International Journal of Bifurcation and Chaos

Nonlinear bifurcations and instabilities of autonomous nonconservative systems, mainly involving the fluid loading of a solid or structure, are reviewed and described in this accessible, pictorial overview. In contrast to the earlier papers in this series (focusing on the instability of elastic deformable systems, and low-order periodically-forced mechanical systems), we focus on a handful of case studies in which the loss of stability is primarily driven by nonconservative forces, i.e. path-dependent forces not associated with a potential. Many systems involving fluid-structure interaction can lose stability under changing conditions in which there is a net flow of energy from the fluid to the structure, sometimes resulting in growing oscillatory behavior. Again, the generic manifestations of instability typically occur within the framework of bifurcation theory. Progression is from simple local bifurcations to more complex global events, and all are related to instructive and intriguing applications. Hopf bifurcations are presented in the context of the galloping and flutter of cables and pipes. Next, Neimark bifurcations appear in aircraft applications involving the free-play fluttering of aerofoils and the wing rock of the Harrier jump-jet. Turning to ships in wind and waves, a homoclinic saddle connection governs the surging and surf-riding of a vessel in stern seas, while an omega flow explosion can compromise the course-keeping of a passenger ferry in a side wind. Recent work on the dynamic step-buckling of a spherical shell illustrates the role of a center manifold, and the paper ends with a careful study of dissipation-generated instabilities, drawing on the historical struggles to understand the evolution of spinning liquid planets.

P53 plays a vital role in DNA repair, and several mathematical models of the p53-Mdm2 feedback loop were used to explain the biological mechanism. In this paper, a p53-Mdm2 model described by a delay reaction–diffusion equation is studied both analytically and numerically. This research aims to provide an understanding of the impact of delay and sustained pressure on the p53-Mdm2 dynamics and tries to explain some biological mechanism. It is found that the type of pattern formation is affected by Hopf bifurcation. Also, the amplitude equation in delay diffusive system is derived and it is shown that sustained stress plays an essential role in the function of p53. Finally, simulation is used to verify the theoretical results.

In this work, we consider a continuous two-dimensional piecewise linear family of maps in which the fixed point and other cycles undergo a center bifurcation, the analogue of a Neimark–Sacker bifurcation, which has a particular structure when occurring in piecewise linear maps. Our goal is to determine and characterize the occurrence of a center bifurcation of supercritical or subcritical type. We show that for the fixed point, a determinant role is played by the virtual fixed point of the system. Also for other $k$-cycles the virtual fixed point plays a determinant role, as well as the saddle cycles related to the cycle which undergoes a center bifurcation. A particular bifurcation point separating the two transitions, supercritical and subcritical, is related to a conservative system (the analogue of the Chenciner point in smooth maps).

In the present study, we analyze the dynamics of a four-dimensional generalized Lorenz system with one variable describing the profile of the magnetic field induced in a convected magnetized fluid. In particular, we identify the subcritical Hopf bifurcation, at which the dimension of the unstable manifold is increased or reduced by two. Moreover, the new four-dimensional system behavior depending on the control parameters is considered and bidirectional bifurcation structures are revealed. The results show the existence of several windows of nonchaotic variation (windows of order), in particular period-3 windows at the edge of which type I intermittency is observed.

This paper studies a dual-channel apparel supply chain faced with strategic consumers and uncertain demands, where the dominant retailer decides its service level and offline price first, and the manufacturer decides the online price later on. This paper develops a framework to examine the impacts of quick response on the system’s short-term profitability and long-term stability. To make comparisons, two scenarios are considered, in which the supply chain system adopts and does not adopt quick response, respectively. The optimal decisions and profits of the firms are first analyzed in two scenarios, and dynamic game models are then developed to study the complex characteristics. Comparisons in supply chain outputs and system stability are further made between these two scenarios. The results show that the imbalance of risks in two channels affects the performance of quick response that tends to benefit the retailer more, while it benefits the manufacturer only when the return rate is extremely high and consumers care much about the purchase channels. However, quick response certainly contributes to the system stability, which not only provides a larger stable region for firms to adjust their decisions, but also makes the system less sensitive to the consumers’ valuation for the product. Along with this, it is easier for the chaotic system with quick response to restore the stability.

A deceptively simple difference system of a bouncing ball is investigated. Applying the center manifold theorem and the normal form analysis, we first give the parameter conditions for the flip bifurcation. Then we discuss the 1:2 resonance. Because the nondegeneracy conditions are not satisfied, we use the approximation of mappings by a flow and change this difference system into an ordinary differential system with dimension two. With the aid of the Melnikov method, the homoclinic bifurcation and the Poincaré bifurcation are analyzed, which imply the existence of the invariant circles for the difference system. Furthermore, we compute the normal forms to provide the parameter conditions for the Chenciner bifurcation and also obtain the stability of the fixed point. Finally, some numerical simulations are presented to verify the obtained results.

