits in phase space in the case of nonlinear oscillators have been a topic of much interest in the recent past. In the current work, a method to go deep inside the limit cycle zone on one side of the bifurcation curve of a 2D non-Lienard biochemical oscillator has been introduced. It is discussed how such an introduction facilitates predicting the boundaries of limit cycles at various points of parameter space, nearly accurately, by the use of perturbative Renormalization Group. Sel’kov model of Glycolytic oscillator has been chosen as the base model to introduce the method.

The ongoing coronavirus disease-2019 (COVID-19) pandemic caused by severe acute respiratory syndrome-coronavirus 2 (SARS-CoV-2) has serious influences on human health and economy. The available clinical data suggest that patients infected by SARS-CoV-2 have the possibility of simultaneous infection of bacteria. In this study, we present a data-driven mathematical model for coinfection of SARS-CoV-2 and bacteria to investigate the dynamics of COVID-19 progress. Specifically, based on the statistical analysis of different clinical data from China and some other countries, a system model with ordinary differential equations (ODEs) in four variables, i.e. SARS-CoV-2, bacteria, neutrophils and lymphocytes, is established. We further validate our model through theoretical analysis and fitting of different clinical data. Moreover, through numerical simulations and bifurcation analysis, we find that bacterial infection and immune-related parameters in certain ranges lead to the system transitions among three steady states, i.e. mild, severe and death. We also analyzed the influence of the time it takes for patients to switch from a high-risk area to a low-risk area on the recovery time. These results reveal that the coinfection of viruses and bacteria can explain the changes in neutrophils and lymphocytes, and that initial bacterial infection and immune-related parameters have great influences on the severity degree and mortality in COVID-19 patients. Together, our model and quantitative analysis suggest that preventing bacterial infection and enhancing immune ability during the early phase of infections could be a potential treatment option for high-risk COVID-19 patients.

Conditions for the emergence of strong Turing–Hopf instabilities in the Lengyel–Epstein CIMA reaction–diffusion model are found. Under these conditions, time periodic spatially inhomogeneous solutions can be induced by diffusive instability of the spatially homogeneous limit cycle emerging at a supercritical Bautin–Hopf bifurcation about the unstable steady state of the reaction system. We report numerical simulations by an Alternating Directions Implicit (ADI) method that show the formation of twinkling patterns for a chosen parameter value, thus confirming our theoretical results.

In this paper, we study the steady-state problem of an S-K-T competition model with a spatially degenerate intraspecific competition coefficient. First, the global bifurcation continuum of positive steady-state solutions from its semitrivial steady-state solution is given, which depends on the spatial heterogeneity and cross-diffusion. Second, two limiting systems are derived as the cross-diffusion coefficient tends to infinity. Moreover, we demonstrate the existence of positive steady-state solutions near the two limiting systems, and show which one of the limiting systems characterizes the positive steady-state solution.

The game characteristics and equilibrium strategies of a triple-channel supply chain under the carbon subsidy policy are studied in this work with three different game power structures. There are simultaneous decision-making, manufacturer-dominated market and retailer-dominated market. The decision mechanism for order quantity of manufacturer and dual-channel retailer, is discussed. Meanwhile, their complex dynamic characteristics are studied. It is found that when the government implements dual low-carbon subsidies, the supply chain system in which the manufacturer dominates the market is more stable. The government should pay attention to the power structure of the market to determine an appropriate subsidy rate. The over-adjustment of order quantity from manufacturer’s direct sales channel and retailer’s online channel both will lead to large periodical fluctuations in the system, and even bifurcation and chaos. These will cause substantial fluctuations and even loss in the profits of supply chain companies. This analysis shows that delayed feedback can effectively stabilize the periodic bifurcation and chaos in the system.

This paper proposes a novel memristive chaotic circuit which originated from a Shinriki oscillator with two flux-controlled memristors of different polarities. This two-memristor-based Shinriki oscillator (TMSO) having a special plane equilibrium is prone to exhibiting the initial-dependent phenomenon of extreme multistability. To investigate its internal dynamics, a third-order dimensionality reduction model is established by utilizing the constitutive relationship of its memristor’s flux and charge. The uncertain plane equilibrium is transfered into some deterministic model that can accurately predict the dynamical evolution of the system, where interesting phenomena of asymmetric bifurcations, extreme multistability and antimonotonicity are detected and analyzed by evaluating the position and stability of the equilibria in the flux–charge model. The simulation is carried out via Multisim to validate the analysis model, and the comparison of the phase trajectories, before and after dimensionality reduction, shows that this oscillator is good for research and practical use.

