The nonlinear dynamics of a recently derived generalized Lorenz model [Macek & Strumik, 2010] 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

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

We provide normal forms and the global phase portraits on the Poincaré disk of all planar Kukles systems of degree 3 with ℤ2-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

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

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

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

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

We study the zero-Hopf bifurcation of the Rössler differential system ẋ=x−xy−z,ẏ=x2−ay,ż=b(cx−z), where the dot denotes the derivative with respect to the independent variable t and a, b, c are real parameters.

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

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

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

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

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

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

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, σ-chaos, on an IFS. Further, we find the necessary conditions to create the chaotic iterated function

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 β-cell dynamic and function for regulating and maintaining bloodstream insulin level. The model includes the insulin degradation due to glucose interaction

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

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

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

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 N1,1(m,n) for fixing k=l=1. Then we obtain N1,1(1,n)=n,n≥1; N1,1(m,1)=m,m≥1 and N1,1(m,n)=m+n−2,m=n≥2.

This paper presents a new physical Sr0.95Ba0.05TiO3 (SBT) memristor-based chaotic circuit. The equilibrium point and the stability of the chaotic circuit are analyzed theoretically. This circuit system exhibits multiple dynamics such as stable point, periodic cycle and chaos by means of Lyapunov exponents spectra, bifurcation diagrams, Poincaré maps and phase portraits, when the initial state or the

In this paper, motivated by [Peletier & Rottschäfer, 2004; Peletier & Williams, 2007], we study the dynamical bifurcation of the modified Swift–Hohenberg equation endowed with an evenly periodic condition on the interval [−λ,λ]. As λ crosses over the critical points, the trivial solution bifurcates to an attractor and some new patterns of solutions emerge. We provide detailed descriptions of all possible

This paper considers the stability of a one-dimensional system during a catastrophic shift described by the Hill function. Because the shifting process goes through a nonequilibrium region, we applied the theory of Kosambi, Cartan, and Chern (KCC) to analyze the stability of this region based on the differential geometrical invariants of the system. Our results show that the Douglas tensor, one of

The paper proposes a novel calculation method on the sensitivity analysis of bifurcation parameters and states in a single-degree-of-freedom (SDoF) impacting system. It presents the causes to (non-) smooth bifurcations in virtue of parameter sensitivity analysis. The derivation of the system’s Poincaré mappings is used to integrate the Floquet matrix. It performs the identifications of the main and

In this paper, steady state bifurcations arising from the reaction–diffusion equations are investigated. Using the Lyapunov–Schmidt reduction on a square domain, a simple, and a double steady state bifurcation caused by the symmetry of spatial region is obtained. By examining the reduced bifurcation equations, complete bifurcation scenario and patterns at simple and double steady state bifurcation

We investigate the bifurcation phenomena for stochastic systems with multiplicative Gaussian noise, by examining qualitative changes in mean phase portraits. Starting from the Fokker–Planck equation for the probability density function of solution processes, we compute the mean orbits and mean equilibrium states. A change in the number or stability type, when a parameter varies, indicates a stochastic

A three-dimensional (3D) Hénon map of fractional order is proposed in this paper. The dynamics of the suggested map are numerically illustrated for different fractional orders using phase plots and bifurcation diagrams. Lorenz-like attractors for the considered map are realized. Then, using the linear fractional-order systems stability criterion, a controller is proposed to globally stabilize the fractional-order

To reduce the global burden of mosquito-borne diseases, e.g. dengue, malaria, the need to develop new control methods is to be highlighted. The sterile insect technique (SIT) and various genetic modification strategies, have a potential to contribute to a reversal of the current alarming disease trends. In our previous work, the ordinary differential equation (ODE) models with different releasing sterile

Micro-macro links between individual psychophysical traits and couple behavior are established. The analysis is carried out through a simple idealized model that mimics couples not influenced by their social environment. The functions describing the traits, i.e. the reactions to the love and appeal of the partner, are quite general. In particular, they allow one to take into account relatively sophisticated

For further increasing the complexity of chaotic attractors, a new method for generating Mirror Symmetry Composite Multiscroll Chaotic Attractors (MSCMCA) is proposed. We take the Lorenz system as an example to explain the mechanism of the method. Firstly, by varying the signs and magnitudes of the nonlinear terms, the Lorenz system generates symmetrical attractors and different-magnitude attractors

The dynamics of the tubular chemical reactor with mass recycle was examined. In such a system, temperature and concentrations may oscillate chaotically. This means that state variable values are then unpredictable. In this paper, it has been shown that despite the chaos, the behavior of such a reactor can be predictable. It has been shown that this phenomenon can occur in two cases. The first case

We study the existence of limit cycles in planar piecewise linear Hamiltonian systems with three zones without equilibrium points. In this scenario, we have shown that such systems have at most one crossing limit cycle.