In this paper, the stability and evolutionary trend of Hopf bifurcations are investigated thoroughly for the double-input SEPIC DC–DC converter with one-cycle control (OCC), which is widely used in hybrid power systems. By means of cycle-by-cycle simulations, the evolutionary trend of bifurcation behaviors is identified. The inherent mechanisms of two types of bidirectional Hopf bifurcations are uncovered based on the proposed averaged model. Moreover, stability boundaries in the parameter space are derived with the help of eigenvalue analysis to optimize the circuit design. Furthermore, some important indices with respect to coupling effects are evaluated to assess the coupling effects in the system. Finally, PSpice circuit experiments are performed to verify the bifurcation analysis.

In this paper, we consider a tritrophic food chain model with Holling functional response types III and IV for the predator and superpredator, respectively. The main attention is focused on the stability and bifurcation of equilibria when the prey has a linear growth. Coexistence of different species is shown in the food chain, demonstrating bistable phenomenon. Hopf bifurcation is studied to show complex dynamics due to multiple limit cycles bifurcation. In particular, normal form theory is applied to prove that three limit cycles can bifurcate from an equilibrium in the vicinity of a Hopf critical point, yielding a new bistable phenomenon which involves two stable limit cycles.

A new higher-dimensional hyperchaotic map, named Sine Chebyshev modulation map (SCMM), is proposed based on the closed-loop modulation coupling (CMC) model. The dynamics of the SCMM are evaluated by attractor diagrams, Lyapunov exponents and bifurcations. By designing a piecewise-linear function, the SCMM is expanded to the grid multi-cavity form. Dynamical behaviors of the two-dimensional grid multi-cavity SCMM are then analyzed. The results show that it has rich dynamical characteristics, including complicated phase space trajectories, hyperchaotic behaviors, large maximum Lyapunov exponent and typical bifurcations. The complexity of the new grid multi-cavity hyperchaotic map is large in the entire parameter space. Finally, digital circuits of 2D-SCMM and 2D grid multi-cavity SCMM are implemented based on the DSP technique. The feasibility of the circuit is verified, and it lays the foundation for applications in chaotic secure communication.

Recently, some field experiments and studies show that predators affect prey not only by direct killing, they induce fear in prey which reduces the reproduction rate of prey species. Considering this fact, we propose a mathematical model to study the fear effect and prey refuge in prey–predator system with gestation time delay. It is assumed that prey population grows logistically in the absence of predators and the interaction between prey and predator is followed by Crowley–Martin type functional response. We obtained the equilibrium points and studied the local and global asymptotic behaviors of nondelayed system around them. It is observed from our analysis that the fear effect in the prey induces Hopf-bifurcation in the system. It is concluded that the refuge of prey population under a threshold level is lucrative for both the species. Further, we incorporate gestation delay of the predator population in the model. Local and global asymptotic stabilities for delayed model are carried out. The existence of stable limit cycle via Hopf-bifurcation with respect to delay parameter is established. Chaotic oscillations are also observed and confirmed by drawing the bifurcation diagram and evaluating maximum Lyapunov exponent for large values of delay parameter.

In this paper, we study the periodic solutions of high order differential delay equations with $2k$ lags. And the $2(2k+1)$-periodic solutions are obtained by using the variational method and the method of Kaplan–Yorke coupling system. This is a new type of differential delay equations compared with all previous researches. And it provides a theoretical basis for the study of differential delay equations. An example is given to demonstrate our main results.

The Lorenz system has attracted increasing attention on the issue of its simplification in order to produce the simplest three-dimensional chaotic systems suitable for secure information processing. Meanwhile, Sprott’s work on elegant chaos has revealed a set of 19 chaotic systems all described by simple algebraic equations. This paper presents a new piecewise-linear chaotic system emerging from the simplification of the Lorenz system combined with the elegance of Sprott systems. Unlike the majority, the new system is a non-Shilnikov chaotic system with two nonhyperbolic equilibria. It is multiplier-free, variable-boostable and exclusively based on absolute value and signum nonlinearities. The use of familiar tools such as Lyapunov exponents spectra, bifurcation diagrams, frequency power spectra as well as Poincaré map help to demonstrate its chaotic behavior. The novel system exhibits inverse period doubling bifurcations and multistability. It has only five terms, one bifurcation parameter and a total amplitude controller. These features allow a simple and low cost electronic implementation. The adaptive synchronization of the novel system is investigated and the corresponding electronic circuit is presented to confirm its feasibility.

