International Journal of Bifurcation and Chaos

Chaos in a DC–DC boost converter operating in discontinuous conduction mode when controlled by sliding mode control is detected. A route to chaos through sliding bifurcations is shown along with the coexistence of the chaotic regime with a stable operating point. A bifurcation analysis in the sliding vector field is carried out with the objective of defining a range of values for which control parameters guarantee the robustness of the system, at least locally. The analysis is based on the study of possible sliding bifurcations, that is, bifurcations inherent to the sliding dynamics. It is rigorously shown that there is an unstable limit cycle in the sliding dynamics due to a subcritical Hopf bifurcation. The existence of this limit cycle and its evolution with parameters turn out to be of great importance for the stability and robust control of this type of power converter.

In this paper, a novel locally active memristor is constructed. Based on the 3D Hindmarsh–Rose (HR) and 2D FitzHugh–Nagumo (FHN) neuron, a heterogeneous neuronal system is constructed by connecting the two neurons with the locally active memristor. The equilibrium point and stability of the system are investigated. The dynamic behavior of the system is numerically and experimentally revealed by utilizing dynamic analyses in terms of interspike interval bifurcation diagram, two-parameter bifurcation diagram and so on. The unique and abundant dynamic behavior is found in the proposed neuronal system by varying the coupling strength, external stimulus current, memristive parameter and other system parameters. Multiple bursting firing groups in the tongue-shaped domains have been discovered for the first time. Finally, in order to validate the numerical simulation, an analog equivalent circuit of the heterogeneous neuronal system is devised, which demonstrates that the system is physically realizable.

In this paper, we investigate an age-structured model of Hand–Foot–Mouth Disease (HFMD) in a contaminated environment. This paper characterizes both age structure and contaminated environment in the HFMD model and discusses their effects on the time delay in the conversion from exposed to infected individuals. By transforming the original model to an abstract nondensely defined Cauchy problem and utilizing the integrated semi-group theory, we establish the well-posedness of solutions. Then the existence and stability of equilibria are obtained on the basis of the basic reproduction number ${\mathcal{ℛ}}_0$. It is proved that the disease-free equilibrium is globally asymptotically stable if ${\mathcal{ℛ}}_0<1$. Meanwhile, the endemic equilibrium is locally asymptotically stable with time delay $\tau =0$. Furthermore, Hopf bifurcation occurs at endemic equilibrium when $\tau >0$. Finally, numerical simulations are performed to validate our theoretical results.

A class of reaction–diffusion models for two interacting species with two memory-based cross-diffusions is considered in this paper. Stability and bifurcation analyses at the steady states are conducted to investigate the joint effects of memory-based diffusion rates and time delays. It is found that memory-based diffusion rates will induce Turing bifurcations, and the first Turing bifurcation curves have more complicated structures than the ones for the classic reaction–diffusion equations, in that they are composed of an asymptotic curve of Turing bifurcation curves with different modes. Furthermore, using time delays as varying parameters, we also find all the stability switching curves, and provide sufficient conditions for determining the shapes of these curves. With the aid of normal form computation, it is proved that Hopf bifurcation will occur on these stability switching curves. Based on these results, predator–prey and competition models with memory-based diffusions are studied, and various temporal or spatial patterns for the models can be observed, including the spatially inhomogeneous steady-state and inhomogeneous periodic solutions.

This paper presents a generalized delayed predator–prey system with habitat complexity. First, the dynamical behaviors of the considered system, including positivity and boundedness, stability property and the existence of Hopf bifurcation and super-critical bifurcation without time delay, are analyzed. Second, the effect of time delay on the dynamical behavior of the presented system is considered in four cases, and sufficient conditions for the existence of Hopf bifurcation are obtained. Moreover, the explicit formulas to determine the direction, stability and period of bifurcating periodic solutions are derived by applying the center manifold theory and normal form theory. Finally, numerical simulations are conducted to verify the corresponding theoretical results.

