International Journal of Bifurcation and Chaos

The nonlinear dynamics of a recently derived generalized Lorenz model [] of magnetoconvection is studied. A bifurcation diagram is constructed as a function of the Rayleigh number where attractors and nonattracting chaotic sets coexist inside a periodic window. The nonattracting chaotic sets, also called chaotic saddles, are responsible for fractal basin boundaries with a fractal dimension near the dimension of the phase space, which causes the presence of very long chaotic transients. It is shown that the chaotic saddles can be used to infer properties of chaotic attractors outside the periodic window, such as their maximum Lyapunov exponent.

The purpose of the present paper is to investigate the pattern formation and secondary instabilities, including Eckhaus instability and zigzag instability, of an activator–inhibitor system, known as the Gierer–Meinhardt model. Conditions on the Hopf bifurcation and the Turing instability are obtained through linear stability analysis at the unique positive equilibrium. Then, the method of weakly nonlinear analysis is used to derive the amplitude equations. Especially, by adding a small disturbance to the Turing instability critical wave number, the spatiotemporal Newell–Whitehead–Segel equation of the stripe pattern is established. It is found that Eckhaus instability and zigzag instability may occur under certain conditions. Finally, Turing and non-Turing patterns are obtained via numerical simulations, including spotted patterns, mixed patterns, Eckhaus patterns, spatiotemporal chaos, nonconstant steady state solutions, spatially homogeneous periodic solutions and spatially inhomogeneous solutions in two-dimensional or one-dimensional space. Theoretical analysis and numerical results are in good agreement for this diffusive Gierer–Meinhardt model.

We provide normal forms and the global phase portraits on the Poincaré disk of all planar Kukles systems of degree $3$ with ${\mathbb{Z}}_2$-equivariant symmetry. Moreover, we also provide the bifurcation diagrams for these global phase portraits.

We solve the inverse problem in the Kuramoto model with three nonidentical oscillators and a phase lag. The model represents a three-cell-in-depth radial profile of the solar meridional circulation in each solar hemisphere and describes the synchronization between the two components (toroidal and poloidal) of the solar magnetic field. We reconstruct natural frequencies of the top and the bottom oscillators from the evolution of their phases when the oscillators are phase-locked. The phase-locking allows to solve the inverse problem when the phase of the middle oscillator is not available. We present the exact solution and investigate its stability. The model reveals a crucial role of the phase lag in the inverse problem solution. We apply the model to the reconstruction of the deep meridional circulation of the Sun and discuss peculiarities of its evolution in terms of solar dynamo.

Using both analytical and numerical methods on the global dynamics, including the existence and uniqueness of solutions, subharmonic bifurcations and dynamic responses, of an elliptically excited pendulum model are investigated in this paper. The heteroclinic orbits, as well as periodic orbits with $O$ and $R$ types of unperturbed systems are obtained analytically. Chaotic vibrations arising from heteroclinic intersections are studied by means of the Melnikov method. The critical curves separating the chaotic and nonchaotic regions are plotted for different system parameters. The chaotic feature on the system parameter $e$, named the ratio between the horizontal and the vertical diameter of the upright ellipse traced out by the pivot during each period, is discussed in detail. The conditions for subharmonic bifurcations with the $O$ type or the $R$ type are also presented with the subharmonic Melnikov method. It is proved rigorously that the system can undergo chaotic motions through finite subharmonic bifurcations with the $O$ type. In addition, chaotic motions can occur through infinite subharmonic bifurcations with the $R$ type. An interesting dynamical phenomenon, i.e. “controllable frequency”, which decreases monotonically with the system parameter $e$, is presented. A number of related numerical simulations are given to confirm the analytical results.

