International Journal of Bifurcation and Chaos

We examine the family of ordinal First Return Maps (FRMs) of a dynamical system time series to model the system’s dynamics. We consider two distinct modeling approaches using the same modeling method and compare their outcomes to ascertain the prerequisites for each to function and which approach is more effective in different circumstances. The first approach involves short-term modeling of the entire flow, followed by identifying the ordinal sections on the predicted flow. The second approach relies on locating the ordinal sections within the reference time series and using the data from each section independently to build a model that can produce a long-term prediction for each section’s FRM. We demonstrate the effectiveness of these approaches on two well-known dynamical systems, the Lorenz and Rössler equations, emulating data deficiency by selecting a low sampling rate. We consider the effect of downsampling on the shape of the FRMs to choose suitable ordinal sequences to use in creating Poincaré sections from the time series. Following that, we propose a dissimilarity measure between the reference and predicted FRMs to assess the model quality. We then compare the reference and predicted FRMs using the introduced dissimilarity criterion to determine which modeling approach performs better in each situation. Ultimately, we classify the FRMs situation into continuous and piecewise continuous scenarios. Based on our observations from the Lorenz and Rössler dynamical systems, we found that when the system’s FRMs are piecewise continuous, it is more effective to model the entire flow. However, when the FRMs are continuous, our choice of modeling approach depends on the criterion that the predicted FRM is more similar to the reference.

The study of chaotic systems with special properties addresses one of the ongoing research topics. In this paper, we introduce a new class of third-order chaotic systems possessing a cyclic symmetry obeying unidirectional or mutual coupling. These new models are built by considering nonlinearities with three, four, or five segments and are distinguished from each other by the topological structure of the associated chaotic attractor. The projection of the novel chaotic attractors on certain planes reveals their multiscroll or multiwing character. For illustration, one of these models is studied in detail using analytical and numerical methods. The mechanism giving rise to the establishment of the chaotic regime starting from the state of equilibrium under the variation of a parameter is described using bifurcation diagrams, phase portraits, basins of attraction as well as the maximum Lyapunov exponent. Several zones of parameters are identified where the model experiences two or more attractors (i.e. multistability). In order to demonstrate the practical feasibility of the proposed circulant models, a series of experimental measurements are carried out using a physical implementation with an Arduino microcontroller.

Let $\tau :I=[0,1]\to [0,1]$ be a piecewise convex map with countably infinite number of branches. In [, ], the existence of Absolutely Continuous Invariant Measure (ACIM) $\mu$ for $\tau$ and the exactness of the system $(\tau ,\mu )$ have been proven. In this paper, we develop an Ulam method for approximation of $f^{\ast }$, the density of ACIM $\mu$. We construct a sequence ${\{{\tau }_n\}}_{n=1}^{\infty }$ of maps ${\tau }_n:I\to I$ s.t. ${\tau }_n$ has a finite number of branches and the sequence ${\tau }_n$ converges to $\tau$ almost uniformly. Using supremum norms and Lasota–Yorke-type inequalities, we prove the existence of ACIMs ${\mu }_n$ for ${\tau }_n$ with the densities $f_n$. For a fixed n, we apply Ulam’s method with k subintervals to ${\tau }_n$ and compute approximations $f_{n,k}$ of $f_n$. We prove that $f_{n,k}\to f^{\ast }$ as $n\to \infty ,k\to \infty ,$ both a.e. and in $L^1$. We provide examples of piecewise convex maps $\tau$ with countably infinite number of branches and their approximations by ${\tau }_n$’s with finite number of branches. For the increasing values of parameter $k,$ we calculate the errors $\parallel f^{\ast }-f_{n,k}{\parallel }_1$.

The Hopf bifurcation problem of a nose landing gear system is investigated. Taking the nearly smooth landing gear system with high-order nonlinear terms as the research object, the nose landing gear system is simplified to a plane dynamic system considering nonlinear factors by using the center manifold simplification method. The first Lyapunov coefficient of the landing gear system is derived by using the classical bifurcation theory, and the Hopf bifurcation type of the system is determined according to its sign. The bifurcation diagram is obtained by solving the differential equations of motion numerically. It is revealed that there is a stable limit cycle in the neighborhood of the supercritical Hopf bifurcation point. The influence of velocity and other factors on the shimmy oscillations of the landing gear system is analyzed. The dynamical behavior in the neighborhood of the Hopf bifurcation point is explored theoretically and numerically. The research results can provide a theoretical basis for the optimal design of aircraft landing gear systems.

