International Journal of Bifurcation and Chaos

Recently, while studying the dynamics of power electronic DC/AC converters we have demonstrated that the behavior of these systems can be modeled by piecewise-smooth maps which belong to a specific class of models not investigated before. The characteristic feature of these maps is the presence of a very high number of switching manifolds (border points in 1D). Obviously, the multitude of control strategies applied in the modern power electronics leads to different maps belonging to this class of models. However, in this paper we show that several models can be studied using the same piecewise-linear approximation, so that the bifurcation phenomena which can be observed in this approximation are generic for many models. Based on the results obtained before for piecewise-smooth models with different kinds of nonlinearities resulting from the corresponding control strategies, in the present paper we discuss the generic bifurcation patterns in the underlying piecewise-linear map.

This paper is devoted to the analysis of bidimensional piecewise linear systems with hysteresis coming from 3D systems with slow–fast dynamics. We focus our attention on the symmetric case without equilibria, determining the existence of periodic orbits as well as their stability, and possible bifurcations. New analytical characterizations of bifurcations in these hysteretic systems are obtained. In particular, bifurcations of periodic orbits from infinity, grazing and saddle-node bifurcations of periodic orbits are studied in detail and the corresponding bifurcation sets are provided. Finally, the study of the hysteretic systems is shown to be useful in detecting periodic orbits for some $3$D piecewise linear systems.

Given a planar quadratic differential system delimited by a straight line, we are interested in studying the bifurcation phenomena that can arise when the position on the boundary of two tangency points are considered as parameters of bifurcation. First, under generic conditions, we find a two-parametric family of quadratic differential systems with at least one tangency point. After that, we find a normal form for this parameterized family. Next, we study two subfamilies, one of them characterized by the existence of two fold points of different nature, and the other one, characterized by the existence of one fold point and one boundary equilibrium point. For the first family, we find sufficient conditions for the existence of stationary bifurcations: saddle-node, transcritical and pitchfork, while for the second family, the existence of the called transcritical Bogdanov–Takens bifurcation is proved. Finally, the results are illustrated with two examples.

Immunotherapy using dendritic cells (DCs) as antigen-presenting cells is widely used in laboratory and animal studies as a promising treatment for advanced prostate cancer. In this paper, the bifurcation analysis of a nonlinear model of prostate cancer with immunotherapy is performed. It is found that the model exhibits complex behaviors such as a saddle-node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and generalized Hopf bifurcation. Moreover, it is shown that the vaccine amount and T-cells killing efficiency of tumor cells have a significant effect on tumor cells, which provides new insights to improve treatment outcomes even for patients with a weak immune system.

This paper investigates the dynamical behaviors of a duopoly model with two content providers (CPs). Competition between two CPs is assumed to take place in terms of their pricing decisions and the credibility of content they offer. According to the CPs’ rationality level, we consider a scenario where both CPs are bounded rational. Each CP in any period uses the marginal profit observed from the previous period to choose its strategies. We compute explicitly the steady states of the dynamical system induced by bounded rationality, and establish a necessary and sufficient condition for stability of its Nash equilibrium (NE). Numerical simulations show that if some parameters of the model are varied, the stability of the NE point is lost and the complex (periodic or chaotic) behavior occurs. The chaotic behavior of the system is stabilized on the NE point by applying control.

A model of two self-sustained oscillators interacting through memristive coupling is studied. The memristive coupling is realized by using a cubic memristor model. Numerical simulation is combined with theoretical analysis by means of quasi-harmonic reduction. It is shown that the specifics of the memristor nonlinearity results in the appearance of infinitely many equilibrium points which form a line of equilibria in the phase space of the system under study. It is established that the possibility to observe the effect of phase locking in the considered system depends on both parameter values and initial conditions. Consequently, the boundaries of the synchronization region are determined by the initial conditions. It is demonstrated that introducing or adding a small term into the memristor state equation gives rise to the disappearance of the line of equilibria and eliminates the dependence of synchronization on the initial conditions.

