International Journal of Bifurcation and Chaos

Threshold policy is more realistic than continuous control for biological system management. Most related works are devoted to studying a single-threshold value for one single population, thereby avoiding complicated mathematical analysis of the nonsmooth differential equations. Based on the fact that numerical simulations play an important role in analyzing and understanding the intrinsic mechanism of a biological experiment and system, we hereby propose a differential linear complementarity system to reformulate the biological system with threshold policy. Using this method, we can transform a biological system with multiple-threshold values for one or more population to a differential linear complementarity system, where the corresponding dynamics can be investigated numerically by various algorithms for the complementarity problem. Firstly, the well-posedness of solutions of the differential linear complementarity system and its discretized method are derived explicitly. Then we illustrate the application of our approach to two systems which are a population harvesting system with threshold policy and an HIV replication system with threshold therapy, respectively. Numerical results demonstrate that those nonsmooth biological systems exhibit much more complex dynamics than the corresponding smooth systems. These results also validate the effectiveness and simplicity of the method that reformulates a common biological system with multiple-threshold policy by a differential linear complementarity system.

This paper presents a global study of the class $QsnS{N}_{11}$ of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the coalescence of a finite singularity and an infinite singularity. This class can be divided into two different families, namely, $QsnS{N}_{11}(A)$ phase portraits possessing a finite saddle-node as the only finite singularity and $QsnS{N}_{11}(B)$ phase portraits possessing a finite saddle-node and also a simple finite elemental singularity. Each one of these two families is given by a specific normal form. The study of family $QsnS{N}_{11}(A)$ was reported in [Artés et al., 2020b] where the authors obtained $36$ topologically distinct phase portraits for systems in the closure $\overline{QsnS{N}_{11}(A)}$. In this paper, we provide the complete study of the geometry of family $QsnS{N}_{11}(B)$. This family which modulo the action of the affine group and time homotheties is three-dimensional and we give the bifurcation diagram of its closure with respect to a specific normal form, in the three-dimensional real projective space. The respective bifurcation diagram yields 631 subsets with 226 topologically distinct phase portraits for systems in the closure $\overline{QsnS{N}_{11}(B)}$ within the representatives of $QsnS{N}_{11}(B)$ given by a specific normal form. Some of these phase portraits are proven to have at least three limit cycles.

Giving customers queue length information about a service system has the potential to influence the decision of a customer to join a queue. Thus, it is imperative for managers of queueing systems to understand how the information that they provide will affect the performance of the system. To this end, we construct and analyze a two-dimensional deterministic fluid model that incorporates customer choice behavior based on delayed queue length information. Reports in the existing literature always assume that all queues have identical parameters and the underlying dynamical system is symmetric. However, in this paper, we relax this symmetry assumption by allowing the arrival rates, service rates, and the choice model parameters to be different for each queue. Our methodology exploits the method of multiple scales and asymptotic analysis to understand how to break the symmetry. We find that the asymmetry can have a large impact on the underlying dynamics of the queueing system.

As image is an important way of information representation, researchers pay more and more attention on image encryption. In order to improve the performance of image encryption, a novel image encryption scheme based on a hybrid cascaded chaotic system and sectoral segmentation is proposed in this paper. Hybrid cascaded chaotic system has a larger key space, higher complexity, more sensitivity to initial conditions. Four chaotic sequences relevant to a plain image are generated by this system, which strengthen plaintext correlation and the randomness. During the scrambling process, sectoral segmentation focuses on how to extract a sequence from the disk storing data, which can not only reduce the correlation between the three components of the image, but also hide image information to a large extent. Further, a DNA algorithm is used in the diffusion process. Simulation shows that the proposed scheme can effectively resist various attacks and improve the encryption performance.

In this work, several new aspects of the dynamics of the well-known TNC hyperchaotic oscillator are investigated. Numerous novelties appear in this work, namely particular structures of space magnetization, the coexistence of bursting patterns, the coexistence of up to four asymmetric different attractors, offset-boosting, and antimonotonicity. In addition to this interesting and particular combination of features in the TNC oscillator, the control of multistability phenomenon is carried out using linear augmentation control scheme. Finally, knowing that the protection of digital images is of particularly great interest, the complexity of pseudo-random hyperchaotic sequences of the TNC oscillator is combined with the similarity that the DNA coding shares with the binary code to build a new image encryption algorithm with strong robustness and high speed. This algorithm is successfully evaluated using cropping attack, noise attack and differential attack. The results obtained demonstrate that the proposed algorithm is efficient and of good quality.

