International Journal of Bifurcation and Chaos

A number of important applications would benefit from the introduction of locally-active memristors, which is defined to be any memristor that exhibits negative differential memristance for at least a voltage or a current applied to the memristor. Two leading examples are emerging nonvolatile memory based on memristor-based crossbar array architectures, and neural networks that exhibit improved computational complexity when operated at the edge of chaos. In this paper, a novel locally-active memristor model is presented for exploring the nonvolatile and switching mechanism of the memristor and the influence of local activity on the complexity of nonlinear circuits. We find that the memristor possesses three locally-active regions in its DC $V$–$I$ plot and two asymptotically stable states (equilibrium points) on its power-off plot (POP) where voltage $v=0$, implying that the memristor is bistable, which can be used as a nonvolatile binary memory or binary switch. We also find the mechanism and the rule of switching between the two stable states by applying a single square voltage pulse of appropriate pulse width and pulse amplitude. We show that it is always possible to switch from one stable state to another of the memristor with an appropriate pulse amplitude and a pulse width, and that there is a trade-off between the voltage pulse amplitude and the pulse width for the faster switching between the two equilibrium points. We also show that fast switching between the two states is possible by using a periodic bipolar narrow pulse sequence. Local activity depends on the capability of a memristor circuit to amplify infinitesimal fluctuations in energy. Based on this principle, we designed a simplest chaotic oscillator that utilizes only three components in parallel: the proposed locally-active memristor, a linear capacitor and an inductor, which can oscillate around an equilibrium point located on its DC $V$–$I$ plot. Its dynamic characteristics are verified by theoretical analyses, simulations and DSP experiments.

Based on the oligopoly game theory, a dynamic duopoly Cournot model with bounded rationality and consumer surplus is established. On the one hand, the type and the stability of the boundary equilibrium points and the stability conditions of the Nash equilibrium point are discussed in detail. On the other hand, the potential complex dynamics of the system is demonstrated by a set of 2D bifurcation diagrams. It is found that the bifurcation diagrams have beautiful fractal structures when the adjustment speed of production is taken as the bifurcation parameter. And it is verified that the area with scattered points in the 2D bifurcation diagrams is caused by the coexistence of multiple attractors. It is also found that there may be two, three or four coexisting attractors. It is even found the coexistence of Milnor attractor and other attractors. Moreover, the topological structure of the attracting basin and global dynamics of the system are investigated by the noninvertible map theory, using the critical curve and the transverse Lyapunov exponent. It is concluded that two different types of global bifurcations may occur. Because of the symmetry of the system, it can be concluded that the diagonal of the system is an invariant one-dimensional submanifold. And it is controlled by a one-dimensional map which is equivalent to the classical Logistic map. The bifurcation curve of the system on the adjustment speed and the weight of the consumer surplus is obtained based on the properties of the Logistic map. And the synchronization phenomenon along the invariant diagonal is discussed at the end of the paper.

We analyze a modified Holling–Tanner predator–prey model where the predation functional response is of Holling type II and we incorporate a strong Allee effect associated with the prey species production. The analysis complements the results of previous articles by Saez and González-Olivares [1999] and Arancibia-Ibarra and González-Olivares [2015] discussing Holling–Tanner models which incorporate a weak Allee effect. The extended model exhibits rich dynamics and we prove the existence of separatrices in the phase plane separating basins of attraction related to coexistence and extinction of the species. We also show the existence of a homoclinic curve that degenerates to form a limit cycle and discuss numerous potential bifurcations such as saddle-node, Hopf, and Bogdanov–Takens bifurcations.

In this paper, we announce a novel 4D chaotic system which belongs to the self-excited attractor and hidden attractor family depending on the parameter values. Lyapunov exponents, bifurcation diagram and bicoherence plot of the CAMO (Camouflage) chaotic system are investigated. Also, fractional-order model of the proposed CAMO system (FOCAMO) is derived and analyzed. FOCAMO chaotic system is then implemented in Field Programmable Gate Array (FPGA) using Adomian decomposition method. Also, power efficiency analysis for various fractional-orders is investigated. The paper helps build a better understanding of chaotic systems with self-excited or hidden attractors.

In this paper, we consider the dynamics of a delayed reaction–diffusion mussel-algae system subject to Neumann boundary conditions. When the delay is zero, we show the existence of positive solutions and the global stability of the boundary equilibrium. When the delay is not zero, we obtain the stability of the positive constant steady state and the existence of Hopf bifurcation by analyzing the distribution of characteristic values. By using the theory of normal form and center manifold reduction for partial functional differential equations, we derive an algorithm that determines the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, some numerical simulations are carried out to support our theoretical results.

