International Journal of Bifurcation and Chaos

One of the most important tasks in the qualitative theory of differential equations in the plane is the study of global asymptotic stability: an equilibrium point that is globally attractive. It is known that if an equilibrium point of a planar vector field is globally asymptotically stable, then the vector field is bounded. A planar vector field is said to be bounded if the forward orbit of every point enters and remains in a compact set. In this paper, we study cubic Kukles systems that are bounded which is the first step toward the characterization of cubic Kukles systems that are globally asymptotically stable. We emphasize that these systems form a seven-parameter family of polynomial differential systems. We obtain a total of 25 cubic Kukles sub-systems that are bounded and we provide 11 phase portraits of such sub-systems in a neighborhood of infinity.

Different species in the environment compete with one another for shared resources and suitable habitats. As a result, it becomes crucial to identify the essential components involved in preserving biodiversity. This paper describes the emergence of a wide range of temporal and spatiotemporal dynamics of an intraguild predation (IGP) model with intraspecies competition and Holling type-II response function between the IG-prey and IG-predator. To explore the dynamics of the temporal model, we study the local stability of all the biologically feasible steady states, global stability of the interior steady state, and Hopf bifurcation. Performing partial rank correlation coefficient (PRCC) sensitivity analysis, we picked more sensitive parameters and plotted the stability regions in two-parameter spaces. The dynamics exhibited by the system can demonstrate some important hallmarks noticed in the environments, such as the existence and nonexistence of oscillatory behavior in the IGP model. The analytical formulations for the stability and bifurcation of the coexistence of the steady-state are not easy to obtain, but we have verified numerically that the limit cycle bifurcates from the coexistence of the steady-state and there eventually emerges a chaotic behavior through period-doubling bifurcations. By plotting the largest Lyapunov exponent (LLE), we have verified that the temporal system experiences chaotic behavior. It is demonstrated that in the case of a diffusive system, diffusion may cause the coexistence of the steady state, which is otherwise stable, to become unstable. Criteria for the occurrence of Turing instability connected to the IGP model are derived. Our numerical illustrations demonstrate that diffusion-driven instability develops different stationary patterns based on the different initial conditions and diffusion parameters. The spatiotemporal patterns are observed in the Turing, Hopf–Turing, and Hopf regions.

In this paper, we propose a digital image encryption scheme based on Tabu Search (TS) algorithm and Chen’s hyperchaos system. First, in order to enhance the security of the algorithm and resist the known-plaintext attack, the key is associated with the ordinary image, and the hash value generated by the ordinary image is used as the initial value of the hyperchaotic system. Moreover, the TS algorithm is used to obtain the optimal subsequence to scramble the sub-block image, which ensures the scrambling effect of the algorithm. In addition, for the sake of minimizing the correlation of neighborhood pixels and strengthening the effect of scrambling, the ordinary image is divided into blocks for scrambling and diffusion. Through simulation and experiments, the key sensitivity, differential attack and pure ciphertext attack are analyzed. Compared with the other encryption schemes, the results verify the effectiveness and reliability of the proposed scheme.

This paper discusses the effect of density-dependent birth rate in a discrete predator–prey system with mixed functional responses of Holling types I and III. We use the Beverton–Holt function to modify the birth rate parameter of the prey species. The steady states and their stability are established. The criteria for flip bifurcation (FB) and Neimark–Sacker bifurcation (NSB) are proposed analytically using the center manifold theorem and normal bifurcation theory. Using the state feedback control method, it is shown that chaotic orbit can be stabilized at an unstable steady-state. This study deduces that the density of predator species is negatively affected by the density-dependent birth rate of prey species. Moreover, density-dependent birth rate can increase the possibility of extinction in prey species, which means that this birth rate may cause Allee effect. Some numerical simulations are given to verify these analytical results.