In the literature, existing Josephson junction based oscillators are mostly driven by external sources. Knowing the different limits of the external driven systems, we propose in this work a new autonomous one that exhibits the unusual and striking multiple phenomena among which coexist the multiple hidden attractors in self-reproducing process under the effect of initial conditions. The eight-term autonomous chaotic system has a single nonlinearity of sinusoidal type acting on only one of the state variables. A priori, the simplicity of the system does not predict the richness of its dynamics. We also find that a limit cycle attractor widens to a parameter controlling coexisting multiple-scroll attractors through the splitting and the inverse splitting of periods. Multiple types of bifurcations are found including period-doubling and period-splitting (antimonotonicity) sequences to chaos, crisis and Hopf type bifurcation. To the best of our knowledge, some of these interesting phenomena have not yet been reported in similar class of autonomous Josephson junction based circuits. Moreover, analytical investigations based on the Hopf theory analysis lead to the expressions that determine the direction of appearance of the Hopf bifurcation, confirming the existence and determining the stability of bifurcating periodic solutions. To observe this latter bifurcation and to illustrate the theoretical analysis, numerical simulations are performed. Chaos can be easily controlled by the frequency of the linear oscillator, the superconducting junction current, as well as the gain of the amplifier or circuit component values. The circuit and Field Programmable Gate Arrays (FPGA)-based implementation of the system are presented as well.

Our start point is a 3D piecewise smooth vector field defined in two zones and presenting a shared fold curve for the two smooth vector fields considered. Moreover, these smooth vector fields are symmetric relative to the fold curve, giving rise to a continuum of nested topological cylinders such that each orthogonal section of these cylinders is filled by centers. First, we prove that the normal form considered represents a whole class of piecewise smooth vector fields. After we perturb the initial model in order to obtain exactly ${ℒ}$ invariant planes containing centers, a second perturbation of the initial model is also considered in order to obtain exactly $k$ isolated cylinders filled by periodic orbits. Finally, joining the two previous bifurcations we are able to exhibit a model, preserving the symmetry relative to the fold curve, and having exactly $k\cdot {ℒ}$ limit cycles.

This paper focuses on the number of limit cycles bifurcating from a symmetrical compound polycycle with three saddles. We use two methods, the Melnikov function method and the method of stability-changing of a homoclinic loop or a double homoclinic loop to study this problem. We find 15 limit cycles and 16 limit cycles respectively with four alien limit cycles under certain conditions.

This paper is concerned with the strong resonance bifurcations with a reflection symmetry i.e. ${\mathbb{Z}}_2$-symmetry in maps. We compute the normal form of $1:2$ resonance and $1:3$ resonance bifurcations with ${\mathbb{Z}}_2$-symmetry. We use standard normal form techniques in order to obtain the reduced map. Then, we will obtain explicit formulae for normal form coefficients of bifurcations with ${\mathbb{Z}}_2$-symmetry. By using critical coefficients, we avoid the computation of the center manifold and the transformation of the linear part of the map into Jordan form. So this method can be used in the study of bifurcations with ${\mathbb{Z}}_2$-symmetry in general problems. To illustrate our results, we will analyze local bifurcations of the strong resonance bifurcations with ${\mathbb{Z}}_2$-symmetry numerically and then we will present some applications from economics and neural networks.

Coupled systems of two canonical-dissipative limit cycle oscillators are considered in the general case and for the case of monofrequency and multifrequency synchronization. Specifically, the oscillator frequency ratios of 1:1, 1:2, and 1:3 are examined and modeled by a hybrid Rayleigh–van der Pol oscillator and an oscillator model suggested by Holt as well as an oscillator model suggested by Fokas and Lagerstrom. It is shown that all three systems exhibit a unique bifurcation diagram that describes limit cycle attractors of monofrequency and multifrequency synchronization. In particular, the relative phase describing the lag between the two oscillators both in the monofrequency and multifrequency case can be tuned by an appropriately defined bifurcation parameter $\eta$. For $\left|\eta \right|<1$ two limit cycle attractors with different relative phases exist that merge at $\eta =\pm 1$ into pitchfork bifurcations and give rise to single limit cycle attractors that continue to exist for $\left|\eta \right|>1$. Similarities and differences to bifurcation diagrams published in previous work of a similar coupled oscillator system, one based on Smorodinsky–Winternitz potentials and exhibiting 1:1 synchronization, are noted.

