International Journal of Bifurcation and Chaos

In this article, we present the influence of a Hamiltonian saddle-node bifurcation on the high-dimensional phase space structures that mediate reaction dynamics. To achieve this goal, we identify the phase space invariant manifolds using Lagrangian descriptors, which is a trajectory-based diagnostic suitable for the construction of a complete “phase space tomography” by means of analyzing dynamics on low-dimensional slices. First, we build a Hamiltonian system with one degree-of-freedom (DoF) that models reaction, and study the effect of adding a parameter to the potential energy function that controls the depth of the well. Then, we extend this framework to a saddle-node bifurcation for a two DoF Hamiltonian, constructed by coupling a harmonic oscillator, i.e. a bath mode, to the other reactive DoF in the system. For this problem, we describe the phase space structures associated with the rank-1 saddle equilibrium point in the bottleneck region, which is a Normally Hyperbolic Invariant Manifold (NHIM) and its stable and unstable manifolds. Finally, we address the qualitative changes in the reaction dynamics of the Hamiltonian system due to changes in the well depth of the potential energy surface that gives rise to the saddle-node bifurcation.

In contrast to traditional viewpoint that positive feedback current always enhances neural firing activities, in the present paper, we identify that the excitatory feedback current mediated by memristor can induce negative responses of bursting patterns, which can be well interpreted with bifurcations. For the Hindmarsh–Rose neuron model without memristor, the period-adding bifurcations of bursting patterns and increase of firing frequency can be induced by increasing the excitatory effect of the background current. After introducing a memristor to simulate the biological synapse or electromagnetic induction effect, inverse period-adding or complex bifurcations of bursting patterns are induced by the excitatory feedback current mediated by the memristor. The number of spikes per burst becomes smaller and the firing frequency becomes lower when increasing the positive feedback gain. Such negative responses of bursting patterns to the positive feedback current are demonstrated in a circuit designed with Digital Signal Processor systems of the MatLab software. Furthermore, the underlying bifurcation mechanism of the negative responses to the positive feedback is acquired with fast–slow variable dissection method. With increasing feedback gain, the initial phase of the burst, which corresponds to a saddle-node bifurcation point of the fast subsystem, delays, while the termination phase of the burst, which corresponds to a saddle-homoclinic bifurcation point, remains unchanged. Therefore, the burst becomes narrower with increasing feedback gain, which leads to decrease in the number of spikes within a burst and decrease in firing frequency. The results present a paradoxical nonlinear phenomenon and the dynamical mechanism, which is helpful for understanding the functions of memristor and roles of the electromagnetic induction current.

Consider the class of reversible quadratic systems

The importance of closed-loop supply chains has been widely recognized both in academic communities and in industrial sectors. This paper starts from the traditional supply chains and the new self-supply chain of GREE to extract realistic problems, to mainly investigating two noncooperative dynamic pricing policies in a dual-channel closed-loop supply chain consisting of a manufacturer and a retailer. Then, it studies the influence of different channel power structures on dynamic decisions and their complexities. Furthermore, the reference price affects the purchase decisions of consumers. Therefore, the model takes into account the influence of reference price of the market demands. Results show that the manufacturer who opens up a direct channel can make a huge profit in the game. In the dynamic game evolution process, the game leader is in a more advantageous position when the system is in a stable region; once entering into the bifurcating region or chaotic region, the game follower needs to adjust his price to follow the leader’s decision in order to make a profit. In addition, the system’s stable region becomes smaller when the market demand becomes more sensitive to the difference between the reference price and the actual price. In this model, if the manufacturer acts as a leader, he is in a more advantageous position when the market is sensitive to channel competition in the stable stage while the result is opposite in the unstable stage.

In the present paper, we investigate the impact of fear in an intraguild predation model. We consider that the growth rate of intraguild prey (IG prey) is reduced due to the cost of fear of intraguild predator (IG predator), and the growth rate of basal prey is suppressed due to the cost of fear of both the IG prey and the IG predator. The basic mathematical results such as positively invariant space, boundedness of the solutions, persistence of the system have been investigated. We further analyze the existence and local stability of the biologically feasible equilibrium points, and also study the Hopf-bifurcation analysis of the system with respect to the fear parameter. The direction of Hopf-bifurcation and the stability properties of the periodic solutions have also been investigated. We observe that in the absence of fear, omnivory produces chaos in a three-species food chain system. However, fear can stabilize the chaos thus obtained. We also observe that the system shows bistability behavior between IG prey free equilibrium and IG predator free equilibrium, and bistability between IG prey free equilibrium and interior equilibrium. Furthermore, we observe that for a suitable set of parameter values, the system may exhibit multiple stable limit cycles. We perform extensive numerical simulations to explore the rich dynamics of a simple intraguild predation model with fear effect.

This paper describes a cholera disease transmission model in the human population through the consumption of zooplankton as food by humans. Here the plankton population is classified into two subpopulations such as phytoplankton and zooplankton. Also, human population is classified into two subpopulations such as susceptible human and infected human. The proposed system reflects the impacts of using time delay in the cholera disease transmission. Different possible equilibrium points of our proposed system have been determined. Here local and global stabilities of our proposed system have been analyzed. The existence of Hopf bifurcation has been studied at the interior equilibrium point. The normal form method and center manifold theorem have been used to test the nature of Hopf bifurcation. It is observed that the interior equilibrium is locally asymptotically stable when the time delay in disease transmission term is large, while the change of stability of positive equilibrium will cause a bifurcating periodic solution at the time delay $\tau$ to be at less than its critical value. Finally, some numerical simulation results have been presented for the better understanding of our proposed system.

