International Journal of Bifurcation and Chaos

The presence of nonlinear feedback is an effective method for generating chaos in dynamical systems. Although physics provides a plethora of nonlinear relationships that can be exploited for this purpose, engineering chaotic electronic circuits with prescribed nonlinear terms remain a formidable challenge. In particular, the implementation of fractional exponentiation represents a particularly challenging task. In this paper, we present a solution to this problem in the form of a new circuit topology, which we have designated as the “444 circuit”, comprising only four transistors, four operational amplifiers, and four resistor pairs. In addition to multiplication and division, this circuit can perform complex algebraic operations, including the calculation of fractional powers. After introducing its operating principles and some conceptual aspects of its implementation, we demonstrate the performance of this unifying circuit structure by realizing a family of Lorenz-like systems.

Introducing the memcapacitor into the Cellular Neural Network (CNN), the Memcapacitor-Cellular Neural Network (MC-CNN) model with infinitely many equilibrium points is constructed. A series of dynamical behaviors of the MC-CNN are investigated by various nonlinear system analysis means. It is shown that the system has a large maximum Lyapunov exponent in a specific parameter range. And with the variation of parameters, the system is able to produce many different phase trajectories of the attractor. Multistability is also found in the system. The pseudo-randomness of the MC-CNN is calculated by Spectral Entropy (SE) complexity algorithm. The final hardware results proves the physical realizability of the system. The MC-CNN model is intended to provide guidance for neural networks and cryptographic strategies based on the memcapacitor.

In this paper, based on the Central Manifold Theorem and Hopf Theorem, the Hopf bifurcation characteristics of a novel 3D-chaotic system with a four-wing attractor are studied. The stability of the equilibria of the system is examined. Additionally, the dynamics near the equilibria at infinity are explored using Poincaré compactification on polynomial vector fields in ${ℝ}^3$. These findings contribute to a deeper understanding of the system’s dynamics and provide insights into the behavior of its solutions.

This paper studies the spatial dynamics in a diffusive model with Ivlev-type functional response and quadratic mortality. Turing instability, steady-state and Hopf bifurcations of the system are studied. The normal forms related to steady-state and Hopf bifurcation are detailedly deduced by employing the normal form theory. Numerical simulations are applied to further explain our analysis and intuitively exhibit where the periodic solutions and steady states arise.

In this paper, a Schistosomiasis model with multi-delays is proposed to study the impacts of delays on disease transmission. First, the exact expression of the basic reproduction number ${{ℛ}}_0$ of the model is obtained, which is proved to be the threshold parameter. More precisely, when ${{ℛ}}_0<1$, the disease is extinct, and when ${{ℛ}}_0>1$, the disease is persistent. Next, the existence of the Hopf bifurcation of the model is discussed with delays as parameters, and the direction and stability of the Hopf bifurcation induced by delays are investigated according to the paradigm theory and the central prevalence theory. In addition, the global extensionality of the local Hopf bifurcation is discussed, and the length of delays is estimated using the Nyquist criterion to determine the stability of the endemic equilibrium. Finally, numerical simulations are presented to demonstrate the main theoretical results and the effect of delays on the disease.

This paper concerns the weak 16th Hilbert problem and considers the Hamiltonian center $ẋ=-y^{2n-1}$, $ẏ=x^{2n-1}$, and we perturb it by all polynomials of degree $2n-1$ for $n=2,3,4,5,6,7,8$. We prove that the maximum number of limit cycles that can bifurcate from the periodic orbits of this center for $n=2,3,4,5,6,7,8$, under the mentioned perturbations and using the averaging theory of first order, is $1,4,3,2,5,6,7$, respectively.

The restoration and maintenance of oscillatory rhythms are integrated to the regular operation of biological and physical processes. In this paper, we explore the revival of oscillations, specifically focusing on the phenomenon of explosive Nontrivial Amplitude Death (NAD) that occurs within N van der Pol oscillators coupled through mean-field attractive–repulsive interactions. By incorporating discrete and distributed delays into the repulsive mean-field, respectively, we provide theoretical and numerical evidence that the explosive NAD can be effectively destabilized, leading to the restoration of the oscillatory behavior. By utilizing Lyapunov stability, we theoretically determine the boundaries of the oscillation region recovered from NAD. For discrete delay, adjusting the magnitude of delay can result in a substantial reduction of the NAD region within the parameter space. Under a specific delay, the NAD region completely becomes an oscillation state. Moreover, in the scenario of uniformly distributed delay, the presence of delay also contributes to a decrease of the NAD area, and as the distribution width increases, the revived oscillation region gradually contracts compared to the case with discrete delay. Importantly, the results from the numerical simulations are in agreement with the theoretical analysis. Our framework establishes a theoretical foundation for comprehending the regeneration of oscillation in mean-field attractive–repulsive coupled systems.

The goal of this work is first to compare the bifurcation diagrams of an electrodynamic electromechanical system when it is subjected to sine, square and triangular periodic excitations. The example considered is the electromechanical pendulum. Then the task is to analyze the dynamical states of the same system using the computation of the Smaller Alignment Index (SALI) along with the Poincaré Surface of Section (PSS). The bifurcation diagrams present abrupt and period-doubling transitions to chaos. It is found that the triangle form of the excitation gives the larger range of the signal amplitude in which the system behaves chaotically. For the SALI and PSS analyses, they are plotted both for the conservative and nonconservative systems with the impact of the monostability and bistability taken into account. It was found that the conservative system with bistable potential is mainly characterized by chaotic dynamics while regular dynamics are predominant in the system with monostable potential. The evolution of the percentage of chaotic orbits based on SALI was represented to support this result. In the nonconservative case, it was noted that when switching from bistability to monostability, the system can undergo a change of dynamics or keep its chaotic dynamics as seen for triangle excitation. The SALI method was successfully used to determine the dynamics of the system driven by the different forms of excitations as predicted by other tools.