Memristor, as a typical nonlinear element, is able to produce chaotic signals in chaotic systems easily. Chaotic systems have potential applications in secure communications, information encryption, and other fields. Therefore, it is of importance to generate abundant dynamic behaviors in a single chaotic system. In this paper, a novel memristor-based chaotic system without equilibrium points is proposed. One of the essential features is the absence of symmetry in this system, which increases the complexity of the new system. Then, the nonlinear dynamic behaviors of the system are analyzed in terms of chaos diagrams, bifurcation diagrams, Poincaré maps, Lyapunov exponent spectra, the sum of Lyapunov exponents, phase portraits, 0–1 test, recurrence analysis and instantaneous phase. The results of the sum of Lyapunov exponents show that the given system is a quasi-Hamiltonian system with certain initial conditions (IC) and parameters. Next, other critical phenomena, such as hidden multi-scroll attractors, abundant coexistence characteristics, are found characterized through basins of attraction and others. Especially, it reveals some rare phenomena in other systems that multiple hidden hyperchaotic attractors coexist. Finally, the circuit implementation based on Micro Control Unit (MCU) confirms theoretical analysis and the numerical simulation.

In this work, we present a novel three-dimensional chaotic system with only two cubic nonlinear terms. Dynamical behavior of the system reveals a period-subtracting bifurcation structure containing all $m$th-order ($m=1,2,3,\dots$) periods that are found in the dynamical evolution of the novel system concerning different values of parameters. The new system could be evolved into different states such as point attractor, limit cycle, strange attractor and butterfly strange attractor by changing the parameters. Also, the system is multistable, which implies another feature of a chaotic system known as the coexistence of numerous spiral attractors with one limit cycle under different initial values. Furthermore, bifurcation analysis reveals interesting phenomena such as period-doubling route to chaos, antimonotonicity, periodic solutions, and quasi-periodic motion. In the meantime, the existence of periodic solutions is confirmed via constructed Poincaré return maps. In addition, by studying the influence of system parameters on complexity, it is confirmed that the chaotic system has high spectral entropy. Numerical analysis indicates that the system has a wide variety of strong dynamics. Finally, a message coding application of the proposed system is developed based on periodic solutions, which indicates the importance of studying periodic solutions in dynamical systems.

Improving the neuron model and studying its electrical activities according to the real biophysical environment are significant in human cognitive brain activity and neural behavior. The complex transmembrane motion of ions on the neuronal cell membrane can establish time-varying electromagnetic fields and affect the transition firing patterns of neurons. In this paper, a threshold memristor is used to describe the electromagnetic induction and magnetic field effects of neuron cell membrane ion exchange to improve the neuron model, and a memristive Morris–Lecar (mM–L) neuron model is proposed. Numerical simulation confirms that different intensities of electromagnetic fields can produce distinct pattern transitions in electrical activities of the neuron, such as periodic bursting, periodic spiking, chaotic bursting. From the perspective of neuron’s interspike interval (ISI), the ISIs bifurcation in the multiparameter planes, ISIs firing periods, the variance of ISIs and other methods are used to find the trend of the mM–L neuron firing pattern transition. Finally, based on the 4D nonlinear differential equation of the mM–L neuron model, the complete electronic implementation of the model is designed. The output of the designed circuit is consistent with the theoretical prediction, which is extremely useful for studying the dynamics of a single neuron.

In this article, we investigate a type of biological tissue formation system with a random structure of reaction or/and diffusion, analyzing the connection with the results obtained in [Rajapakse & Smale, 2017a] for the corresponding deterministic systems and showing the major difference from these results. Interestingly, we find a dynamical phenomenon leading to morphogenesis or emergence in such a system. Also we find their transitions in this system, while only one type of dynamical behavior occurs for the deterministic systems that satisfy typical conditions. Using the stability theory of stochastic systems, we quantitatively elucidate how such a phenomenon is emergent in complex networks with random structures. We believe that our analytical results could be beneficial to understand the underlying mechanisms of complexity-induced functions in tissue formation within real environments.

The article considers the question of local dynamics of the logistic equation with delays and the delay coefficients being the nonlinear functions. It is supposed that one of the parameters, characterizing the delay value, is sufficiently large. The stability criterion is formulated and critical cases in the problem of balance state stability are defined. The main content of the paper is focused on studying the local dynamics in cases close to critical. In nature, these critical cases are infinite dimensional. Special nonlinear boundary-value problems of the parabolic type are constructed, which play the role of normal forms. Their nonlocal dynamics defines the behavior of the solutions from the small neighborhood of the balance state of the initial equation. The conclusions on the role of nonlinearity in the delay coefficients, in the behavior of the dynamic properties of the solutions are summarized.

Machine learning and deep learning techniques are contributing much to the advancement of science. Their powerful predictive capabilities appear in numerous disciplines, including chaotic dynamics, but they miss understanding. The main thesis here is that prediction and understanding are two very different and important ideas that should guide us to follow the progress of science. Furthermore, the important role played by nonlinear dynamical systems is emphasized for the process of understanding. The path of the future of science will be marked by a constructive dialogue between big data and big theory, without which we cannot understand.