International Journal of Bifurcation and Chaos

This paper focuses on the magneto-aero-elastic nonlinear resonances, bifurcations and chaos for superharmonic and subharmonic responses of conductive circular plates spinning in air and magnetic fields. The electromagnetic forces acting on in-plane are presented by using electromagnetic field theory, and a rotational dissipative damping aerodynamic model is employed to characterize the aerodynamic forces acting on the plates. Considering the effects of the geometric nonlinearity and centrifugal tensioning, the magneto-aero-elastic governing equations of spinning conductive circular plates are derived by the Hamiltonian principle. Based on the mode shapes with Bessel functions, the transverse nonlinear vibration differential equations of spinning plates with clamped-edge are derived by the Galerkin method. Amplitude-frequency equations for subharmonic and superharmonic responses are obtained by using the approach of multiple scales, respectively. Numerical calculations indicate the influences of important parameters (e.g. magnetic field intensity, excitation amplitude and rotational speed) on amplitude frequency characteristics. Moreover, it is demonstrated by the numerical results that system responses are sensitive to the variation of these important parameters, where the complex intermittency, period-doubling and chaotic dynamical behaviors exist.

In this paper, the peculiarities of a quadratic map are investigated, unveiling the hidden relationship with the classical logistic map. Moreover, the improvements in generating odd-periodic oscillations and the related intermittent behavior using the shifted logistic formulation are shown from an experimental point of view.

The unexpected subharmonic oscillation with an evident asymmetric feature is one of the most salient dynamical behaviors in the grid-connected inverter, which will bring disaster to normal system operation. In view of time-varying equilibrium points and complex dynamical response of the system, this asymmetrical subharmonic oscillation is difficult to predict through traditional analysis methods. In this paper, a ripple-based current model is proposed to describe the special subharmonic oscillation of single-phase grid-connected DC–AC inverter with the One-Cycle Compensation (OCC) method. Based on the proposed model, the influence of dynamic response and the time-varying equilibrium points on the stability of the system are comprehensively analyzed, and the internal mechanism of asymmetric subharmonic oscillation is revealed. Furthermore, the critical points of the subharmonic oscillation at the beginning and the end of the whole circuit cycle are predicted. Particularly, a global stable operation boundary is identified from the ripple-based analysis to optimize circuit design. Finally, circuit experiments are performed to verify these numerical and theoretical results.

In this work, a sufficiently simple quorum sensing model allows one to perform detailed analytic studies to gain insights into the dynamical mechanisms in Pseudomonas aeruginosa. It is shown that an optimal rate of model parameters is essential to induce oscillations without considering time delays. Theoretical analysis and numerical simulation reveal that the delays can induce subcritical Hopf bifurcation and oscillation hysteresis. By using the center manifold and normal form theory, the explicit formulas for determining the stability and direction of periodic solutions bifurcating from Hopf bifurcations are derived. Numerical results show that the global periodic solutions bifurcating from the equilibrium exist when the delay is faraway from the first critical value. Moreover, the length of the delay can determine the amplitudes and the periods of the oscillations. A two-parameter diagram of delays is given to illustrate their crucial roles in coordinating and regulating oscillatory dynamics of the system. These results may help to further understand the dynamics of quorum sensing system in Pseudomonas aeruginosa and provide beneficial guidelines in the process of bacterial delivery of drugs.

Any singular irreducible cubic curve (or simply, cubic) after an affine transformation can be written as either $y^2=x^3$, or $y^2=x^2(x+1)$, or $y^2=x^2(x-1)$. We classify the phase portraits of all quadratic polynomial differential systems having the invariant cubic $y^2=x^2(x+1)$. We prove that there are $63$ different topological phase portraits for such quadratic polynomial differential systems. We control all the bifurcations among these distinct topological phase portraits. These systems have no limit cycles. Only three phase portraits have a center, 19 of these phase portraits have one polycycle, three of these phase portraits have two polycycles. The maximum number of separartices that have these phase portraits is 26 and the minimum number is nine, the maximum number of canonical regions of these phase portraits is seven and the minimum is three.