The effect of retardation along second vector of angular velocity on a new attitude system of quad-rotor unmanned aerial vehicle (QUAV) is examined in this paper. Catastrophic and hovering conditions of rotor dynamics are verified through bifurcation analysis. It is analyzed that disturbed rotor dynamics during yaw maneuvering lead towards the existence of oscillations and hamper the efficiency of attitude system. We have categorically signified the situation where remote controller of attitude system cannot prevent flipping and consequently, shortens the life of QUAV. This type of situation arises in a strong wind friction zone where positive drag force of rotor dynamics is impeded by the magnitude of rotational moment of inertia. Moreover, subcritical and zero-Hopf bifurcations aroused due to retardational effect in attitude system are analyzed with the aid of normal form and averaging theory. Numerical simulations are also provided for validation of analytical results.

In this paper, we study Poincaré bifurcation of a class of piecewise polynomial systems, whose unperturbed system has a period annulus together with two invariant lines. The main concerns are the number of zeros of the first order Melnikov function and the estimation of the number of limit cycles which bifurcate from the period annulus under piecewise polynomial perturbations of degree $n$.

Complexity is an important feature of complex time series. In this paper, we construct a weighted dispersion pattern and propose a new entropy plane using past Tsallis entropy and past Rényi entropy by using weighted dispersion pattern (PTEWD and PREWD, respectively), to quantify the complexity of time series. Through analyzing simulated data and actual data, we have verified the reliability of the entropy plane method. This entropy plane successfully distinguishes American and Chinese stock indexes, as well as developed and emergent stock markets. We introduce PTEWD and PREWD into multiscale settings, which could also well distinguish different stock markets. The results show that the new entropy plane could be used as an effective tool to distinguish financial markets.

By using center manifold theory, Poincaré–Bendixson theorem, spatiotemporal spectrum and dispersion relation of linear operators, the spatiotemporal dynamics of an activator-substrate model with double saturation terms under the homogeneous Neumann boundary condition are considered in the present paper. It is surprising to find that the system can induce new dynamics, such as subcritical Hopf bifurcation and the coexistence of two limit cycles. Moreover, Turing instability in equilibrium mainly generates stripe patterns, while homogeneous periodic solutions mainly generate spot patterns or spot-stripe patterns, where the pattern formations are enormously consistent with the theoretical results. Interestingly, Turing instability can create equilibrium and periodic solution simultaneously in the subcritical Hopf bifurcation, which is the new finding of the diffusion-driven instability. In fact, those theoretical methods are also valid for finding the patterns of other models in one-dimensional space.

Carbon nanotubes (CNTs) are used in various nano-electromechanical systems (NEMS), and the parameters (including the system parameters and the excitation parameters) may result in chaos in these systems. Thus, understanding the mechanism of the chaos arising from NEMS is vital for CNT’s applications. Motivated by this need, the chaotic properties of a single-walled carbon nanotube system resulting from parametric excitation and external excitation are investigated in this paper. The criteria for the existence of the chaotic behavior in the system with periodic and quasi-periodic perturbations are obtained by the homoclinic Melnikov and the second-order average methods. Furthermore, in order to show the connection between periodic motion and complex behavior, the subharmonic periodic solutions, inside and outside the homoclinic loop, are analyzed. The global structure and the saddle-node bifurcation of the unperturbed averaged system are also considered. Finally, the Poincaré section and the transversal intersection of the unstable and stable manifolds are presented to verify the occurrence of chaos or subharmonic solution. The simulation results confirm the correctness of the theoretical analysis.

This paper investigates chaos suppression in widely used indirect field oriented controlled induction machine (IFOCIM) drive system using fixed-time synergetic control method. The complex eighth-order IFOCIM drive system shows chaotic oscillations during a specific range of the integral gain of the PI speed controller. The conventional synergetic control method is improved by employing the fixed-time theory, the characteristics of the designed controller are chattering free and the convergence time is limited within a fixed-time upper bound depending on the controller parameters. The fixed-time synergetic control method is used to converge a selected macro variable to the origin within the fixed time upper bound. The controller can stabilize the chaotic system dynamics in a good way within a short time.

In this paper, we consider a closed-loop supply chain (CLSC) consisting of two suppliers, one manufacturer, one risk-averse retailer and one fair-caring third-party in the presence of supply disruption. We focus on establishing a dynamic Stackelberg game model with bounded rational expectation and analyzing the game evolution process. The effects of key parameters on the Nash equilibrium solutions and their stability are investigated, as well as the complex dynamical behaviors of the CLSC system are explored by using the stability region, bifurcation graph, the largest Lyapunov exponent (LLE), strange attractors, etc. Moreover, the performance of channel members under different values of parameters is researched by utilizing the (average) expected profits or utilities index. The analysis results reveal that the excessive fast adjustment speed of the manufacturer will lead to the system losing stability and falling into chaos. Also, the retailer’s risk aversion and the third party’s fairness concerns have a destabilization effect on the Nash equilibrium point, while the possibility of supply disruption has different effects on the scope of the adjustment speed of decision variables of the manufacturer. Furthermore, in most cases, an over the top adjustment speed of the manufacturer is disadvantageous to all the channel members for more expected profits, but the third-party can achieve a better performance when the system is in periodic state. Finally, the time-delay feedback control method is proposed to eliminate the system chaos.