In this paper, an attempt has been made to understand the role of predator’s interference and additional food on the dynamics of a diffusive population model. We have studied a predator–prey interaction system with mutually interfering predator by considering additional food and Crowley–Martin functional response (CMFR) for both the reaction–diffusion model and associated spatially homogeneous system. The local stability analysis ensures that as the quantity of alternative food decreases, predator-free equilibrium stabilizes. Moreover, we have also obtained a condition providing a threshold value of additional food for the global asymptotic stability of coexisting steady state. The nonspatial model system changes stability via transcritical bifurcation and switches its stability through Hopf-bifurcation with respect to certain ranges of parameter determining the quantity of additional food. Conditions obtained for local asymptotic stability of interior equilibrium solution of temporal system determines the local asymptotic stability of associated diffusive model. The global stability of positive equilibrium solution of diffusive model system has been established by constructing a suitable Lyapunov function and using Green’s first identity. Using Harnack inequality and maximum modulus principle, we have established the nonexistence of nonconstant positive equilibrium solution of the diffusive model system. A chain of patterns on increasing the strength of additional food as spots$\to$stripes$\to$spots has been obtained. Various kind of spatial-patterns have also been demonstrated via numerical simulations and the roles of predator interference and additional food are established.

In this paper, we propose a diffusive predator–prey model with herd behavior in prey and pack predation behavior in predator under the homogeneous Neumann boundary condition. Due to the pack predation-herd behavior, the predator–prey interaction occurs only at the outer edge of each group. First, we analyze the existence and local stability of the equilibria of temporal model using vector field analysis and characteristic equations. Second, we deduce the conditions under which Turing instability occurs with the help of linear stability analysis. Then, using standard multiple-scale analysis, we derive the amplitude equations for the excited modes. By numerical simulations, we find that the model exhibits complex pattern replication by varying the value of parameters $a$ and $d_{21}$, and study the relationships between the average density of both prey and predator and these two parameters.

Responses to drugs play key roles in exploring how drug toxicity affects the evolution of tumour cells. We model pulsed comprehensive therapies using an impulsive tumour-immune model, in which the application of comprehensive therapies is dependent on a threshold tumour size. By employing the definitions and properties of the Poincaré map, we show that the effector cell eradication periodic solution is globally stable under threshold conditions. In the light of bifurcation theorems, it is shown that transcritical bifurcations can occur with respect to many treatment parameters including depletion rate, chemotherapeutic drug concentration, a medicine toxicity coefficient and the accumulation rate of effector cells. Then we provide conditions for the existence of order-$k$ $(k\ge 1)$ periodic solutions. The results indicate that the threshold $R_0$ is sensitive to treatment parameters and the proposed system exists with very complex dynamics when treatment parameters are chosen as bifurcation parameters. Moreover, a therapeutic protocol with a smaller chemotherapy drug dosage and more frequent applications is more effective for maintaining a high tumour cell depletion rate.

Chen [2016a, 2016b] studied global dynamics of the Filippov systems $ẍ+(\alpha +\beta x^2)ẋ+x\pm sgn(x)=0$, respectively. To study the global dynamics of $ẍ+(\alpha +\beta x^2)ẋ\pm x\pm sgn(x)=0$ completely, since the dynamics of $ẍ+(\alpha +\beta x^2)ẋ-x-sgn(x)=0$ is very simple, we are only interested in the global dynamics of $ẍ+(\alpha +\beta x^2)ẋ-x+sgn(x)=0$ in this paper. Firstly, we use Briot–Bouquet transformations and normal sector methods to discuss these degenerate equilibria at infinity. Secondly, we discuss the number of limit cycles completely. Then, the sufficient and necessary conditions of existence of the heteroclinic loop are found. To estimate the upper bound of the heteroclinic loop bifurcation function on parameter space, a result on the amplitude of a unique limit cycle of a discontinuous Liénard system is given. Finally, the complete bifurcation diagram and all global phase portraits are presented. The global dynamic property of system $ẍ+(\alpha +\beta x^2)ẋ-x+\mathrm{\text{sgn}}(x)=0$ is totally different from systems $ẍ+(\alpha +\beta x^2)ẋ+x\pm \mathrm{\text{sgn}}(x)=0$.

