International Journal of Bifurcation and Chaos

The main objective of this study is to investigate the chaotic dynamics and analog circuit implementation of a nonautonomous single inertial neuron. First, the dimensionless mathematical model of such neuron is established. The equilibria with three kinds of stabilities are then depicted, and system symmetry and Hamilton energy function are analyzed, respectively. In numerical simulations, by using the two-dimensional/one-dimensional bifurcation diagrams, Lyapunov exponent spectra, phase plots, and basins of attraction, bifurcation of coexisting bubbles as well as bi-stable patterns are revealed, where the generated coexisting attractors are symmetric about the origin. In addition, we confirm that the complexity of the basins of attraction deteriorates with the increase of the frequency parameter. Finally, analog circuit experiments with few analog circuit elements are deployed, via which the bi-stable patterns of the model are well verified.

In our earlier research [, , , , , , , ], we developed two new approaches for building up dividing surfaces in the phase space of Hamiltonian systems with three or more degrees of freedom. These surfaces were derived either from periodic orbits or from 2D or 3D generating surfaces in the phase space. Our previous work extended the realization of these dividing surfaces into more intricate forms, such as tori or cylinders, situated within the constant energy manifold of the Hamiltonian system. In those studies, we utilized the above-mentioned surfaces in the setting of a three-degrees-of-freedom quadratic normal form Hamiltonian system. This series of papers extends our findings to 3D generating surfaces for a three-degrees-of-freedom quartic normal form Hamiltonian system. The focus of this paper is to introduce the first method for constructing 3D generating surfaces.

In our previous studies [, , , , , , , ], we presented two methods for building up dividing surfaces based on either periodic orbits or 2D/3D generating surfaces, specifically for Hamiltonian systems with three or more degrees of freedom. These papers extended these dividing surface constructions to allow for more complex forms, such as tori or cylinders, embedded within the energy surface of the Hamiltonian system. These studies were applied to a quadratic normal form Hamiltonian system with three degrees of freedom. This series of papers extends our findings to 3D generating surfaces for three degrees of freedom quartic Hamiltonian systems. This paper focuses on the second approach for constructing these 3D generating surfaces.

In this paper, we study the stability of the equilibrium point and investigate the periodicity properties of a class of difference equations $x_{n+1}=\frac{A_n{x}_n^3+B}{C{x}_{n-1}}$, $n\in \mathbb{N},$ whose parameters and initial values are positive. In particular, $A_n$ is a two-periodic coefficient with respect to n. Moreover, we depict the phase transition of orbits and analyze the deviation of the quasi-periodic orbits from some periodic orbits related to the arithmetic property of $A_n$, including Diophantine and Liouvillean numbers. We associate the response solution problem of differential equations with the fixed point problem of the area-preserving mapping related to multiperiodic difference equations, and pose some future work. The techniques are based on Moser’s twist mapping theorem, reversibility and some invariant structures embedded in the system.

Media has a notable impact on reducing disease prevalence, while sanitation measures and heightened awareness can effectively control epidemics by diminishing bacterial growth rates and limiting direct contact with infected individuals. In this study, we propose and analyze an epidemic model to explore how media and sanitation practices influence the dynamics of diseases transmitted through direct contact between susceptible and infected individuals, as well as via bacteria in the environment. Our study entails a combined approach involving both analytical and numerical analyses of the system. We observe that the disease-free and endemic equilibria of the system are interconnected through a forward transcritical bifurcation. We estimate the most important model parameter using authentic cholera data from Sudan for calibration. Our numerical findings suggest that regulating disease transmission through direct contact and environmental bacteria can significantly decrease disease prevalence. Additionally, we note that the growth rate of social media advertisements, along with efforts made by government officials and informed individuals to eliminate bacteria through sanitation coverage, introduces destabilizing effects. However, system stability is reestablished when the baseline number of social media advertisements exceeds a specific threshold. The dissemination of awareness among susceptible individuals, as well as the rate of transfer of informed people to the susceptible class, initially leads to destabilization but eventually stabilizes the system. Disease eradication becomes feasible when the rate of transfer of informed individuals to the susceptible class is very low. Moreover, higher initial values for awareness programs and the dissemination rate of awareness could also eliminate the disease from society. Furthermore, we see that increasing the treatment rate of infectives plays a significant role in achieving disease eradication. Moreover, we investigate an optimal control problem that integrates sanitation interventions and awareness protocols.

