International Journal of Bifurcation and Chaos

In this article, we have applied the Weyl differential operator in an epidemic model with the standard incidence rate to study pattern formation among species with superdiffusive movement in space. A thorough linear stability analysis predicts the various Turing pattern regions. Further, the analysis shows the relationship between the wavenumber of the Turing pattern and the superdiffusive exponent, which are supported by numerical results. A Fourier spectral method in space and a fourth-order exponential time differentiating Runge–Kutta method are used for numerical simulation. Simulations are done for the Turing pattern regions for 2D and 3D problems, showing the only quantitative change in patterns for varying superdiffusive exponents.

This paper develops a spatial platform for economic agglomerations that can represent a hierarchical structure of cities, towns, and so on. A global system models geographical distribution of a system of cities, while population size and local geography of each city are modeled by an individual square lattice network. The mechanism of economic agglomeration is described by economic geography models with the replicator dynamics. As a major theoretical contribution of this paper, we elucidate the bifurcation mechanism of the mono-centric distribution in a square domain. Such bifurcation expresses how satellite places appear around a large city. This bifurcation behavior, called sustaining bifurcation, is different from symmetry-breaking bifurcation studied in nonlinear mathematics, and is not given much attention in economic geography, despite its importance. An air-network of cities with the seven largest hub airports in USA is employed as a realistic example, and is modeled as a global–local system comprising a series of local square lattices with different sizes connected by an equidistant economy. This system for an economic geography model displays successive sustaining bifurcations occurring many times leading to gradual emergence of satellite places around a large city.

In this study, a uniformly distributed mathematical model for the self-heating process within compost piles is formulated and investigated. It consists of mass balance equations for oxygen and energy as well as the heat-generation processes due to both biological and chemical activities. This model is an extension of the study of Luangwilai et al. [2013] which consists of only biological activity. The singularity and degenerate Hopf bifurcation theories are used to determine the loci of different singularities: the isola, cusp, double-limit points and boundary limit set as well as the double-Hopf, generalized Hopf (Bautin) and Bogdanov–Takens bifurcations. In this investigation it is found that these loci separate the secondary parameter plane into twenty-two regions of different steady-state solution behaviors, whereas the model in the earlier study reported only eight regions. With more topological detail in the parameter space, a clearer understanding of the thermal behavior of a compost pile can be obtained, thus assisting compost operators in controlling the temperature within the pile more effectively: for example, achieving desirable elevated temperature range for ideal composting conditions via periodic solutions and S-shaped solution branch, or alternatively, understanding conditions, if not monitored carefully, that can also increase the likelihood of spontaneous ignition within the pile.

In this paper, Myc/E2F/miR-17-92 network under Gaussian white noise is studied. Taking the time delay as the parameter, the Hopf bifurcation of the system is obtained, which causes the protein concentration to oscillate periodically. Under the influence of time delay and noise, the stochastic D-bifurcation of the system is obtained. It is worth noting that the occurrence of stochastic P-bifurcation is successfully captured. Thus a pattern of coexistence of high and low protein concentrations is founded in the network. The specific research methods of this paper are as follows: firstly, the system is reduced to a finite dimensional system by using stochastic center manifold and normal form theory. Then, using the stochastic averaging method, the Fokker–Planck–Kolmogorov equation of the system is constructed in which the statistical response in the stationary state is the probability density. Finally, the stochastic bifurcation analysis and numerical simulation are carried out. The agreements between the analytical method and those obtained numerically validate the effectiveness of the analytical investigations.

Dispersal of organisms between patches is a common phenomenon in ecology and plays an important role in predator–prey population dynamics. We propose a nonsmooth Filippov predator–prey model in a two-patch environment characterized by a generalist predator-driven intermittent refuge protection of an apprehensive prey along with a balanced dispersal of the prey between refuge and nonrefuge areas. By employing qualitative techniques of nonsmooth dynamical systems, we see that the switching surface is a repeller whenever the interior equilibria are virtual, causing long-term population fluctuations. We find that the level of prey vigilance and the rate of prey dispersal play pivotal roles in the total harvesting yield. We observe that a sustainable high harvesting yield is possible when the prey is less vigilant and obtain the harvesting efforts for maximum sustainable total yield (MSTY). We further modify the model by considering a continuous threshold predator-driven prey dispersal and show that the model exhibits a Hopf bifurcation when the level of prey vigilance exceeds some critical threshold value. By comparing the dynamics of the two models we see that for a sustainable high harvesting yield of the system with continuous threshold dispersal, the prey needs to be highly vigilant compared to that of the system with intermittent dispersal of the prey. Further, we find numerically that the estimated MSTY from both models remains the same.

In this paper, we study the number of limit cycles for discontinuous piecewise differential systems formed by two differential systems separated by a straight line, when these differential systems are quadratic isochronous centers in a half-plane and the cubic isochronous centers in the other one. We deal with eight classes of discontinuous piecewise differential systems formed by these types of isochronous centers, and we provide an upper bound for the maximum number of limit cycles for these classes that can exhibit. Also, we show that the corresponding maximum number of limit cycles in four classes is reached.