In this paper, chaotic dynamics of a class of partial difference equations are investigated. With the help of the coupled-expansion theory of general discrete dynamical systems, two chaotification schemes for partial difference equations with polynomial maps are established. These controlled equations are proved to be chaotic either in the sense of Li–Yorke or in the sense of both Li–Yorke and Devaney. One example is provided to illustrate the theoretical results with computer simulations for demonstration.

The timing of interventions plays a central role in managing and exploiting biological populations. However, few studies in the literature have addressed its effect on population stability. The Seno equation is a discrete-time equation that describes the dynamics of single-species populations harvested according to the proportional feedback method at any moment between two consecutive censuses. Here we study a discrete-time equation that generalizes the Seno equation by considering the management and exploitation of populations through the target-oriented chaos control method. We investigate the combined effect of timing, targeting, and control on population stability, focusing on global stability. We prove that high enough control values create a positive equilibrium that attracts all positive solutions. We also prove that it is possible to determine parameter values to stabilize the controlled populations at any preset population size. Finally, we investigate the parameter combinations for which the management and exploitation are optimized in different scenarios.

A fractional-order model is proposed to describe the dynamic behaviors of the velocity of blood flow in cerebral aneurysm at the circle of Willis. The fractional-order derivative is used to model the blood flow damping term that features the viscoelasticity of the blood flow behaving between viscosity and elasticity, unlike the existing fractional models that use fractional-order derivatives to replace the integer-order derivatives as mathematical/logical generalization. A numerical analysis of the nonlinear dynamic behaviors of the model is carried out, and the influence of the damping term and the external power supply on the nonlinear dynamics of the model is investigated. It shows that not only chaos via period-doubling bifurcation is observed, but also two additional small period-doubling-bifurcation-like diagrams isolated from the big one are observed, a phenomenon that needs further investigation.

Many natural phenomena can be modeled as discontinuous dynamical systems separated by a nonregular line. The number and distribution of limit cycles in discontinuous linear systems are important topics for research. In this paper, we focus on the limit cycles created by discontinuous planar piecewise linear systems separated by a nonregular line of center–center type, and prove that such systems have at most two limit cycles, which can be reached. Furthermore, the two limit cycles are nested and intersect the separation line at two points or four points, that is, either both intersect the separation line at two points or one intersects the separation line at two points and the other one at four points.

An autonomous system of ordinary differential equations on the plane with a center-saddle bifurcation is considered. The influence of a class of time damped perturbations is investigated. The particular solutions tending to the fixed points of the limiting system are considered. The stability of these solutions is analyzed by Lyapunov function method when the bifurcation parameter of the unperturbed system takes critical and noncritical values. Conditions that ensure the persistence of the bifurcation in the perturbed system are described. When the bifurcation is broken, a pair of solutions tending to a degenerate fixed point of the limiting system appears in the critical case. It is shown that, depending on the structure and the parameters of the perturbations, one of these solutions can be stable, metastable or unstable, while the other solution is always unstable. The proposed theory is applied to the study of autoresonance capturing in systems with quadratically varying driving frequency.

A directed projection graph of the $n$-dimensional hypercube on the two-dimensional plane is successfully created. Any $n$-variable Boolean function can be easily transformed to an induced subgraph of the projection. Therefore, the discussions on $n$-variable Boolean functions only need to focus on a two-dimensional planar graph. Some mathematical theories on the projection graph and the induced subgraph are established, and some properties and characteristics of a balanced linearly separable Boolean function (BLSBF) are uncovered. In particular, the sub-hypercube decompositions of BLSBF is easily represented on the projection, and meanwhile, the enumeration scheme for counting the number of $n$-variable BLSBFs is developed by using equivalence classification and conformal transformation. With the aid of the directed projection grap constructed in this paper, one can further study many difficult problems in some fields such as Boolean functions and artificial neural networks.

Two-dimensional unstable manifolds of the modified Chen system are constructed at equilibrium solution by “expanding up” along the unstable eigen-direction, hence it is tangent to the unstable eigenspace. In general, unstable manifold expands to the attraction basin of the corresponding limit cycle or attractor. With the introduction of time delay, the two-dimensional unstable manifold of an unstable focus is simulated via expanding solution orbits with restriction condition on the associated foliations. The simulated unstable manifold coincides with the attraction basin of the limit cycle of the delay differential equations.