In online social networks (OSNs), the spread of negative information can cause social shocks, endanger public safety and harm public interests. In this paper, to reduce the impact of negative information on the network ecosystem as much as possible, we consider a competitive strategy of mainstream media coverage and construct a competitively regulated reaction–diffusion model of negative information transmission. For this model, we obtain the basic reproduction number of information propagation, the invade numbers of negative information and positive information. Besides, the local and global stabilities of equilibrium states are discussed. Further, steady bifurcation analyses from a positive information equilibrium and a negative information equilibrium are carried out. Finally, we present some numerical simulations to verify the theoretical analysis results and discuss how spatial heterogeneity affects the process of information diffusion. The conclusions are as follows: Increasing the spread of positive information, controlling the spatial heterogeneity of the transmission rate and reducing the diffusion coefficient when spatial heterogeneous transmission rate occurs are beneficial for suppressing the spread of negative information.

In the dynamic bifurcation and dynamic hysteresis of a system, our first consideration is the tiny changes in time that can lead to variations in the vector field, potentially causing the occurrence of system dynamic bifurcation and dynamic hysteresis. This paper presents a novel method for analyzing the global bifurcation and hysteresis of Functionally Graded Material (FGM) truncated conical shell structure under aerodynamics and in-plane force along meridian near internal resonances. This method involves introducing the ratio of vector fields as a periodic perturbation parameter in the system’s nonlinear second-order ordinary differential governing equations. This allows for the analysis of the nonlinear ordinary differential equations as algebraic equations, yielding algebraic expressions that only involve parameters and do not contain state variables. Subsequently, the global bifurcation set and global hysteresis set of FGM truncated conical shell structure are obtained, providing the parameters interval ranges for the stability and instability of FGM truncated conical shell structure. By utilizing MAPLE software for numerical simulation, the images of the global bifurcation set and global hysteresis set about the amplitude of in-plane load are first simulated numerically. Subsequently, the amplitude graphs of the equilibrium points are numerically simulated for validation. The results demonstrate a perfect alignment between the images of the global bifurcation set and global hysteresis set and the data from the amplitude graphs of the equilibrium point. Due to the periodicity of the periodic perturbation parameter, it shuttles between different persistent regions as other parameters change, leading to the generation of chaotic solutions in the governing equations. The validity of the obtained results is confirmed through comparisons with existing literature.

The impact of pests on crop yield is a significant worry for farmers, and finding an effective strategy to control insect population growth has become a pressing matter. This study explores the dynamic analysis of a model that incorporates natural predators to control crop pests. In the research, an analysis is conducted on the model’s positivity and boundedness, followed by an examination of the existence and stability of the equilibrium points. The study focused on analyzing bifurcations at a biologically feasible equilibrium point. The parameters considered for bifurcation are the predation rate by pests and the consumption rate of natural predators. In addition, an optimal control technique has been employed to enhance the growth of the plant population through the utilization of Pontryagin’s maximum principle. Moreover, the impact of the nine parameters on the model is examined through sensitivity analysis using the partial rank correlation coefficient method. Numerical simulations are conducted to validate the analytical findings, revealing the occurrence of bifurcation and the positive impact of control on plant population growth.

Drawing inspiration from the paradigm of engineering projects, this paper presents a novel real-time chaotic video encryption scheme based on multithreaded parallel computing and multiround confusion–diffusion architecture. It utilizes a contractor thread to retrieve and assign frames to worker threads, which operate concurrently to perform DNA coding, confusion, and diffusion operations on their designated frames for encryption, with the resulting frames processed by a dealer thread. To evaluate the performance of the proposed algorithm, it is implemented on a workstation equipped with an Intel Xeon Gold 6226$@$2.9GHz CPU and 64GB of memory. The statistical and security analyses demonstrate that the proposed strategy exhibits exceptional statistical properties and provides robust resistance against various attacks. The encryption speed evaluations show that the deployed cryptosystem achieves delay-free $512\times 512$ 24FPS video encryption, with an average encryption time of 34.69ms, despite the execution of four rounds of DNA coding, five rounds of confusion, and three rounds of diffusion operations on each frame. In comparison to existing real-time chaotic video encryption schemes based on parallel computing, our method attains superior versatility and heightened security by processing each frame with an independent worker thread and incorporating DNA coding technology.