The resonant chaotic dynamics of a symmetric cross-ply composite laminated plate are studied using the exponential dichotomies and an averaging procedure for the first time. The partial differential governing equations of motion for the symmetric cross-ply composite laminated plate are derived by using Reddy’s third-order shear deformation plate theory and von Karman type equation. The partial differential governing equations of motion are discretized into two-degree-of-freedom nonlinear systems including the quadratic and cubic nonlinear terms by using Galerkin method. There exists a fixed point of saddle-focus in the linear part for two-degree-of-freedom nonlinear system. The Melnikov method containing the terms of the nonhyperbolic mode is developed to investigate the resonant chaotic motions of the symmetric cross-ply composite laminated plate. The obtained results indicate that the nonhyperbolic mode of the symmetric cross-ply composite laminated plate does not affect the critical conditions in the occurrence of chaotic motions in the resonant case. When the resonant chaotic motion occurs, we can draw a conclusion that the resonant chaotic motions of the hyperbolic subsystem are shadowed for the full nonlinear system of the symmetric cross-ply composite laminated plate.

Tip growth is a growth stage which occurs in fungal cells. During tip growth, the cell exhibits continuous extreme lengthwise growth while its shape remains qualitatively the same. A model for single celled fungal tip growth is given by the Ballistic Aging Thin viscous Sheet (BATS) model, which consists of a five-dimensional system of first-order differential equations. The solutions of the BATS model that correspond to fungal tip growth arise through a codimension-1 global bifurcation in a two-parameter family of solutions. In this paper we derive a toy model from the BATS model. The toy model is given by two-dimensional system of first-order differential equations which depend on a single parameter. The main achievement of this paper is a proof that the toy model exhibits an analogue of the codimension-1 global bifurcation in the BATS model. An important ingredient of the proof is a topological method which enables the identification of the bifurcation points. Finally, we discuss how the proof may be generalized to the BATS model.

In this work, we present the dynamical study and the bifurcation structures of the $\gamma$-Ricker population model. Resorting to the Lambert $W$ function, the analytical solutions of the positive fixed point equation for the $\gamma$-Ricker population model are explicitly presented and conditions for the existence and stability of these fixed points are established. The main focus of this work is the definition and characterization of the Allee effect bifurcation for the $\gamma$-Ricker population model, which is not a pitchfork bifurcation. Consequently, we prove that the phenomenon of Allee effect for the $\gamma$-Ricker population model is associated with the asymptotic behavior of the Lambert $W$ function in a neighborhood of zero. The theoretical results describe the global and local bifurcations of the $\gamma$-Ricker population model, using the Lambert $W$ function in the presence and absence of the Allee effect. The Allee effect, snapback repeller and big bang bifurcations are investigated in the parameters space considered. Numerical studies are included.

This paper studies the bifurcations of phase portraits for the regularized Saint-Venant equation (a two-component system), which appears in shallow water theory, by using the theory of dynamical systems and singular traveling wave techniques developed in [Li & Chen, 2007] under different parameter conditions in the two-parameter space. Some explicit exact parametric representations of the solitary wave solutions, smooth periodic wave solutions, periodic peakons, as well as peakon solutions, are obtained. More interestingly, it is found that the so-called $u$-traveling wave system has a family of pseudo-peakon wave solutions, and their limiting solution is a peakon solution. In addition, it is found that the $u$-traveling wave system has two families of uncountably infinitely many solitary wave solutions and compacton solutions.

The paper proposes a novel input–output approach to characterize the dynamical properties of a class of circuits composed by a linear time-invariant two-terminal element coupled with one of the ideal memelements (memory elements) introduced by Prof. L. O. Chua, i.e. memristors, memcapacitors, and meminductors. The developed approach permits to readily determine the conditions under which the dynamics of any circuit of the class admits a first integral. It is also shown that the circuit dynamics can be obtained by collecting the dynamical behaviors displayed by a canonical reduced-order input–output system subject to a constant input of any amplitude. Notably, the reduced-order system exactly describes the dynamics of a circuit forced by a constant generator and with a nonlinear memoryless element in place of the memelement. The relation between the proposed input–output approach and the available state space ones (e.g. Flux-Charge Analysis Method (FCAM)) is also addressed. The main result is that explicit expressions of the invariant manifolds can be directly obtained in the voltage–current state space. Finally, it is shown how the approach also applies to circuits which contain forcing generators. It is believed that the proposed input–output approach can be a useful alternative to state space methods for studying multistability and control issues.