In this paper, we study the dynamic behaviors of a predator–prey system with a general form of nonmonotonic functional response. Through analysis, it is found that the system exists in extinction equilibrium, boundary equilibrium and two positive equilibria, one or no positive equilibrium. Furthermore, the conditions are given such that the boundary equilibrium is a saddle, node or a saddle-node point of codimension 1, 2 or 3. Then, some conditions are obtained so that the unique positive equilibrium of the system is a cusp point of codimension 2, 3 and higher or a saddle-node one of codimension 1 or 3, or a nilpotent saddle-node of codimension 4. When there are two positive equilibria in the system, the equilibrium with a large number of preys is a saddle point. For the other one, the system may undergo Hopf bifurcation. To verify our conclusion, we consider the functional response function in the literature [; ]. Finally, we give a brief discussion and numerical simulation.

The main purpose of this paper is to explore the bursting oscillations as well as the mechanism of a parametric and external excitation Filippov type system (PEEFS), in which different types of bursting oscillations such as fold/nonsmooth fold (NSF)/fold/NSF, fold/NSF/fold and fold/fold bursting oscillations can be observed. By employing the overlap of the transformed phase portrait and the equilibrium branches of the generalized autonomous system, the mechanisms of the bursting oscillations are investigated. Our results show that the fold bifurcation and the boundary equilibrium bifurcation (BEB) can cause the transitions between the quiescent states and repetitive spiking states. The oscillating frequencies of the spiking states can be approximated theoretically by their occurring mechanisms, which agree well with the numerical simulations. Furthermore, some nonsmooth evolutions are investigated by employing differential inclusions theory, which reveals that the positional relationship between the points of the trajectory interacting with the nonsmooth boundary and the related sliding boundary of the nonsmooth system may affect the nonsmooth evolutions.

In this paper, we consider a diffusive Leslie–Gower model with weak Allee effect in prey. We first study the global existence and non-negativity of the solutions by using the upper-lower solution method. We analyze the local stability of the positive constant steady state and prove the global attractivity by means of the LaSalle’s invariance principle. Additionally, we derive the parameter region that makes the positive constant steady state stable, and conclude that the boundary of this region contains a Hopf bifurcation curve and countable Turing curves. Thus, we get the existence of Turing–Hopf bifurcation and Turing–Turing bifurcation. Moreover, we calculate the normal form of the Turing–Hopf singularity on the center manifold. Our theoretical analysis shows that the system may produce a pair of spatially inhomogeneous steady states, spatially homogeneous periodic solutions, transient spatially inhomogeneous periodic solutions or even other solutions near the Turing–Hopf singularity. Finally, we carry out some numerical simulations for illustrating the analytical results.

We study the zero-Hopf bifurcation of the Rössler differential system

This paper deals with a stage-structured predator–prey system which incorporates cannibalism in the predator population and harvesting in both population. The predator population is categorized into two divisions; adult predator and juvenile predator. The adult predator and prey species are harvested via hypothesis of catch-per-unit-effort, whereas juveniles are safe from being harvested. Mathematically, the dynamic behavior of the system such as existing conditions of equilibria with their stability is studied. The global asymptotic stability of prey-free equilibrium point and nonzero equilibrium point, if they exist, is proved by considering respective Lyapunov functions. The system undergoes transcritical and Hopf–Andronov bifurcations. The impacts of predator harvesting rate and prey harvesting rate on the system are analyzed by taking them as bifurcation parameters. The route to chaos is discussed by showing maximum Lyapunov exponent to be positive with sensitivity dependence on the initial conditions. The chaotic behavior of the system is confirmed by positive maximum Lyapunov exponent and non-integer Kaplan–Yorke dimension. Numerical simulations are executed to probe our theoretic findings. Also, the optimal harvesting policy is studied by applying Pontryagin’s maximum principle. Harvesting effort being an emphatic control instrument is considered to protect prey–predator population, and preserve them also through an optimal level.

Memristor is a natural synapse because of its nanoscale and memory property, which influences the performance of memristive artificial neural networks. A three-variable memristor model is simplified with 15 kinds of properties, including the learning experience, the forgetting curve, the spiking time-dependent plasticity (STDP), the spiking rate dependent plasticity (SRDP), and the integration property. Through the analysis of the model, one more unobserved property called pseudo-polarity reversibility property is predicted by assuming the long-term memory is independent of memductance.