Recently, an image encryption scheme with new chaotic signals has been proposed. Some limitations of this scheme will be pointed out. It adopts a “diffusion–permutation” architecture. The diffusion encryption is linear and can be simplified. Bit-level permutation significantly decreases the encryption efficiency. To solve the above problems, an improved scheme is proposed. The new scheme employs a Feistel-like network with four rounds of encryption. The dynamic substitution box (S-box) generated by Fisher–Yates permutation increases the nonlinearity of encryption. The optimized diffusion process is applied to obtain the avalanche effect. The experimental results show that the improved scheme performs well. In addition, the resistance against various attacks is analyzed from the perspective of a cryptanalyst.

The aim of this study is to introduce a novel threshold-free recursive analysis method for friction-induced signals and explore its potential application in the identification of running-in state. To address some issues from recursive analysis results being influenced by the recursive threshold, a grayscale threshold-free recursive metric is investigated to measure the recursive characteristics in a tribosystem. Additionally, the visualization and quantitative measure extraction of the gray-scaled recursive matrix are discussed. Afterward, the friction-induced torque signals throughout the running-in process are extracted and then characterized by the proposed method. The results demonstrate that the morphology and quantitative features of the gray-scale recursive plots effectively reveal tribological behaviors and are closely related to the frictional running-in states. Therefore, the proposed method can be considered a reliable approach for characterizing the recursive properties of a tribosystem and identifying running-in states.

In this work, we present a method to find periodic orbits for state-dependent delay differential equations. This method is based on the Newton–Kantorovich algorithm and is illustrated in the case of the one-dimensional Cubic Ikeda Map. Though this work is mainly numerical, the techniques developed are aimed to be a posteriori used in the frame of computer-assisted proof.

With increasing remote work and online learning, there is a growing demand for image encryption algorithms with sufficient capacity and efficiency. In this paper, we propose an effective multiimage encryption algorithm. By coupling the Sine map and some useful mathematical functions, we design a two-dimensional hyperchaotic system (2D SFCM). After evaluating different chaos indicators, it is found that the Lyapunov exponent and the permutation entropy of 2D SFCM are significantly higher than those of other chaotic mappings. Additionally, its sample entropy has an average value of 1.916. These results indicate that the chaotic sequences generated by 2D SFCM exhibit a high level of randomness. The pseudo-random sequences generated by this chaotic system are combined with image encryption algorithms to shuffle 3D images by accessing pixels from different planes. Using randomly determined pixel points, cyclic diffusion is performed on each plane to ensure that the pixel information at any point extends to the whole 3D image. Experimental results demonstrate that the proposed algorithm exhibits good key sensitivity and robustness. Numerical results show that the algorithm has considerable capabilities in resisting common attacks. Compared to related algorithms, our algorithm shows certain advantages in terms of information entropy and resistance to differential attacks.

In this paper, we investigate a predator–prey model with Holling-II functional response, Allee effect and constant-yield predator harvesting, by comparing the differences between $h_1=0$ and $h_1>0$, where $h_1$ denotes the harvesting rate of predators. For system without harvesting, the Allee effect leads to population extinction. The system has at most one positive equilibrium and has a supercritical Hopf bifurcation which depends on the natural mortality rate of predators. Besides, by using normal form theory, we show that the system with $h_1>0$ reveals rich dynamic properties, including saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation, where numerical simulations are presented to demonstrate the Bogdanov–Takens bifurcation of codimension 2 with a limit cycle and a homoclinic cycle. The system can generate up to two positive equilibria with the changes of $h_1$, which indicates that appropriate predator harvesting can assist in regulating the ecosystem. We then give the optimal harvesting strategy by using Pontryagin’s maximum principle. Finally, numerical simulations are performed to validate the functions of Allee effect and harvesting. Theoretical studies and numerical simulations demonstrate that the Allee effect can lead to species extinction and highlight the role of appropriate harvesting in controlling the stability of the system.