We consider third-order autonomous continuous piecewise differential equations in the variable x. For such differential equations with nonlinearities of the form xm, we investigate their periodic solutions using the averaging theory. We remark that since the differential system is only continuous we cannot apply to it the classical averaging theory, that needs that the differential system be at least

In this paper, the effects of low and fast response speeds of neuron activation gradient of a simple 3D Hopfield neural network are explored. It consists of analyzing the effects of low and high neuron activation gradients on the dynamics. By considering an imbalance of the neuron activation gradients, different electrical activities are induced in the network, which enable the occurrence of several

In this paper, we investigate limit cycles induced by threshold nonlinearity of piecewise linear (PWL) differential systems, which are node-focus type or node-center type with the focus or the center being virtual or boundary. To get the number and stability of limit cycles, we adopt a new displacement function with a better configuration than usual. For a given parameter subregion, we exhibit the

We propose a new simple three-dimensional continuous autonomous model with two nonlinear terms and observe the dynamical behavior with respect to system parameters. This system changes the stability of fixed point via Hopf bifurcation and then undergoes a cascade of period-doubling route to chaos. We analytically derive the first Lyapunov coefficient to investigate the nature of Hopf bifurcation. We

In this work, we examine the solitary wave solutions of the mKdV equation with small singular perturbations. Our analysis is a combination of geometric singular perturbation theory and Melnikov’s method. Our result shows that two families of solitary wave solutions of mKdV equation, having arbitrary positive wave speeds and infinite boundary limits, persist for selected wave speeds after small singular

We characterize the families of periodic orbits of two discontinuous piecewise differential systems in ℝ3 separated by a plane using their first integrals. One of these discontinuous piecewise differential systems is formed by linear differential systems, and the other by nonlinear differential systems.

A two-species predator–prey plankton model is studied, where the grazing pressure of microzooplankton on phytoplankton is controlled through external infochemical mediated predation. The system stability is analyzed in order to explain the conditions for phytoplankton bloom formation and to explore system bifurcations. The interplay between the level of infochemical-mediated external predation and

A new three-dimensional (3D) memristive HR neuron model is presented, which is improved from an existing memristive HR neuron model using a memristor synapse with sine memductance to substitute the original one. The improved memristive HR neuron model has no equilibrium but hidden firing activities can emerge with discrete memristor initial-offset boosting. Treating the neuron model as a two-dimensional

This paper reports numerical experiments done on a two-parameter family of vector fields which unfold an attracting heteroclinic cycle linking two saddle-foci. We investigated both local and global bifurcations due to symmetry breaking in order to detect either hyperbolic or chaotic dynamics. Although a complete understanding of the corresponding bifurcation diagram and the mechanisms underlying the

In smooth systems, the form of the heteroclinic Melnikov chaotic threshold is similar to that of the homoclinic Melnikov chaotic threshold. However, this conclusion may not be valid in nonsmooth systems with jump discontinuities. In this paper, based on a newly constructed nonsmooth pendulum, a kind of impulsive differential system is introduced, whose unperturbed part possesses a nonsmooth heteroclinic

A new 4D chaotic system with infinitely many equilibria is proposed using a linear state feedback controller in the Sprott C system. Although the new 4D chaotic system has only two nonlinear terms, it has rich dynamic characteristics, such as hidden attractors and coexisting attractors. Besides, the freedom of offset boosting of a variable is achieved by adjusting a controlled constant. The dynamic

In this paper, we study the maximum number of limit cycles for the unfolding of codimension-3 planar singularities with nilpotent linear parts. In [J. Math. Anal. Appl.499 (2017)], the authors proved that when parameter a∈(−1,1)\{0,−2/3} is rational, the corresponding problem could be transformed to solving semi-algebraic systems. At the same time, it is pointed out that when a=−2/3, the logarithmic

Advertising competition among firms may give rise to very complex dynamical behaviors. In this article, under the assumption of spillover effects, a two-stage dynamical Cournot game of advertising competition between two firms, which produce homogeneous products, is developed. Then, the local stability of the equilibrium points is discussed, and stability conditions of the equilibrium points are obtained

In this paper, we exploit an idea of nonlinear modal series method to investigate the effects of modal interaction in the one-cycle controlled (OCC) double-input SEPIC DC–DC converter. Based on the proposed nonlinear averaged model, the analytical approximate solutions are obtained to characterize the dynamic response characteristic of the transient behaviors in the double-input SEPIC converters. The

This paper discusses the sensitivity and chaotic properties of nonautonomous dynamical systems. For the first time we obtain sufficient conditions of some stronger forms of sensitivity for nonautonomous dynamical systems. Then, some chaotic relations between two nonautonomous dynamical systems are proved.