In this paper, we prove that for a nonuniformly hyperbolic system $(f,\tilde{\mathrm{\Lambda }})$, and for every convex and compact subset $K\subset {{ℳ}}_{\mathrm{\text{inv}}}(\tilde{\mathrm{\Lambda }})$ with the same hyperbolic rate, the packing entropy of saturated set $G_K$ equals the supremum of the metric entropy of all measures in K. Here ${{ℳ}}_{\mathrm{\text{inv}}}(\tilde{\mathrm{\Lambda }})$ denotes the set of all $f$-invariant Borel probability measures supported on $\tilde{\mathrm{\Lambda }}$. As an application, we describe the size of the set of mean Li–Yorke pairs.

Cryptanalysis of key expansion algorithms in AES and SM4 revealed that (1) there exists weaknesses in their S-Boxes, and (2) the round key expansion algorithm is reversible, i.e. the initial key can be recovered from any round key, which may be explored by the attacker. To solve these problems, first we constructed a nondegenerate 2D exponential hyper chaotic map (2D-EQCM), derived a recursion formula to calculate the number of S-Boxes, and designed a strong S-Box construction algorithm without such weakness. Then based on 2D-EQCM and S-Box, we designed an irreversible parallel key expansion algorithm, which could transform the initial key to any number of relatively independent round keys. Security and statistical analysis demonstrated the flexibility and effectiveness of the proposed irreversible parallel key expansion algorithm.

The effect of multiplicative stochastic perturbations on planar Hamiltonian systems is investigated. It is assumed that perturbations fade with time and preserve a stable equilibrium of the limiting system. The paper investigates bifurcations associated with changes in the stability of the equilibrium and with the appearance of new stochastically stable states in the perturbed system. It is shown that depending on the structure and the parameters of the decaying perturbations, the equilibrium can remain stable or become unstable. In some intermediate cases, a practical stability of the equilibrium with estimates for the length of the stability interval is justified. The performed stability analysis is based on a combination of the averaging method and the construction of stochastic Lyapunov functions.

Cellular neural network (CNN) is one of the best artificial neural network models due to its hardware implementability and many applications for image processing. The memristor-based cellular neural network (MCNN) combines the distinct benefits of memristors and neural networks, and it excels in mimicking chaotic systems under the effect of electromagnetic induction. In this work, a novel CNN chaotic system based on a flux-controlled memristor (MCNN-CS) with electromagnetic induction effects is constructed. Dynamical behaviors are examined by controlling the magnetic flux leakage strength and the intensity of the electromagnetic induction related to the memristor. Based on stability analyses and numerical simulation, forward and reverse period-doubling bifurcations, various coexisting attractors, multiperiods, chaos, several periodic windows, and chaotic crisis are observed. Besides, a number of complicated phenomena, including transient chaos, intermittent chaos, sustained chaos and bistability are also observed, demonstrating that this chaotic system has a wide range of dynamic properties. Finally, we generate pseudo-random number sequences based on this new system and apply them to image encryption. The obtained results show that this chaotic system offers superior randomness and is hence appropriate for image encryption applications.

The Multiscale Entropy (MSE) is an effective measure to quantify the dynamical complexity of complex systems, which has many successful applications in physiological and physical fields. It uses different scales to mean-coarse-grain the original series, and then calculates the sample entropy for each coarse-grained series. Inspired by the MSE, we in this paper propose the Multi-Moment Multiscale Local Sample Entropy (MMMLSE), which considers both mean-coarse-grained and standard-deviation-coarse-grained characteristics of the original series for each scale, to quantify the dynamical complexity of complex systems. We use simulated data ($1/f$ noise, white noise and logistic map) to test the performance of our proposed method, with results showing that the MMMLSE can accurately and effectively characterize these complex systems. The ability to preserve nonlinear dynamics of the proposed method is also proved by surrogate data and nonlinearity test experiment. Furthermore, we apply the MMMLSE to analyze physiological signals, and the MMMLSE reveals that the ill individuals have lower dynamical complexity at larger scales than the healthy ones, and the elder individuals have lower dynamical complexity at larger scales than the younger ones, which are consistent with the reality.

To disclose the pathophysiological mechanism of schizophrenia from the general connectivity among distinct brain regions, the computational model proposed by Szalisznyó et al. is generalized by incorporating the effects of the hippocampus. Based on the center manifold theorem and normal form theory, the emergence of abnormal alpha oscillations, which are directly associated with symptoms of schizophrenia, is analyzed in this network with Hopf bifurcation as a precursor. It is found that the clinically significant hypoconnectivities between the anterior cingulate and the dorsolateral prefrontal cortex or from the dorsolateral prefrontal cortex to the hippocampus are helpful in inhibiting the abnormal alpha oscillations, while disconnection of the coupling from the anterior cingulate cortex to the striatum can induce schizophrenia. Moreover, polynomial chaos expansion and variance-based sensitivity analysis are applied to determine the contributions of these connection strengths to the variance of the neuronal population activity. Our model results are consistent with experimental observations and thus might be meaningful for clinical interference.