This paper proposes an innovative approach to designing image encryption hardware by leveraging the Negative-resistance Memristor-based Hopfield Neural Network (NMHNN) model. In this method, the conventional Hopfield Neural Network (HNN) is modified by substituting one of its synaptic weights with a negative-resistance memristor model. This modification demonstrates enhanced complex dynamics while maintaining a simplified structure. The NMHNN generates a chaotic sequence, which serves as a key for image encryption and decryption through a confusion–diffusion process. As a result, this approach significantly improves the efficiency of both image encryption and decryption processes. To validate the practical feasibility of this memristor-based neural network for encryption, a 3-neuron HNN circuit with one ${\mathrm{\text{HfO}}}_2$ memristor is constructed and tested. The hardware experiment demonstrates strong resilience against statistical analysis and entropy attacks. Notably, the operational efficiency of the proposed method is demonstrated to be 23 times greater than that of the Advanced Encryption Standard (AES), highlighting its substantial effectiveness and potential for hardware implementation in the field of image encryption.

A saddle–node bifurcation is demonstrated in a model of mosquito population growth. This bifurcation helps us understand the dynamics of two stages in the model: the aquatic stage (which includes the egg, larva, and pupa) and the nonaquatic stage (the mature mosquito). We calculated the basic offspring number using the next-generation matrix method and displayed it as the value of the saddle–node bifurcation. This means that the stable extinction state becomes unstable, while a survival state emerges as dominant in the dynamics, along with another coexistence state as a transition to surveillance. Numerical computations verify how factors such as limited logistic growth, the Allee effect, and the mortality density rate modify the mosquito dynamics.

In this paper, we study the traveling wave solutions of a generalized reaction–diffusion system based on the classical Fisher-type system. Through qualitative analysis and blow-up techniques, we prove the existence of traveling fronts of the system. Moreover, we detect the limit points (i.e. $\omega$-limit points or $\alpha$-limit points) of all traveling wave solutions by analyzing the global topological phase portraits of the equivalent system.

Teixeira singularity (TS) is the transverse intersection point of invisible quadratic tangency lines in 3D piecewise-smooth systems, which significantly impacts both local and global dynamics. This paper considers the 3D nonlinear Filippov system composed of Yang system and Sprott-C system with a symmetric pair of TSs. In parameters space, this Filippov system exhibits rich bifurcation phenomena, such as the boundary equilibrium bifurcations and a symmetric pair of sliding transcritical bifurcations. By constructing the Poincaré map and analyzing the number of fixed points, we present the conditions for the existence of crossing limit cycles. Furthermore, we observe the phenomenon of compound bifurcation occurring at TS, namely the TS bifurcation. This bifurcation results from the combination of transcritical bifurcation of sliding dynamics and Bogdanov–Takens bifurcation of crossing dynamics. Finally, a new chaotic attractor, which consistently surrounds two TSs, is found through numerical analysis.

Resource allocation is a pivotal concern in swarm robotic systems. This study delves into the chaotic dynamics within the yield game process of resource-providing robots in such systems, analyzing system bifurcation and stability across multiple parameters. We propose a market-based model for swarm robotic systems, employing game theory to formulate a yield game model among resource-providing robots. Through numerical analysis and simulation, we identify stable regions of system parameters and present bifurcation diagrams under varied parameters. Our findings indicate that increasing cost parameters or decreasing decision parameters stabilizes the yield and utility of resource-providing robots, while the opposite destabilizes them. Moreover, we examine the intricate influence of cost parameters on system stability by comparing linear and exponential resource cost functions. Results reveal that exponential cost functions lead to heightened system chaos and parameter sensitivity. These insights offer crucial theoretical groundwork and practical directions for the development and deployment of swarm robot systems.

This study presents 2D-PECLSM, a novel hyperchaotic map that integrates the 2D Logistic and the Sine map to enhance chaotic behavior. We propose a new image encryption algorithm (IEA) that leverages this map alongside fractal theory and global cross-coupled diffusion techniques. The 2D-PECLSM generates two essential chaotic sequences for secure encryption and based on this map associated with fractal theory processes. Each pixel of the plain image is effectively confused using a fractal matrix and globally diffused, ensuring robust protection of visual data. Experimental results demonstrate that the 2D-PECLSM-based IEA significantly outperforms the existing state-of-the-art IEAs, highlighting its potential as a powerful tool for enhancing image security against digital threats. This work contributes valuable insights to the field of cryptography, emphasizing the need for innovative approaches to safeguard sensitive visual information.

Fractional chaotic maps have been recently proposed to describe long-memory effects. However, their bifurcation analysis is complicated. This paper considers a linearized system and introduces necessary and sufficient conditions of Neimark–Sacker (N–S) bifurcation by complex analysis method. Both critical values of fractional order and the system’s parameters are determined when the fractional order maps undergo sudden changes. This paper provides an efficient method for the N–S bifurcation analysis of fractional discrete-time systems with time delays.