In this paper, a simple four-wing chaotic attractor is first proposed by replacing the constant parameters of the Chen system with a periodic piecewise function. Then, a new 4D four-wing memristive hyperchaotic system is presented by adding a flux-controlled memristor with linear memductance into the proposed four-wing Chen system. The memristor mathematical structure model is simple and easy to implement

This paper investigates a finance system with nonconstant elasticity of demand. First, under some conditions, the system has invariant algebraic surfaces and the analytic expressions of the surfaces are given. Furthermore, when the two surfaces coincide and become one surface, the dynamics on the surface are analyzed and a globally stable equilibrium is found. Second, by using the normal form theory

Dynamic behavior of a discrete-time prey–predator system with Leslie type is analyzed. The discrete mathematical model was obtained by applying the forward Euler scheme to its continuous-time counterpart. First, the local stability conditions of equilibrium point of this system are determined. Then, the conditions of existence for flip bifurcation and Neimark–Sacker bifurcation arising from this positive

In this paper, we study ultimate dynamics and derive tumor eradication conditions for the angiogenic switch model developed by Viger et al. This model describes the behavior and interactions between host (x); effector (y); tumor (z); endothelial (w) cell populations. Our approach is based on using the localization method of compact invariant sets and the LaSalle theorem. The ultimate upper bound for

Previous studies assumed that the reaction processes in the chemical Brusselator model are memoryless or Markovian. However, as long as a reactant interacts with its environment, the reaction kinetics cannot be described as a memoryless process. This raises a question: how do we predict the behavior of the chemical Brusselator system with molecular memory characterized by nonexponential waiting-time

The coexistence of random periodic solutions with various periods (i.e. subharmonics) is proved to random differential equations on a circle with random impulses of all integer orders. One of the theorems is also extended to random differential inclusions on a circle with multivalued deterministic impulses. These results can be roughly characterized as a further application of the randomized Sharkovsky

The randomly forced Rulkov neuron model with the discontinuous 2D-map is considered. We study the phenomena of the stochastic excitement: (i) noise-induced spiking in the parametric zone where the equilibrium is a single attractor; (ii) stochastic generation of the spiking in bistability zone; (iii) noise-induced bursting in the parametric zone where the deterministic model exhibits the tonic spiking

The dynamic behavior of many physical, biological, and other systems, are organized according to the synchronization of chaotic oscillators. In this paper, we have proposed a new method with low sensitivity to noise for detecting synchronization by mapping time series to complex networks, called the ordinal partition network, and calculating the permutation entropy of that structure. We show that this

The planar version of the equilateral restricted four-body problem, with three unequal masses, is numerically investigated. By adopting the grid classification method we locate the coordinates, on the plane (x,y), of the points of equilibrium, for all possible values of the masses of the primaries. The linear stability of the libration points is also determined, as a function of the masses. Our analysis

The nonlinear dynamics of a multigear pair with the time-varying gear mesh stiffness are investigated using an enhanced compliance-based methodology. In the proposed approach, Lagrangian theory and Runge–Kutta method are used to derive the equation of motion of the multigear pair and solve its dynamic response for various values of the gear mesh frequency, respectively. The simulation results obtained

The Cardiac Purkinje Fiber (CPF) is the last branch of the heart conduction system, which is meshed with the normal ventricular myocyte. Purkinje fiber plays a key role in the occurrence of ventricular arrhythmia and maintenance. Does the heart Purkinje fiber cells have the same memory function as the cerebral nerve? In this paper, the cardiac Hodgkin–Huxley equation is taken as the object of study

The curse of dimensionality looms over many studies in science and engineering. Low-order systems provide conceptual clarity but often fail to reveal the extent of possible complexity, whereas high-order systems present a host of daunting challenges to the analyst, not least the classification and visualization of typical behavior. In this paper, we detail the behavior of systems that fall somewhere