In this paper, we consider the properties of a Leslie–Gower predator–prey system under two basic assumptions that the prey growth is affected by an addictive Allee effect, and the intrinsic growth rate of prey is much bigger than that of predator. Using the geometric singular perturbation theory and blow up technique, we show that this system can have rich dynamics, such as subcritical Hopf bifurcation, canard explosion and relaxation oscillations. Numerical simulations are also carried out to validate our analytical results.

In this paper, the existence and bifurcations of higher-order complex periodic motions in a nonlinear, electromagnetically tuned mass damper system are studied through period-3, period-9, and period-12 motions. For such a nonlinear system, there exist three branches of period-3 motions, one branch of period-9 motions, and one branch of period-12 motion. The corresponding stability and bifurcations of periodic motions are determined. The frequency-amplitude characteristics for bifurcation routes of such higher-order periodic motions are presented. Numerical illustrations are presented for complex periodic motions. Higher-order unstable and stable periodic motions exist in nonlinear systems, which are difficult to control for energy harvesting and vibration reduction. The unstable higher-order periodic motions make the system responses more complicated. The higher-order stable and unstable periodic motions have been studied comprehensively. This study on periodic motions can help people better understand the dynamical behaviors of an electromagnetically tuned mass damper system under different excitation frequencies. The study of higher-order periodic motion stability and bifurcations will also benefit the new development and design of vibration reduction and energy harvesting systems.

A shockwave signal has the characteristics of short duration and wide dynamic range, so the wireless test system needs to collect it at a continuous high sampling rate, resulting in a short node life cycle and a large amount of redundant data during the test process. Moreover, test data with confidentiality, if it is still recorded and transmitted in plaintext, will challenge information security. Therefore, a compressed and encrypted acquisition framework based on hyperchaotic compressed sensing and semi-tensor is proposed in this paper. Firstly, a new four-dimensional hyperchaotic system is used to generate a compressed sensing measurement matrix to improve the encryption and anti-noise performance. Secondly, for saving memory space and improving computing efficiency, the semi-tensor product is introduced to reduce the dimension of the measurement matrix exponentially. The experimental results show that the proposed framework can realize the compressed and encrypted acquisition of transient signals with small memory space and low computational complexity. And the acquisition results can be directly transmitted to meet the requirements of real-time wireless transmission bandwidth and information security.

In this paper, we propose a delayed mathematical model describing oncolytic virotherapy treatment of a tumour that proliferates according to the logistic growth function, incorporating viral lytic cycle. The tumour population cells are divided into uninfected and infected cell sub-populations and the virus spreading is supposed to be in a direct mode (i.e. from cell to cell). Depending on the time delay, we analyze the positivity and boundedness of solutions and the stability of tumour, infected and uninfected free equilibria (TFE, IFE, UFE) and uninfected–infected equilibrium (UIE) is established. We prove that, delay can lead to “Jeff’s phenomenon” observed in a laboratory which causes oscillations in tumour size whose phase and period change over time. With nonlinear dependence of UIE equilibrium on time delay, we develop a more general algorithm determining the stability/instability of the oscillating periodic solutions bifurcating from the UIE equilibrium. Finally, we present numerical simulations illustrating our theoretical results.

Constant prey refuge with immigration and harvesting in two species would result in significant diversity in the dynamics of a prey–predator population. The phrase “refuge” increases the likelihood of prey population survival in the face of a predator population. Based on these findings, we created and examined a two-species prey–predator system with immigration and harvesting factors, including refuge to only prey population. All ecologically possible equilibrium points are studied for the proposed system. Routh–Hurwitz stability criterion is used for local stability analysis. Global stability of the interior equilibrium point is examined with a suitable Lyapunov function. Local bifurcation of the proposed system, such as saddle-node bifurcation, is analyzed. The conditions for the emergence of this bifurcation at the critical threshold near the nonhyperbolic equilibrium point are established by utilizing Sotomayor’s theorem. The transversality condition is validated for the occurrence of Hopf-bifurcation. The first Lyapunov number is exploited for determining the nature of Hopf bifurcating periodic solution. Finally, numerical simulations are illustrated to validate our theoretical predictions.

Understanding the effects of cooperative hunting among the predators is an important topic in ecology. We explore a prey–predator model with cooperative hunting among predators modeled by a Holling type-II functional response having saturated encounter rate. Bifurcation analysis is carried out on the temporal model to detect local bifurcations such as transcritical, saddle-node, Hopf bifurcation and global homoclinic bifurcation. A comprehensive numerical bifurcation analysis is performed on the Turing solutions of the corresponding diffusive model. Stable and unstable branches of mode specific Turing solutions and their dependence on parameters are identified. For suitable values of parameters, spatio-temporal chaos via period doubling cascade is observed when a given parameter traverses the Turing–Hopf region.