A novel versatile image encryption scheme about hologram technology and chess rules is proposed in this paper. This scheme can encrypt images by transferring them from the spatial domain to the frequency domain. First, the theory of computer-generated holograms is applied to record phase and amplitude features by imitating the optical hologram. The original image is transformed to a hologram and the phase values of the points in the image are recorded. Second, the parameters associated with the plaintext image are obtained by a Hash function and used as the initial values of the chaotic system, and then the pawns on the board are regarded as pixel points of the image, where the movement direction of the pawns is the movement direction of the pixel points of the image. The hologram is confused by combining the pawns’ movement rules with the chaotic sequences. Finally, the encrypted image is gained through a combination of the chaotic sequence and the selective diffusion method. The test results show that the scheme is able to resist various attacks and has good security performance in encryption.

With the development of Internet, the security of information transmission has drawn widespread attention. In order to enhance the security of audio data communication, we propose an audio encryption algorithm using a two-dimensional chaotic map and dynamic S-box. To address the drawbacks of single control parameters and small chaotic space in one-dimensional chaotic map, we integrate Sine map and Fuchunao map to generate a new two-dimensional chaotic map 2D-SIFN. This map has two control parameters and a larger chaotic space. Then we generate chaotic sequences using the 2D-SIFN, construct dynamic S-boxes using these sequences for permutation operations, and finally apply chaotic sequences for diffusion operations to confuse audio data. The simulation experiment results show that the proposed algorithm performs excellently in metrics such as Number of Sample Change Rate (NSCR) and Normalized Cross-correlation (NCC) analysis, proving the feasibility of applying S-boxes to audio data encryption algorithms and proposing a feasible new approach in the field of secure communication.

In order to explore the influences of the strong Allee effect on population evolution in two-patch environments, this paper constructs a population model with discrete diffusion and analyzes its dynamics. Using the center manifold theorem, we discuss the codimension-one bifurcation, such as flip bifurcation, of the model. The qualitative structures and stability of the model at the degenerate fixed point are obtained from the approximation of flow. Based on the theoretical analysis results, we investigate the influences of the strong Allee effect and diffusion on population evolution. The qualitative structures of degenerate fixed point and numerical simulation show that the strong Allee effect will make the population tend to a stable size when the two initial population patch sizes are equal and exceed a threshold. But the stable size of population does not exceed the carrying capacity, depending on the constant Allee effect and carrying capacity, which differ from the conclusion of logistic model that population size tends to the carrying capacity. However, if the two initial population patches are equal in size and below this threshold, or if the sizes of the initial population patches are unequal, the population size will decrease. The fixed point becomes degenerate with respect to the strong Allee effect causing the population tend to a stable size below carrying capacity, which exacerbates the complexity of population evolution.

In this paper, two newly proposed generalized Hamiltonian Conservative Chaotic Systems (CCSs) based on integrated 4D Euler equations are first discussed. Second, another generalized Hamiltonian CCS is also proposed. Furthermore, a class of generalized Hamiltonian CCSs based on integrated 4D Euler equations is given and studied. Subsequently, numerical analysis for the class of generalized Hamiltonian CCSs is further investigated to show their advantages over most of the existing CCSs, where the corresponding Lyapunov exponents’ diagrams, bifurcation diagrams and phase portraits are all given to show their multistability and complex dynamics. Finally, the class of generalized Hamiltonian CCSs is implemented by using FPGA technology, and the results observed in FPGA implementation are consistent with those observed in the numerical analysis. All these results not only show multistability from a physical point of view, but also provide new physical models for chaos applications.

Earlier, the research was mainly focused on the classic diffusive two-species Lotka–Volterra competitive system (without advection). Averill et al. [] regarded a diffusive two-species Lotka–Volterra competitive system with advection term (with one species having a combination of random dispersal and biased movement upward along the resource gradient, while the other species adopts purely random dispersal). In this paper, we consider a more general diffusive two-species Lotka–Volterra competitive system with advection term (involving both species with a combination of random dispersal and biased movement upward along their respective resource gradients). By utilizing the theory of monotone dynamical systems, spectral analysis, methods of super- and sub-solutions and some nontrivial analytic skills, we finally obtain a rather deep understanding on the global dynamics. Moreover, our results reveal that the dynamical behavior of the system is very complex: exclusion, coexistence and bistability all may happen depending closely on the interspecific competition intensities. More interestingly, by bifurcation theory, we find that the system always admits multiple coexistence steady states when the interspecific competition intensities are near the critical point.