The dynamics of a discrete nonlinear prey–predator model is studied. The local dynamical results are obtained on asymptotic properties of fixed points and Neimark–Sacker bifurcations. Then global dynamics is studied by finding invariant sets. Also some achievements on attraction are shown for certain trivial invariant sets including a shadowing type result, and estimates on boundedness of orbits. Some ergodic type results are also derived. Certain issues are extended to systems with impulses by showing the influence of impulses on dynamics. Moreover, backward dynamics is investigated as well. All these results are derived analytically and numerical computations are presented to support them.

We give the complete classification of irreducible invariant algebraic curves in quadratic systems from family $(I)$ of the Chinese classification, that is, of differential system

Thanks to the simplicity and robustness of its calculation methods, algorithmic (or Kolmogorov) complexity appears as a useful tool to reveal chaotic dynamics when experimental time series are too short and noisy to apply Takens’ reconstruction theorem. We measure the complexity in chaotic regimes, with and without extreme events (sometimes called optical rogue waves), of three different all-solid-state lasers: Kerr lens mode locking femtosecond Ti:Sapphire (“fast” saturable absorber), Nd:YVO₄ $+$ Cr:YAG (“slow” saturable absorber) and Nd:YVO₄ with modulated losses. We discuss how complexity characterizes the dynamics in an understandable way in all cases, and how it provides a correction factor of predictability given by Lyapunov exponents. This approach may be especially convenient to implement schemes of chaos control in real time.

Recently, an effective method called return maps is proposed for the parameter estimation of chaotic systems. However, high time-consumption limits practical applications. In this paper, we focus on this problem, and an improved return maps method is proposed. It combines the differential evolution algorithm with the return maps method, and simplifies the calculation process of Euclidean distance. Numerical simulations are carried out on two fractional-order chaotic systems, and the other five methods are used as the comparison. Results show that the improved method can accurately estimate the parameters of chaotic systems, and it saves much time than does the classical return maps method. Furthermore, the proposed method also exhibits good anti-noise performance.

This paper deals with an age-structured HIV infection model with logistic growth for target cells and both virus-to-cell and cell-to-cell infection routes. Based on the existence of the infection-free and infection equilibria and some rigorous analyses for the considered model, we study the asymptotic stability of these equilibria via determining the distribution of eigenvalues. We also address the persistence of the solution semi-flow by proving the existence of a global attractor. Furthermore, Hopf bifurcation occurring at the positive steady state is exploited. At last, some numerical examples are provided to illustrate the obtained results.

In this paper, we propose an image encryption algorithm based on random walk and two hyperchaotic systems. The random walk method is adopted to scramble the position of pixels within a block. Furthermore, the permutation operation between blocks is presented to enhance the scramble effect. Thus, high correlation among pixels of original image is broken by permutation. Moreover, the chosen plaintext attack is used to test the anti-attack ability of the proposed algorithm. By analyzing experimental results and comparing with other image encryption algorithms, we show that the proposed algorithm has better performance and higher security.

In this paper, we study linear cellular automata (CAs) on Cayley tree of order $k$ over the field ${𝔽}_p$ (the set of prime numbers modulo $p$). After revealing the rule matrix corresponding to cellular automata on Cayley tree with the null boundary condition, we analyze the reversibility problem of these cellular automata for some given values of $a,b,c,d\in {𝔽}_p$ and the levels $n$ of Cayley tree. The necessary and sufficient conditions for determining whether a CA is reversible or not are demonstrated. Furthermore, we compute the measure-theoretical entropy of the cellular automata which we define on Cayley tree. We show that for CAs on Cayley tree, the measure entropy with respect to uniform Bernoulli measure is infinity.

To improve the robustness and imperceptibility of the existing coverless image information hiding, a generative coverless image information hiding algorithm based on fractal theory is proposed in this paper. Firstly, four fractal image generation methods are analyzed, and the relationship between the coverless information hiding and these methods is discussed. Secondly, based on the fractal image generation algorithm, secret information is hidden by controlling pixel rendering during the generation process. The robustness, imperceptibility, and capability of resisting steganalysis are balanced by adjusting the rendering distance. As it directly generates stego images, this can resist the detection of most existing steganalysis methods. Meanwhile, different capacities can be achieved by adjusting the size of the generated image. Experimental results and analysis show that the proposed scheme can effectively resist steganalysis and has good robustness against various image attacks. Furthermore, it can achieve large capacity, and it has broad prospects for covert communication.

We identify an unusual route to the creation of a strange nonchaotic attractor (SNA) in a quasiperiodically forced interval map. We find that the smooth quasiperiodic torus becomes nonsmooth due to the grazing bifurcation of the torus. The nonsmooth points on the torus increase more and more with the change of control parameter. Finally, the torus gets extremely fractal and becomes a SNA which is termed the grazing bifurcation route to the SNA. We characterize the SNA by maximal Lyapunov exponents and their variance, phase sensitivity exponents and power spectra. We also describe the transition between a torus and a SNA by the recurrence analysis. A remarkable feature of the route to SNAs is that the positive tails decay linearly and the negative tails exhibit recurrent fluctuations in the distribution of the finite-time Lyapunov exponents.

In this paper, we present a criterion for determining the formal Weierstrass nonintegrability of some polynomial differential systems in the plane ${ℂ}^2$. The criterion uses solutions of the form $y=f(x)$ of the differential system in the plane and their associated cofactors, where $f(x)$ is a formal power series. In particular, the criterion provides the necessary conditions in order that some polynomial differential systems in ${ℂ}^2$ would be formal Weierstrass integrable. Inside this class there exist non-Liouvillian integrable systems. Finally we extend the theory of formal Weierstrass integrability to Puiseux Weierstrass integrability.