Tumor prevention and control, taking into account the load of tumor cells or immune cells, are often described by Filippov systems, usually with a smooth function on the right-hand side describing the tumor immunity. However, these systems only consider the load of tumor cells or immune cells as a threshold function, which suggest that patients will only be treated if the load of tumor cells or immune cells exceeds a certain threshold. As a result, there may be an excessive growth of tumor cells or a rapid decrease in immune cells, which can lead to undesirable results. Considering this problem, we propose a weighted switching system using the load of tumor cells and its change rate as the threshold. Theoretical analysis and numerical simulation of the system are performed by changing the threshold value. We find that in the proposed nonlinear weighted threshold control method for the system, on the one hand, the system can produce rich dynamical behaviors, including many possible cases of bistability and sliding segments of the system, etc., and on the other hand, it is also very important for the prevention and control of tumor.

In this paper, we consider piecewise smooth differential systems with subsystems having cubic reversible isochronous centers at the origin, where the separation line is a straight line passing through the origin. By computing Lyapunov constants, we obtain Hopf cyclicity at the origin with necessary and sufficient conditions for the origin to be a center in this type of piecewise smooth differential systems.

A chaotic system without equilibrium points is a focal subject in recent years, and its unique dynamic characteristics have attracted extensive research. In this paper, different chaotic systems with no equilibrium points are constructed by modifying the Sprott-A system. The test results show that the first system is conservative. After investigating the hidden dynamical behavior of the first system, the rotation attractor phenomenon is revealed by the rotation factor. Next, we make a second transformation of the Sprott-A system. By introducing a trigonometric function, we get a new system that can generate infinitely many coexisting grid attractors. After calculation, we find that the new system still belongs to the chaotic system without equilibrium points and the coexistence phenomenon of attractors is created by adjusting the initial value continuously. The chaotic system without equilibrium points which can realize the coexistence of attractors has potential applications in some related fields, which is worthy of further study.

This study investigates the generalized Nosé–Hoover system. The original version of the system is a chaotic system designed to represent the interaction between a harmonic oscillator and a heat bath maintained at a constant temperature. Despite its simplicity in just three dimensions, it exhibits complex and unusual dynamics. This investigation focuses on studying local bifurcations, including Saddle-Node and Hopf bifurcations, of the generalized Nosé–Hoover system. In terms of cyclicity, the Lyapunov quantities technique is used to demonstrate that three periodic orbits can bifurcate from the Hopf point. This mathematical research contributes to understanding the equilibrium points, their stability and the dynamics of the nonlinear model when some of the parameters are varied.

In earlier research, we developed two techniques designed to expand the construction of a periodic orbit dividing surface for Hamiltonian systems with three or more degrees of freedom. Our methodology involved transforming a periodic orbit into a torus or cylinder, thereby elevating it to a higher-dimensional structure within the energy surface (refer to [, , , , ]). Recently, we introduced two new methods for creating dividing surfaces, which do not rely on periodic orbits. Instead, we used 2D surfaces (geometric entities) or 3D surfaces in a Hamiltonian system with three degrees of freedom (see [, , , ]). In these studies, we applied these surfaces within a quadratic normal-form Hamiltonian system with three degrees of freedom. This series of two papers (this paper and [, ]) extends our results to 2D-generating surfaces for quartic Hamiltonian systems with three degrees of freedom. This paper focuses on presenting the second method of constructing 2D-generating surfaces.

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

The conditions for the synchronization of three interconnected generalized Lorenz systems are given. The interconnection topology contains a loop. It is shown that, under certain conditions on the strength of the coupling of the systems, the full synchronization of all three systems is guaranteed. The results are illustrated by an example.