In this paper, the hybrid-scale bifurcation evolutionary phenomena in one-cycle controlled (OCC) single-inductor dual-output (SIDO) Buck DC–DC converter are investigated thoroughly. First, the single-cycle and crossing-cycle iterative mapping models of the OCC SIDO converter in four operation modes are derived to describe the dynamical information under different topology switching sequences. Second, the evolutionary trend of hybrid-scale bifurcation behaviors is identified by numerical simulation. Third, bifurcation analysis based on the proposed models is performed by the trajectories of the eigenvalues, and significantly, the occurrence mechanism of hybrid-scale bifurcations in the evolutionary phenomena is explained in detail. And then, the participation factor is used to analyze the relationship between the variables and the eigenvalues. Especially, the relative parameter sensitivity of the iterative mapping model is proposed to evaluate the influence of such key parameters as load resistors on the eigenvalues so that plenty of design-oriented information will be provided to avoid the hybrid-scale bifurcation behaviors. Finally, PSpice circuit simulation and hardware experiments are carried out to verify the analysis results.

We consider a Kuramoto model of four-coupled oscillators of nonidentical initial frequencies. Under the influence of coupling, the oscillators fall into a full phase locked state of a common frequency when the coupling strength surpasses a threshold value. We find numerically the parameters that control this distinguishable coupling constant at the moment the oscillators transit into an entire frequency synchronization when a complete phase lock state takes place. We are able to set apart a recognizable phase condition at the fully locked state. This phase condition helps to derive an analytic formula to calculate the coupling factor as soon as the oscillators depart to a full phase locking state. The explicit expression of the edge coupling factor is given in terms of the initial frequencies of the four oscillators. The formula valid for calculating the distinct coupling allows to find mathematical expressions to calculate the phase differences when the four-coupled phase oscillators are strictly at the full phase locked state and have a common frequency synchronization.

It is of critical importance to comprehend the biological environment and core tumor populations when trying to design successful therapeutic solutions for fighting cancers. In several diseases, G9a has been recognized as a novel epigenetic therapeutic target, and its blockage can shift tumor cells (TCs) toward tumor propagating cells (TPCs). This study combines mathematical modeling based on ordinary differential equations and dynamical analysis to quantitatively and qualitatively understand the interactions among G9a, TCs, and TPCs, denoted as G9a-TC-TPC. We propose four different dynamical systems with the impact of the strong Allee effect, named the Hill–Hill system, Logistic–Logistic system, Hill–Logistic system and Logistic–Hill system, to simulate different biological processes through the Hill functions and the Logistic functions that are often used in the models of biological systems. Based on theoretical analysis of these models, including the positivity, boundedness and stability of equilibria, we find that the Hill–Logistic system can display bistable states that correspond to the wild-type tumors and the aggressive tumors. Consequently, we use bifurcation analysis and numerical simulations to illustrate the complicated dynamical behavior of this system. It has been shown that under a specific therapy that changes the relative apoptotic rate of TCs (G9a suppresses the apoptosis of TCs), which can affect the bistability and instability of the system, the wide-type state can be obtained. We also discover that the relative handling time of TCs and TPCs can cooperatively enhance bistability, whereas the cooperative coefficient of feedback can contribute to all tumor cells moving from high-level monostability to bistable states in a restricted region, then to low-level stable states. These results offer new insights for more precisely understanding epigenetic therapy treatments with G9a.

Solitary wave solutions of two-component Drinfel’d–Sokolov–Wilson system with Kuramoto–Sivashinsky perturbation are considered. We first employ geometric singular perturbation theory to reduce the higher-dimensional system of equations to the perturbed planar system. We then further exploit the Melnikov method to explore the persistence of one homoclinic orbit, and the generation of a new homoclinic orbit, indicating the existence of single- and double-peak solitary waves. Of particular interest is the appearance of the double-peak solitary wave solution. Finally, we include the numerical simulations to verify the theoretical results.

In this paper, a Leslie–Gower model with weak Allee effect on prey and fear effect on predator is proposed. Compared with the case without fear effect on predator, the new model undergoes richer dynamic behaviors such as saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Also, different from the strong Allee effect on prey, the system with weak Allee effect has bistable attractors which are a largely stable limit cycle and a stable positive equilibrium, two stable equilibria, or a stable limit cycle and a stable trivial equilibrium. When the Allee effect coefficient is intermediate, fear effect on the predator can protect the prey and the predator from being extinguished. The results in this paper can be seen as a complement to those in the literatures about the Leslie–Gower model with Allee effect and fear effect.