This paper is concerned with the spatial patterns of a predator–prey system with Beddington–DeAngelis functional response, in which the parameter $k$ measuring the mutual interference between predators can play an essential role. By using the bifurcation theory and implicit function theorem we first consider the positive steady state solution bifurcating from the semitrivial steady state solution set of the system and prove that the positive steady state solution is constant. Then we show that nonconstant positive steady state solution may bifurcate from the constant positive steady state solution when $k$ is neither small nor large. Finally, we show that spatially nonhomogeneous periodic orbits may also bifurcate from the constant positive steady state solution as $k$ is not large.

In this paper, a diffusive predator–prey model is considered in which the predator and prey populations both exhibit schooling behavior. The system’s spatial dynamics are captured via a suitable threshold parameter, and a sequence of spatiotemporal patterns such as hexagons, stripes and a mixture of the two are observed. Specifically, the linear stability analysis is applied to obtain the conditions for Hopf bifurcation and Turing instability. Then, employing the multiple-scale analysis, the amplitude equations near the critical point of Turing bifurcation are derived, through which the selection and stability of pattern formations are investigated. The theoretical results are verified by numerical simulations.

Time-delay-induced synchronous behaviors and synchronization transitions have been widely investigated for coupled neurons, and they play important roles for physiological functions. In the present study, time-delay-induced synchronized subthreshold oscillations were simulated, and the bifurcations underlying the synchronized behaviors were identified for a pair of coupled FitzHugh–Nagumo neurons. Multiple transitions between in-phase and anti-phase synchronizations induced by the time delay were simulated for the inhibitory and excitatory couplings. Subcritical or supercritical Hopf bifurcations and the stability of the Hopf-bifurcating periodic subthreshold oscillations were acquired using center manifold reduction and normal form theory. The in-phase or anti-phase synchronizations of the stable periodic subthreshold oscillations, which appear for multiple values of the time delay, were interpreted with the related eigenspace. The distributions of the different dynamical behaviors, including the synchronizations and bifurcations in the two-parameter plane of the time delay and coupling strength, were acquired for both types of synapses, and the different roles of the inhibitory and excitatory couplings on the synchronization transitions were compared.

Functionally graded (FG) graphene reinforced composite (GRC) is a new class of advanced composite materials. In GRC, several layers of graphene platelets (GPLs) are randomly or uniformly dispersed in matrix. These GPLs have uniform arrangement, or are arranged with gradient, in the direction of thickness in accordance with three different graphene distribution rules. In this study, the nonlinear dynamic analysis of FG GRC truncated conical shells, subjected to a combined action of transverse excitation and axial force, is performed using the first shear deformation theory (FSDT). Estimation of equivalent Young’s modulus of the composites is calculated using a modified Halpin–Tsai model. In addition, a partial differential equation model is developed based on the Hamilton principle and nonlinear strain-displacement relationship. The Galerkin method and the fourth-order Runge–Kutta method are used to solve the equation. The dimensionless linear natural frequency of an FG GRC truncated conical shell is calculated by the Rayleigh–Ritz method and compared with available results in the literature to verify the accuracy of the present model. Simultaneously, significant effects of the different parameters, such as the total layer numbers, semi-vertex angles, GPLs weight fractions, distribution patterns and the length-to-thickness ratios, on the nonlinear dynamics including bifurcation and chaos of FG GRC truncated conical shells are investigated.

In this work, Monte Carlo method is used for the probability calculation of the appearance of spatial chaos in MIMO cascade nonlinear systems. The obtained results are given in tables and presented graphically using probability histograms. Their validity is confirmed by comparison with the results of spatial chaos appearance analysis that uses bifurcation diagrams, Lyapunov diagrams and spatial phase portraits.

This paper explores the bifurcation control of a fractional predator–prey system with an active extended delayed feedback controller. Delay-induced bifurcations criteria for such an uncontrolled system are firstly derived by selecting time delay as a bifurcation parameter. Then, an extended delayed feedback controller is cleverly devised to control Hopf bifurcation for the proposed system. It means that the bifurcation dynamics can be efficaciously controlled for a given system with the adjustment of the fractional order, feedback gain and extended feedback delay provided that the remnant parameters are fixed. The obtained results significantly extend the preceding studies concerning bifurcation control of delayed fractional-order systems. To verify the correctness of the established theory, some numerical results are presented.