In this paper, we analyze the monotonicity and the number of critical periods of the period function associated with a family of piecewise planar continuous potential systems. In particular, we describe the bifurcation diagram of the period function, determining the regions of the parameter space where it is monotonically increasing or decreasing and where it has at most one simple critical period. In order to prove the main result, we rewrite such a function in terms of the two period functions of the uncoupled planar Hamiltonian systems matched by the separation line.

We consider the Sprott cubic conservative jerk differential equation $\stackrel{\mathrm{...}}{x}-a(1-x^2)x+x^2ẋ=0,$ with $a\in ℝ$. It is known that this differential equation exhibits chaotic motion for some values of the parameter a. Here, we study when this differential equation has no chaotic motion, i.e. when it has first integrals, and then we describe its dynamics.

In this paper, we apply the method of Lagrangian descriptors as an indicator to study the chaotic and regular behaviors of trajectories in the phase space of the classical double pendulum system. To successfully quantify the degree of chaos with this tool, we first derive Hamilton’s equations of motion for the problem in nondimensional form and demonstrate how they can be expressed compactly using matrix algebra. Once the dynamical equations are obtained, we conduct a parametric study based on the system’s total energy and other key parameters such as the lengths and masses of the pendulums, as well as gravity, to determine the extent of the chaotic and regular regions in the phase space. Our numerical results reveal that, for a given mass ratio, the maximum chaotic fraction of phase space trajectories is attained when the pendulums have equal lengths. Furthermore, we characterize how chaos grows and decays within the system in terms of the model parameters, and explore the hypothesis that the chaotic fraction follows an exponential law over different energy regimes.

Machine Learning (ML) is highly data dependent. But all data contain noise. Especially for chaotic systems, due to the famous “butterfly-effect”, small disturbances can grow exponentially to the same order of magnitude as the physical solution so that numerical simulations of chaos become badly polluted. A fundamental and open problem is how data noise in chaotic systems influences short and long-term predictions of ML based on such badly polluted data. Ultra-chaos, whose statistics are sensitive to small disturbances, is a new concept in chaos theory. In this paper, the influence of data noise on ML of chaos is investigated in detail by comparing predictions with three ML models based on a “clean” dataset given by Clean Numerical Simulation (CNS), and a “badly polluted” dataset given by the traditional Runge–Kutta method. It is found that neither data noise nor ML prediction deviation influence the long-term statistics of ML predictions for normal-chaos. However, data noise has a significant influence on the statistics of ML predictions for ultra-chaos: this might pose a great challenge for ML that is a part of AI.

Spatial patterns are an important characteristic of ecological epidemic reaction–diffusion models, as they help predict the spatial dynamics of epidemic spread. However, there is currently a significant shortage in research on control strategies for generating and regulating these patterns. In this paper, we establish a Susceptible–Infectious (SI) model with Proportional-Derivative (PD) control and cross-diffusion. We analyze the stability of the model without diffusion and provide the conditions for Turing instability driven by diffusion. Using the cross-diffusion coefficient as the bifurcation parameter, we derive the amplitude equation of the two-dimensional Turing pattern at the Turing bifurcation threshold based on the multiple scale method, which determines the selection and stability of pattern formation. Numerical simulations in two-dimensional space show that the bifurcation parameter varies with changes in control parameters, and the PD control not only effectively alters the structure of Turing patterns but also suppresses Turing instability. Furthermore, it is found that more complex spatial patterns in three-dimensional space emerge under PD control.

Due to their widespread applications in modeling natural issues, the study of piecewise linear differential systems has gained popularity in recent years. It is well known that the qualitative theory of piecewise linear differential systems heavily relies on limit cycles. By now, most studies have only considered planar systems by examining the presence and maximum number of limit cycles for piecewise differential systems. However, there have been few studies on this issue in ${ℝ}^3$. We recall the problem of the existence and the maximum number of limit cycles for planar discontinuous piecewise differential systems formed by linear differential centers separated by one or two parallel straight lines that have at most one limit cycle, respectively. Although, in ${ℝ}^3$, the maximal number of limit cycles for the same problem is 0 when the separation surface is a plane and at most four if the separation surface is two parallel planes. In this paper, we mainly focus on the problem of the existence and the maximum number of limit cycles in ${ℝ}^3$, when the separating surface is formed by two intersecting half-planes. First, we prove that when the entire space is divided into two regions, this family can have at most five limit cycles, where one limit cycle intersects the separation surface at two points and the remaining four limit cycles intersect the separation surface at four points. Second, when the entire space is divided into three regions, we prove that the maximum number of limit cycles intersecting the separation surface at three points and four points simultaneously is at most eight.