In order to improve the efficiency in computing the global properties of nonlinear systems, some effective methods have been proposed for the global analysis of these systems. However, there are only few investigations focusing on the numerical algorithms of the system response trajectories. This paper presents a higher-efficiency geometric numerical integration method with simple cell mapping for solving the basins of attraction of nonlinear systems. This numerical algorithm is based on the rule of Lie derivative, using it to calculate the trajectories of the nonlinear systems. Then, the global structure is studied by the numerical algorithm associated with a simple cell mapping method. Compared with the traditional Runge–Kutta methods of orders 4 and 5, it is demonstrated that the proposed Lie derivative iterative algorithm has significant advantages in efficiency.

We have developed a predator–prey model driven by higher-order interactions to investigate their potential in stabilizing ecological community dynamics and promoting species coexistence. Initially, the stability of the equilibrium is analyzed under diffusion-free conditions, and its local bifurcation behavior is systematically examined. In alignment with real ecological scenarios, higher-order interactions are modeled as random simplicial complexes. Moreover, a network dynamics approach is employed to study the pattern dynamics induced by higher-order interactions. Perturbation analysis is conducted to identify parametric conditions that lead to Turing instability. The results indicate that higher-order interactions play a pivotal role in this process, whereas first-order interactions alone are insufficient to induce Turing instability. Specifically, higher-order interactions contribute to a transition from regions of high abundance to regions of low abundance. In addition, the mean-field approximation offers critical insights into the mechanism by which higher-order interactions induce Turing instabilities, primarily by increasing the nodes’ degrees within the network. Meanwhile, the findings underscore that higher-order interactions play a pivotal role in enhancing the stability of ecological community dynamics through the facilitation of pattern formation. This highlights the crucial importance of higher-order interactions in sustaining ecosystem diversity and stability.

In this paper, we investigate the existence of limit cycles in piecewise linear systems separated by two parallel straight lines, where in one of these straight lines, the piecewise differential system is continuous, and in the other, it is discontinuous. Based on the methods of Poincaré map and first integral, we prove that these systems can have at most two limit cycles. Additionally, taking into account the type of singular points in the zones, we present examples with one or two limit cycles for all possible combinations.

Traditional image encryption methods typically make a trade-off between privacy protection and visual usability. To address this issue, a novel thumbnail-preserving encryption scheme is proposed based on a hybrid of decimal and binary pixel values in three pixel groups, utilizing the excellent dynamic characteristics of a generalized coupled logistic map. Theoretically, it is proved that the core algorithm of this scheme is sum-preserving. Experimentally, it is demonstrated that this scheme not only excels in efficiency and resistance to attacks, effectively safeguarding image privacy, but also retains high-quality thumbnails after encryption, achieving both objectives of image privacy protection and visual usability.

The problem of analyzing the bifurcation mechanisms of complex stochastic oscillations in population dynamics is considered. We study this problem on the basis of the modified Leslie–Gower prey–predator model with randomly forced Holling-II functional response. The paper focuses on the effects of noise in the Canard explosion zone. The phenomenon of noise-induced splitting of Canard cycles is discovered and studied in terms of stochastic P-bifurcations. In parametric analysis, we use the stochastic sensitivity technique with the apparatus of confidence domains to find the most noise-sensitive Canard. For the phenomenon of stochastic splitting, an underlying deterministic mechanism using critical curves near the cycle orbit and sub-/super-critical zones is revealed.

The goal of this paper is to introduce a hybrid technique, combining two existing methods conveniently, to analyze the stability and bifurcations of equilibrium points and periodic orbits in delay-differential equations of retarded type. One method is called the frequency-domain approach, an analytical tool based on control theory allowing to detect and represent periodic solutions emerging from the Hopf bifurcations, via Fourier series and harmonic balances. The second one, the so-called semi-discretization method, provides a numerical scheme to approximate the monodromy operator of a periodic solution, thus permitting to establish the stability and bifurcations of limit cycles as well as analyzing the equilibrium points. It is shown that a proper combination of these methods provides a straightforward strategy to study both local and global bifurcations in time-delay systems. The usefulness of the proposed approach is shown by three examples, where the results are compared with those given by the software DDE-BIFTOOL.