International Journal of Bifurcation and Chaos

In this paper, we explore the rich dynamics in the Chen system using Runge–Kutta method and Deep Neural Networks (DNNs). Compared with the Lorenz system, we find that the first return map $P$ of the Chen system exhibits a more complex structure of continuous regions in the Poincaré section. Moreover, the existence of six crossing blocks with respect to the second return map implies that there is a closed invariant set $\mathrm{\Lambda }$ in the Poincaré section such that $P^2|\mathrm{\Lambda }$ is semi-conjugate to a 6-shift map, thus demonstrating remarkably chaos in the Chen system.

This paper studies sliding homoclinic bifurcations in a class of symmetric three-zone three-dimensional piecewise affine systems. The systems have one parameter and the unperturbed systems have a pair of sliding homoclinic orbits to a saddle. Based on the analysis of the one-dimensional Poincaré maps, two types of sliding cycles are obtained from the sliding homoclinic bifurcations of the systems. In addition, two examples of sliding homoclinic orbits and sliding cycles are provided with simulations to illustrate the effectiveness of the theorems.

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.

This paper introduces a methodology for computing expected values associated with compression networks resulting from the application of compression algorithms to independent and identically distributed random time series. Our analysis establishes a robust correspondence between the calculated expected values and empirically derived results obtained from constructing networks using nondeterministic time series. Notably, the ratio of the average indegree of a network to the computed expected indegree for stochastic time series serves as a versatile metric. It enables the assessment of inherent randomness in time series and facilitates the distinction between nondeterministic and chaotic systems. The metric demonstrates high sensitivity to nondeterminism in both synthetic and real-world datasets, highlighting its capacity to detect subtle disturbances and high-frequency noise, even in series characterized by a deficient sample rate. Our results extend and confirm previous findings in the field.

In this paper, we study a class of nonlinear stochastic Duffing oscillators with multidelay feedback. We propose an effective reduction approach with the help of center manifold theory and stochastic averaging method. Taking the initial time-delay $\tau$ as the parameter, we reduce the original system to a one-dimensional averaged Itô equation. Our analysis reveals that the original system exhibits stochastic bifurcations, including stochastic D and P bifurcations. Once we have a clear understanding of the bifurcation structure, we can use this knowledge to choose appropriate system parameters and place the system in the desired state. For instance, by adjusting the initial time-delay $\tau$ of the control system, we can stabilize the system and achieve the desired outcome. Numerical simulations also verify the theoretical results. With appropriate parameter choices, multiple time delays can destabilize the equilibrium and promote chaotic behaviors, and can also lead to more stable dynamical behavior. Remarkably, we discovered that increasing the interval of time delays and feedback numbers can enhance system stability. It may potentially serve as a novel mechanism for stabilizing stochastic systems. The study provides a solid theoretical foundation for exploring stochastic systems subject to complex time-delay feedback control, and offers a valuable framework for related fields.

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.

In this paper, the stability and local bifurcations of a composite laminated circular cylindrical shell with radially pre-stretched membranes are explored by using analytical and numerical methods. On the basis of the four-dimensional averaged equation in the case of 1:1 internal resonance, three types of critical points are studied in detail. They are characterized as (1) one pair of purely imaginary eigenvalues and two negative eigenvalues; (2) a simple zero and one pair of purely imaginary eigenvalues; (3) two pairs of purely imaginary eigenvalues in nonresonant case. With the aid of normal form theory and Maple software, the steady-state solutions and the stability regions of the initial equilibrium solutions are obtained. The explicit expressions for the critical bifurcation curves leading to static bifurcation and Hopf bifurcation are also presented. The presence of Hopf bifurcation indicates that the circular cylindrical shell will flutter. The results contribute to the design of reasonable structure parameters to avoid flutter. Finally, numerical simulations are also presented to demonstrate the good agreement with the analytical predictions.

The establishment of a financial system should not only consider the current situation, but also need to refer to the past. Due to the memory of the fractional derivative, a fractional-order system can more effectively describe the historical significance of the financial system. Most scholars use the prediction–correction scheme to study fractional-order systems. This paper provides a higher-precision numerical method for the financial system, which more effectively simulate the system. Based on the definition of the Grünwald–Letnikov fractional derivative, the integer-order system with nonconstant demand elasticity is extended to the fractional-order setting, and its dynamic behavior is studied, with some novel chaotic attractors found. The research results are helpful for improving the understanding of the financial system and the financial market and for predicting financial risks.