In this paper, the Fuzzy Generalized Cell Mapping (FGCM) method is developed with the help of the Adaptive Interpolation (AI) in the space of fuzzy parameters. The adaptive interpolation on the set-valued fuzzy parameter is introduced in computing the one-step transition membership matrix to enhance the efficiency of the FGCM. For each of initial points in the state space, a coarse database is constructed at first, and then interpolation nodes are inserted into the database iteratively each time errors are examined with the explicit formula of interpolation error until the maximal errors are just under the error bound. With such an adaptively expanded database on hand, interpolating calculations assure the required accuracy with maximum efficiency gains. The new method is termed as Fuzzy Generalized Cell Mapping with Adaptive Interpolation (FGCM with AI), and is used to investigate codimension-two bifurcations in two-dimensional and three-dimensional nonlinear dynamical systems with fuzzy noise. It is found that global changes in fuzzy dynamics are dominated by the underlying deterministic counterparts, and the fuzzy attractor expands along the unstable manifold leading to a collision with a saddle when a bifurcation occurs. The examples show that the FGCM with AI has a thirtyfold to fiftyfold efficiency over the traditional FGCM to achieve the same analyzing accuracy.

In this paper, we consider a diffusive predator–prey model with Monod–Haldane functional response. We study the Turing instability and Hopf bifurcation of the coexisting equilibriums. We investigate the Turing–Hopf bifurcation through some key bifurcation parameters. In addition, we obtain a normal form for the Turing–Hopf bifurcation. Finally, we show numerical simulations to illustrate the theoretical results. For parameters around the critical value of the Turing–Hopf bifurcation, we demonstrate that the predator–prey model exhibits complex spatiotemporal dynamics, including spatially homogeneous periodic solutions, spatially inhomogeneous periodic solutions, and spatially inhomogeneous steady-state solutions.

Since the electromagnetic field of neural networks is heterogeneous, the diffusion phenomenon of electrons exists inevitably. In this paper, we investigate the existence of Turing–Hopf bifurcation in a reaction–diffusion neural network. By the normal form theory for partial differential equations, we calculate the normal form on the center manifold associated with codimension-two Turing–Hopf bifurcation, which helps us understand and classify the spatiotemporal dynamics close to the Turing–Hopf bifurcation point. Numerical simulations show that the spatiotemporal dynamics in the neighborhood of the bifurcation point can be divided into six cases and spatially inhomogeneous periodic solution appears in one of them.

This paper focuses on the investigation of a vibro-impact (VI) system based upon time-delayed feedback control (TDFC) and visco-elastic damping (VED) under bounded random excitations. A pretreatment for the TDFC and VED is necessary. A further simplification for the system is achieved by introducing the mirror image transformation. The averaging approach is adopted to analyze the above system relying on a parametric principal resonance consideration. By means of the first kind of a modified Bessel function, explicit asymptotic formulas for the maximal Lyapunov exponent (MLE) are given to examine the almost sure stability or instability of the trivial steady-state amplitude solution. Besides, the steady-state moments (SSM) of the nontrivial solutions of the system’s amplitude are derived by the application of the moment method and Itô’s calculus. Finally, the stability and its critical situations of the trivial solution are explored in detail through the important system parameters, i.e. embodying the TDFC parameters, the VED parameters, the restitution coefficient, the excitation amplitude and the random noise intensity. They are tested by numerical simulations. Additionally, the exploration of the steady-state moments involves the emergence of the general frequency response curve and the frequency island, discussions of conditions satisfied by the unstable boundary, and variations of the time-delayed island. Stochastic jumps and bifurcations are observed for the stationary joint transition probability density of the system’s trivial and nontrivial solutions based on parameter schemes of VED and TDFC.

Based on a hybrid neuronal model of the Hindmarsh–Rose (HR) model (the slow negative feedback dynamics) with the Wilson model (the fast variable dynamics), the discharge modes of neurons are studied by using bifurcation analysis. When the hybrid neuronal model is driven by external stimulus current, the hybrid neuronal model is capable of simulating an extensive range of bursting patterns with different systemic parameters. With the increase of external stimulus, the multiple discharge patterns of neurons arise, and the electrical modes of the neurons appear for successive transitions. When the hybrid neuronal model is driven by electromagnetic induction, the electrical activities of the neurons have different responses to the variations in the amplitude and the angular frequency of high and low frequency electromagnetic radiations. It is found that the neuronal discharge activity is transformed from chaotic state to periodic state by increasing the angular frequency, and the firing pattern of neuron can be shifted from the bursting state to chaotic state by increasing the amplitude of the low frequency signal.

We consider the one degree-of-freedom Hamiltonian system defined by the Morse potential energy function (the “Morse oscillator”). We use the geometry of the level sets to construct explicit expressions for the trajectories as a function of time, their period for the bounded trajectories, and action-angle variables. We use these trajectories to prove sufficient conditions for chaotic dynamics, in the sense of Smale horseshoes, for the time-periodically perturbed Morse oscillator using a Melnikov type approach.