This paper constructs a nonlinear dynamical model of a mixed ownership market duopoly game for environmental tax collection within the framework of bounded rationality expectations. A stability analysis of the proposed model is conducted, and a Nash equilibrium stability region is estimated using the Jury criterion. The effects of the degree of privatization and environmental tax rate on the stability are investigated through numerical simulations. Subsequently, the comprehensive global analysis is carried out using the composite cell coordinate system method. Two types of crisis phenomena, namely, interior and boundary crises, have been identified. The former is caused by the collision between chaotic or periodic attractors and interior chaotic saddle within the basin of attraction, and the latter is caused by the collision between chaotic attractor and periodic saddle on the basin boundary. Meanwhile, the global dynamic behavior of two private enterprises engaged in synchronized output adjustment is investigated. When the output adjustment speed varies, the structure of the basin of attraction undergoes a transformation as changes are made to the number of coexisting attractors. The sudden disappearance of one attractor is due to the collision of the chaotic attractor with the saddle, on the basin boundary, merging into a larger saddle and resulting in a boundary crisis. The emergence of numerous “holes” can also be observed. It is found that the presence of periodic or chaotic saddles enriches the dynamical phenomena. Finally, the chaotic phenomenon occurring in the mixed ownership market is controlled using the time-delay feedback control method, stabilizing the system in a stable state, which implies a steady-state of total market output. The study of this model can provide theoretical guidance for the decision-making process of enterprises, helping minimize profit losses while stabilizing the market.

Acetylcholine, as an important neurotransmitter, regulates neural excitability and discharge property distinctly, whose deficiency in the hippocampus can deteriorate cognitive and memory functions in Alzheimer’s Disease (AD). By establishing a hippocampal small-world network of pyramidal neurons and interneurons, this work aims to explore how acetylcholine deficiency influences the hippocampal neural dynamics from the perspective of neurocomputation. Based on some experimental results that acetylcholine can inhibit several $K^{+}$ channels in the pyramidal neuron via muscarinic receptors, we simulate acetylcholine deficiency by increasing the conductance $g_M$ of M-type (or $g_A$ of A-type) $K^{+}$ channel. The numerical results indicate that the enhancement of $g_M$ or $g_A$ can affect the pyramidal neuron dynamics in the individual node and network levels significantly. For the individual pyramidal neuron, its excitability and mean firing rate gradually decrease upon increasing $g_M$ or $g_A$, in particular, the neuron transitions from a firing state to a resting state when $g_M$ or $g_A$ increases over a threshold. Through dynamic bifurcation analysis, we find that the underlying dynamical mechanism behind these changes is attributed to sub-critical Andronov–Hopf or saddle-node bifurcations induced by acetylcholine deficiency. For pyramidal neurons in the network scale, acetylcholine deficiency can shift the network encoding mode from collective regular firing to bursting, and more importantly, the averaged mean firing rate and sample entropy of the network decrease significantly with severe acetylcholine deficiency. These results are consistent with previous studies reported in the electrophysiological experiments of AD. This work may help to deeply understand the AD pathogenesis of cholinergic hypothesis.

The herd behavior of prey plays an important role in the predator–prey system. In this paper, we analyze the dynamical properties of a discrete predator–prey model with herd behavior. We found that due to the introduction of herd effects, this model exhibits complex dynamical behavior such as codimension-1 bifurcation (Flip and Neimark–Sacker bifurcations) and codimension-2 bifurcation (1:2, 1:3, and 1:4 strong resonance). Using the center manifold theorem and normal form theory, we obtained the critical conditions that may lead to changes in the stability of the system and even the emergence of new stable states or oscillating behaviors. Numerical simulations not only validate theoretical proofs but also reveal the complex and rich dynamical behavior of the discrete model.

To find the exact explicit solution of the concatenation model, the corresponding differential system of the amplitude component is used, which is a planar dynamical system with a singular straight line. In this paper, by using the techniques from dynamical systems and singular traveling wave theory developed by Li and Chen [], its corresponding traveling wave system is solved and analyzed, obtaining the corresponding phase portraits and showing the dynamical behavior of the amplitude component. Under different parameter conditions, exact explicit solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions, compacton solutions, as well as peakons and periodic peakons, are found explicitly.

This paper is dedicated to studying the maximum number of crossing limit cycles of discontinuous planar piecewise-linear differential systems with three regions separated by a nonregular line and formed by linear centers and linear Hamiltonian systems without equilibria. There are two types of crossing limit cycles: one that intersects the nonregular line at three points and the other that intersects the nonregular line at four points, named three-point crossing limit cycles and four-point crossing limit cycles, respectively. We prove that the maximum numbers of three-point and four-point crossing limit cycles are three and one, respectively, and both reach their upper bounds. Furthermore, we show by concrete examples that one four-point crossing limit cycle and two three-point crossing limit cycles can exist simultaneously.