Filippov systems are a representative class of piecewise smooth dynamical systems with sliding motion. It is known that such systems can exhibit complex dynamics, but how they generate chaos remains to be further studied. This paper establishes three Shilnikov-type heteroclinic theorems for 3-dimensional (3D) Filippov systems divided by a smooth surface, which admit heteroclinic cycles sliding on the switching surface. These theorems correspond to two typical scenarios of sliding heteroclinic cycles: (i) connecting two saddle-foci; (ii) connecting one saddle and one saddle-focus. In the presence of a sliding heteroclinic cycle, the corresponding Filippov system can be analytically proved to have a chaotic invariant set nearby the singular cycle under some assumed conditions. These results provide a reasonable explanation for the chaotic behaviors of 3D Filippov systems. Two numerical examples are presented to validate the theorems.

The relationship between the producer and the consumer is important in grassland ecosystems. Most producer-consumer models only consider food quantity and focus on the above ground part of producers in producer-consumer interactions, while food quality and the below ground part of the producer can be an important factor in these models. We developed a stoichiometric producer-consumer model, where the producer represents a primary production in terrestrial plant with above ground and below ground parts subject to stoichiometric constraints of carbon and phosphorus. The analysis shows that the dynamic behavior of the model is richer than that of the traditional producer-consumer models. The solution curves of the model can be used to explain the paradox of enrichment and reflect the rapid growth of the vegetation. Our findings help understand and interpret the relationship between the producer and the consumer in grassland ecosystems and provide guidance for ecosystem management on maintaining ecological equilibrium.

Compared with most known singularly degenerate heteroclinic cycles consisting of two different equilibria of a line or a curve, or two parallel lines of semi-hyperbolic equilibria, little seems to be noticed about the one that connects two perpendicular lines of semi-hyperbolic equilibria, i.e. $E_z=(0,0,z)$ and $E_x=(x,ex,k+ef)$, $z,x\in ℝ$, which is found in the mathematical chaos model: $ẋ=a(y-x)+dxz$, $ẏ=kx+fy-xz$, $ż=-e{x}^2+xy+cz$ when $c=0$ and $e(a+fd)=(a-kd)$. Surprisingly, apple-shape attractors are also created nearby that kind of singularly degenerate heteroclinic cycles in the case of small $c>0$. Further, some other rich dynamics are uncovered, i.e. global boundedness, Hopf bifurcation, limit cycles coexisting with one chaotic attractor, etc. We not only prove that the ultimate bound sets and globally exponentially attracting sets perfectly coincide under the same parameters, but also illustrate four limit cycles coexisting with one chaotic attractor, the saddle in the origin, and other two stable nontrivial node-foci, which are also trapped in the obtained globally exponentially attracting set, extending the recently reported results of the Lü-type subsystem. In addition, combining theoretical analysis and numerical simulation, the bidirectional forming mechanism of that kind of singularly degenerate heteroclinic cycles is illustrated, and their collapses may create three-scroll/apple-shape attractors, or limit cycles, etc. Finally, conservative chaotic flows are numerically found in the new system. We expect that the outcome of the study may provide a reference for subsequent research.

In this paper, we investigate the spatiotemporal dynamics of the Sel’kov–Schnakenberg system. The stability of the positive constant steady state is studied by the linear stability theory. Hopf bifurcation and Turing–Hopf bifurcation are generated by varying two parameters in the model. The normal form near the Turing–Hopf singularity is calculated to explore the complex dynamics of the system. Finally, numerical simulations are carried out to verify the theoretical results. Our results show that the Sel’kov–Schnakenberg system exhibits complex dynamics near the Turing–Hopf singularity, including the existence of inhomogeneous steady states, homogeneous periodic solutions and inhomogeneous periodic solutions.

In this paper, a Beddington–DeAngelis prey–predator model with fear effect, refuge and harvesting is investigated. First, the positivity of solutions and boundedness of system are given. Then, the existence and local stability of equilibria of such system are obtained. Next, not only different codimension-one bifurcations, such as saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation take place, but also Bogdanov–Takens bifurcation of codimension-two occurs as predicted by the center manifold theorem and bifurcation theory. Finally, some numerical simulations are carried out to confirm our theoretical results.

For the cubic Camassa–Holm type equation, by using the techniques from dynamical systems and singular traveling wave theory developed by Li and Chen [2007] to analyze its corresponding traveling wave system, it was found that under different parameter conditions, its bifurcation portraits exhibit all possible exact explicit bounded solutions (solitary wave solutions, periodic wave solutions, peakon as well as periodic peakons). A total of 19 explicit exact parametric representations of the traveling wave system of the Camassa–Holm type equation are presented.