Based on the classic quadratic map (CQM) with abundant bifurcations and periodic windows, a new 3D improved coupling quadratic map (3D-ICQM) is constructed, and its phase diagram, Lyapunov exponent (LE) and randomness testing by TestU01 demonstrated that it has better ergodicity, more complex nonlinear behavior, larger chaotic range and better randomness. To investigate its application in cryptography, a 3D-ICQM based color image encryption algorithm with round key expansion is designed. To further enhance the randomness of key stream, the hash value of intermediate cipher of the previous round is converted into new initial values and fed back into 3D-ICQM, to generate the key stream for the current round. The security and statistical analyses demonstrated that the encryption scheme has better security performance than several existing algorithms.

This paper is mainly devoted to the investigation of discrete-time fractional systems in three aspects. Firstly, the fractional Bogdanov map with memory effect in Riemann–Liouville sense is obtained. Then, via constructing suitable controllers, the fractional Bogdanov map is shown to undergo a transition from regular state to chaotic one. Meanwhile, the positive largest Lyapunov exponent is calculated by the Jacobian matrix algorithm to distinguish the chaotic areas. Finally, the Grassberger–Procaccia algorithm is employed to evaluate the correlation dimension of the controlled fractional Bogdanov system under different parameters. The main results show that the correlation dimension converges to a fixed value as the embedding dimension increases for the controlled fractional Bogdanov map in chaotic state, which also coincides with the conclusion driven by the largest Lyapunov exponent. Moreover, three-dimensional fractional Stefanski map is considered to further verify the effectiveness and generality of the obtained results.

In this paper, we improve an Ignorant-Lurker-Spreader-Removal (ILSR) rumor propagation model as in [Yang et al., 2019] in social networks with consideration to Logistic growth and two discrete delays. First, we prove the existence of equilibrium points by calculating the basic reproduction number according to the next generation matrix. Regarding the two discrete delays as bifurcating parameters, the local asymptotical stability and Hopf bifurcation of the positive equilibrium point are discussed for six different scenarios by analyzing the characteristic equations of linearized systems. Applying the normal form theory and the center manifold theorem, some important conclusions about the stability and direction of bifurcating periodic solution are given when the two time delays are equal. Subsequently we study the global stability of the equilibrium points by constructing Lyapunov functions when the two delays disappear. Finally, we verify the conclusions through numerical simulations and perform sensitivity analysis on the basic reproduction numbers.

Short-term wind speed prediction has its special significance in wind power industry. However, due to the characteristics of the wind system itself, it is not easy to predict the short-term wind speed accurately. In order to solve the problem, this paper studies the chaotic characteristics and prediction of short-term wind speed time series. The short-term wind speed data at four time scales are collected as the research object. The predictability of short-term wind speed time series is determined by the Hurst exponent. The chaotic characteristics of short-time wind speed at different time scales are analyzed by the 0–1 test method for chaos and the maximum Lyapunov exponent method. The results show that the short-term wind speed time series has chaotic characteristics at different time scales. The phase-space reconstruction technology is introduced; delay time is determined by the C–C method; embedding dimension is obtained by the G–P method. Echo state network is improved to suppress the influence of input noise on prediction performance. At the same time, an improved grey Wolf optimization algorithm is proposed to optimize the parameters of reserve pool of the echo state network. The results of a case study show that, compared with state-of-the-art methods, the proposed prediction method improves the prediction accuracy and reduces the predictive errors.

In the present paper, we study chaos in iterated function systems (IFS), namely dynamical systems with several generators. We introduce weak Li–Yorke chaos, chaos in branch, and weak topological chaos to perceive the role of branches to create chaos in an IFS. Moreover, we define another type of chaos, $\sigma$-chaos, on an IFS. Further, we find the necessary conditions to create the chaotic iterated function systems.