Chaos Optimization Algorithms (COAs) have emerged as a potent global optimization technique, exhibiting remarkable capabilities in tackling complex optimization challenges. In recent years, a plethora of COAs and their variants have been developed and employed to address a wide array of practical problems in engineering and science. Despite the proliferation of these algorithms, existing reviews often suffer from being outdated or lacking in robust classification criteria. This deficiency impedes the accurate assessment of the latest advancements and obstructs researchers’ ability to swiftly comprehend the current state of research in chaos optimization. To bridge this gap, this paper offers a review by classifying chaos optimization into five distinct categories: chaos map-based optimization algorithms, chaos metaheuristic optimization algorithms, chaos game optimization algorithms, hybrid optimization algorithms, and others. Each category is meticulously analyzed, with a thorough discussion of the inherent strengths and weaknesses of the respective algorithms. This analysis not only clarifies the unique features of each algorithm but also enhances understanding by contrasting their various applications. The review extends to the practical deployment of chaos algorithms across specific problems, illustrating their versatility and effectiveness. Conclusively, this paper delineates potential future research evolution of COAs, providing a clear and structured guide to forthcoming explorations in this dynamic field. This survey aims to empower researchers within the optimization community with a deeper and more comprehensive understanding of the landscape of COAs, paving the way for innovative research and applications.

In a recent paper [, ], we proved that the bifurcation structure of a quadratic noninvertible map persists when the parameter increases from zero and the map turns into an invertible multidimensional Henon map. In this paper, we consider a similar problem for a generalized map which combines the Henon-type maps, the Poincaré return map for Shilnikov bifurcation of saddle-focus homoclinic orbit, the Lurie discrete time system, etc. We have obtained the expected result about the persistence of periodic orbits and their bifurcations when passing from a One-Dimensional (1D) endomorphism to the generalized map as a small parameter becomes nonzero. We have revealed the precise mechanism of change of homoclinic orbits and splitting of unstable manifolds as a result of the transition of 1D endomorphism to multidimensional map. Thereby we have derived the reconstruction rules of nonwandering set of orbits and bifurcations of the generalized map from those of 1D endomorphism.

High-resolution remote sensing images with complex contents have garnered significant attention for practical applications, which also pose substantial challenges related to space occupation, security, and information privacy. Compression is frequently coupled with encryption to balance the above-mentioned issues. Most existing methods for joint compression and encryption are primarily designed for common low-resolution images. Toward this end, we propose a novel compression-and-encryption algorithm for remote sensing images that maintains the JPEG format compatibility while virtually eliminating the block effects in the encrypted images. A novel block segmentation strategy that is friendly to simple images is first introduced, in which the original image is divided into nonoverlapping sub-blocks that are greater than the blocks in JPEG compression. The key that is utilized to encrypt the values of DC coefficients is generated through a fusion of the 2D Logistic-Sine-Coupling Map and SHA-512. After that, a parallel diffusion method is proposed for all nonzero AC coefficients, which significantly enhances the encryption efficiency. The global blocks permutation is conducted to further improve the security. Extensive experiments are carried out to demonstrate that our encryption scheme is highly efficient, provides enhanced security, and maintains format compliance for JPEG. Meanwhile, the presented method can resist various attacks to a certain extent.

We study the existence of heteroclinic solutions for discontinuous Kurland–Levi differential equations with slowly varying coefficients. Our main results follow from our recent general theorem concerning heteroclinic solutions in singularly perturbed discontinuous differential equations. We apply the new analytical results to concrete examples and give numerical simulations.