We revisit the Kahan–Hirota–Kimura discretization of a quadratic vector field. The corresponding discrete system is generated by successive iterations of a birational map $F_h$. We include a proof of a formula for the Jacobian of this map. In the following, we essentially focus on the case of the Lotka–Volterra system. We discuss the notion of focal points and prefocal lines of the map $F_h$ and of its inverse $F_h^{-1}$. We show that the map $F_h$ is the product of two involutions. The nature of the fixed points of $F_h$ is studied. We introduce the notion of asymptotic focal and prefocal sets. We further provide a new proof of the theorem of Sanz-Serna. We show that the mapping $F_h$ is integrable for $h=1$ and that it preserves a pencil of conics (generic hyperbolas). To conclude, we provide several numerical simulations for $0<h<1$.

Memristor synapses have been widely introduced into neuronal models to investigate the effects of external magnetic fields. However, there is a relative lack of research on the external-induced electric fields in neurons. In this paper, a 4D-memristive Hindmarsh–Rose neuron model is constructed by introducing a memristor and an electric field variable, which can generate complex neural firing. Notably, numerical simulations reveal that the initial conditions of the memristor can induce different firing patterns, exhibiting a unique fractal structure in the basin of attractions. Remarkably, the offset parameters of the internal variables of the neuron can be canceled out so that the offset boosting of the variables can be achieved according to the initial values, giving rise to an uncountably many hidden attractors with homogeneous multistability. This model provides the first example of generating uncountably many attractors in a memristive neuron model without relying on trigonometric functions, significantly advancing our understanding of neuronal dynamics. Finally, a digital circuit is designed and implemented on the RISC-V platform to verify the numerical simulation and theoretical analysis. The findings of this study have a certain implication for the development of advanced neuromorphic computing systems and the understanding of complex neuronal behaviors in the presence of external electric fields.

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.

In this study, we discuss the global dynamics of the Holling-II amensalism model for a strong Allee effect of harmful species. We discuss the existence and stabilization of the extinction equilibria, exclusion equilibria, coexistence equilibria, and infinite singularities by analyzing the presence and stabilization of the system characteristics in terms of the possibilities and correspondences in the model when the death rate of the injured species is used as a threshold value. Also, we find that the two equilibrium points in the first quadrant are effective in proving that the model does not have globally stabilizing features and obtain two critical conditions and their corresponding global phase diagrams. Finally, we explore the weak Allee effect of the victim species, and using the analysis from numerical simulations, we recapitulate the analysis and dynamics of the model in equilibrium.

The limited availability of formal jobs in developing nations always heightens the challenge for unemployed individuals in securing regular employment. Temporary employment in the informal sector serves as a source to fulfill their basic needs and enhance their employable skills. In this paper, we introduce a nonlinear mathematical model to study the effect of informal and formal jobs on the dynamics of unemployment. For the model formulation, we categorize the labor force into three classes: unemployed, temporary employed, and regularly employed. A separate dynamical variable is used to represent the available temporary vacancies. It is assumed that temporarily employed individuals may transition into regular employment or self-employment. Furthermore, self-employed individuals contribute to generating temporary vacancies within the informal sector. The long-term behavior of the proposed system is analyzed using the qualitative theory of differential equations. A quantity known as the reproduction number of the system is derived, and it is found that the occurrence of multiple bifurcations for the proposed system is influenced by the value of this threshold quantity. Furthermore, we validate our analytical findings numerically. The findings of this study illustrate that an increase in the shifting rate of individuals from temporary to regular employment is not always effective in increasing the number of regularly employed individuals. Additionally, an increase in the transition of temporarily employed individuals into self-employment, coupled with their involvement in creating more temporary jobs, proves beneficial in reducing unemployment.

Speech Emotion Recognition (SER) is a challenging task characterized by the diversity and complexity of emotional expression. Due to its powerful feature extraction capabilities, Transformer Network (TN) demonstrates advantages and potential in SER. However, the limited size of available datasets and the difficulty of decoupling emotional features restrain its performance and present challenges in implementing SER on edge devices. To address these issues, we present a Memristor-based Progressive Hierarchical Conformer Architecture (MPCA) and design a conformer submodule that leverages convolution to mitigate TN’s limitations in SER. We propose attention-based feature decoupling, employing hierarchical extraction to decouple speaker characteristics and retain the relevant components, thereby obtaining reliable emotional features. Furthermore, we propose a reconfigurable circuit implementation scheme for MPCA based on operator multiplexing achieving flexible modules that can be dynamically adjusted based on the resources of edge devices, and the stability of the designed circuit is analyzed by simulation experiments with PSPICE. We show that the suggested MPCA demonstrates state-of-the-art performance in SER while significantly reducing system power consumption, offering a solution for SER implementation on edge devices.

We investigate novel bifurcation phenomena in the generalized KdV–mKdV-like equation. Unlike previous phase diagram studies, we choose variables related to wave speed as the coordinate axis, which led us to discover some interesting bifurcation phenomena. Our method can also be extended to study some other bifurcation phenomena in equations.