Mathematical modeling is very helpful for noninvasive investigation of glucose-insulin interaction. In this paper, a new time-delay mathematical model is proposed for glucose-insulin endocrine metabolic regulatory feedback system incorporating the $\beta$-cell dynamic and function for regulating and maintaining bloodstream insulin level. The model includes the insulin degradation due to glucose interaction. The dynamical behavior of the model is analyzed and two-dimensional bifurcation diagrams with respect to two essential parameters of the model are obtained. The results show that the time-delay in insulin secretion in response to blood glucose level, and the delay in glucose drop due to increased insulin concentration, can give rise to complex dynamics, such as periodic oscillation. These dynamics are consistent with the biological findings and period doubling cascade and chaotic state which represent metabolic disorder that may lead to diabetes mellitus.

The influence of random fluctuations on the recruitment of effector cells towards a tumor is studied by means of a stochastic mathematical model. Aggressively growing tumors are confronted against varying intensities of the cell-mediated immune response for which chaotic and periodic oscillations coexist together with stable tumor dynamics. A thorough parametric analysis of the noise-induced transition from this oscillatory regime to complete tumor dominance is carried out. A hysteresis phenomenon is uncovered, which stabilizes the tumor at its carrying capacity and drives the healthy and the immune cell populations to their extinction. Furthermore, it is shown that near a crisis bifurcation, such transitions occur under weak noise intensities. Finally, the corresponding noise-induced chaos-order transformation is analyzed and discussed in detail.

It is not common in applied sciences to realize simulations which depict fractal representation in attractors’ dynamics, the reason being a combination of many factors including the nature of the phenomenon that is described and the type of differential operator used in the system. In this work, we use the fractal-fractional derivative with a fractional order to analyze the modified proto-Lorenz system that is usually characterized by chaotic attractors with many scrolls. The fractal-fractional operator used in this paper is a combination of fractal process and fractional differentiation, which is a relatively new concept with most of the properties and features still to be known. We start by summarizing the basic notions related to the fractal-fractional operator. After that, we enumerate the main points related to the establishment of proto-Lorenz system’s equations, leading to the $n$th cover of the proto-Lorenz system that contains $n$ scrolls ($n\in \mathbb{N}$). The triple and quadric cover of the resulting fractal and fractional proto-Lorenz system are solved using the Haar wavelet methods and numerical simulations are performed. Due to the impact of the fractal-fractional operator, the system is able to maintain its chaotic state of attractor with many scrolls. Additionally, such attractor can self-replicate in a fractal process as the derivative order changes. This result reveals another great feature of the fractal-fractional derivative with fractional order.

In this paper, we study the dynamics of an ecoepidemic competition system where the individuals of one population gather together in herds with a defensive strategy, showing social behavior, while another predator population is subject to a transmissible disease and behaves individually. By analyzing the existence and stability of the equilibria of the system, we find that the relatively isolated population can be eradicated, or the population with group defense can live alone eventually under some constraints. Infected individuals end up in two possible situations. In the first case, the disease is eventually eliminated, meaning that only healthy and group-defense individuals in the system can survive. In the other case, the spread of the disease is controlled and eventually all three individuals can coexist. We also conduct a correlation analysis using competition parameter and recovery rate of disease as birfurcation parameters in order to study the transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation. The long-term dynamics of the boundary and interior equilibria are demonstrated by numerical simulations.

In this paper, we consider some piecewise smooth Liénard systems and study the Hopf bifurcation of limit cycles from the origin after a perturbation. We define the cyclicity $N(k,l,m,n)$ of the piecewise smooth Liénard systems at the origin and denote $N_{1,1}(m,n)$ for fixing $k=l=1$. Then we obtain $N_{1,1}(1,n)=n,$ $n\ge 1$; $N_{1,1}(m,1)=m,$ $m\ge 1$ and $N_{1,1}(m,n)=m+n-2,$ $m=n\ge 2$.