The negative or hyperpolarization pulse stimulation induces action potential, i.e. the post-inhibitory rebound spike, which has been widely observed in various single neurons with hyperpolarization-activated cation current ($I_{\mathrm{\text{h}}}$) in neuroscience and is suggested to be evoked from a focus near the Hopf bifurcation according to the traditional viewpoint of nonlinear dynamics. In the present paper, a novel viewpoint that post-inhibitory rebound spike can be evoked from a stable node near the saddle-node bifurcation on invariant circle (SNIC) is proposed, which can be well interpreted with hyperpolarization activation characteristic of $I_{\mathrm{\text{h}}}$ current, bifurcation analysis, and threshold. Especially, the boundary between the subthreshold and suprathreshold initial values which respectively evoke subthreshold potential and action potential is acquired to be a threshold surface containing the saddle. $I_{\mathrm{\text{h}}}$ current after the negative pulse stimulation for small conductance $g_{\mathrm{\text{h}}}$ of $I_{\mathrm{\text{h}}}$ is low enough to evoke just a subthreshold potential while for large $g_{\mathrm{\text{h}}}$ is high enough to evoke a post-inhibitory rebound spike. For small $g_{\mathrm{\text{h}}}$, the pulse induces the decrease of membrane potential $V$ and then the phase trajectory always stays within the subthreshold initial value region locating lower to the threshold surface with a nearly fixed $V$ value. For large $g_{\mathrm{\text{h}}}$, the threshold surface changes and is composed of two parts: one part with a nearly fixed $V$ value and the other with a nearly fixed value of $H$ variable to describe $I_{\mathrm{\text{h}}}$ inactivation probability. Although the negative pulse stimulation induces the decrease of $V$, $H$ increases to a level high enough and then the phase trajectory runs across the part with a nearly fixed $H$ value to form a post-inhibitory rebound spike. The appearance of the novel $H$ threshold is the internal dynamical mechanism for the generation of post-inhibitory rebound spike, and the external cause is that the negative pulse stimulation induces the phase trajectory to run across the $H$ threshold surface. The results present a novel nonlinear phenomenon and the corresponding dynamical mechanism related to post-inhibitory rebound spike induced by $I_{\mathrm{\text{h}}}$ current near the SNIC bifurcation point.

In this paper, we propose a discrete Lotka–Volterra predator–prey model with Holling type-I and -II functional responses. We investigate the stability of the fixed points of this model. Also, we study the effects of changing each control parameter on the long-time behavior of the model. This model contains a lot of complex dynamical behaviors ranging from a stable fixed point to chaotic attractors. Finally, we illustrate the analytical results by some numerical simulations.

In this paper, a method for pattern analysis in networks of diffusively coupled nonlinear systems of Lur’e form is presented. We consider a class of nonlinear systems which are globally asymptotically stable in isolation. Interconnecting such systems into a network via diffusive coupling can result in persistent oscillatory behavior, which may lead to pattern formation in the coupled systems. Some of these patterns may coexist and can even all be locally stable, i.e. the network dynamics can be multistable. Multistability makes the application of common analysis methods, such as the direct Lyapunov method, highly involved. We develop a numerically efficient method in order to analyze the oscillatory behavior occurring in such networks. We focus on networks of Lur’e systems in which the oscillations appear via a Hopf bifurcation with the (diffusively) coupling strength as a bifurcation parameter and therefore display sinusoidal-like behavior in the neighborhood of the bifurcation point. Using the describing function method, we replace nonlinearities with their linear approximations. Then we analyze the system of linear equations by means of the multivariable harmonic balance method. We show that the multivariable harmonic balance method is able to accurately predict patterns that appear in such a network, even if multiple patterns coexist.

If we change the upper and lower densities in the definition of distributional chaos of a continuous linear operator on a Banach space $X$ by the Banach upper and Banach lower densities, respectively, we obtain Li–Yorke chaos. Motivated by this, we introduce the notions of reiterative distributional chaos of types $1$, $1^{+}$ and $2$ for continuous linear operators on Banach spaces, which are characterized in terms of the existence of an irregular vector with additional properties. Moreover, we study its relations with other dynamical properties and present the conditions for the existence of a vector subspace $Y$ of $X$, such that every nonzero vector in $Y$ is both irregular for $T$ and distributionally near zero for $T$.

To explore the feasibility of physiological manipulation of autaptic structures, the effects of autaptic connections on an FHN-ML neuronal system with phase noise stimulation are studied systematically. Firstly, according to the dynamic analysis of the FHN-ML neuron model, a saddle-node bifurcation can occur on an invariant circle. Under the action of external oscillatory current with phase noise, the neuronal firing activity is sensitive to phase noise with less intensity, and an appropriate noise intensity can induce a significant stochastic resonance phenomenon. Secondly, the chemical autaptic function can effectively regulate the neuronal discharge activity. An inhibitory autapse can not only induce the transition from depolarized resting to periodic spiking, but can also induce the FHN-ML neuron suppressed by strong phase noise to generate a pronounced intermittent high-level burst-like discharge mode when the autaptic conductance is greater than 0.1. Finally, for a two-dimensional regular FHN-ML neuronal network, a small amount of autaptic structures can induce some special waveforms to restore the propagation of nerve impulses interrupted by phase noise disturbance. This indicates the significant regulation of autapses on spatial patterns of the FHN-ML neuronal network. The study can provide some theoretical guidance for building autaptic structures in local areas to modulate the dynamic behaviors of biological neuronal systems.

This article introduces a new hyperchaotic system of four-dimensional autonomous ordinary differential equations, with only cubic cross-product nonlinearities, which can respectively display two hyperchaotic attractors with only nonhyperbolic equilibria line. Several issues such as basic dynamical behaviors, routes to chaos, bifurcations, periodic windows, and the compound structure of the new hyperchaotic and chaotic system are investigated, either theoretically or numerically. Of particular interest is the fact that the two coexisting attractors of the new hyperchaotic system are symmetrical, and this hyperchaotic system can generate plenty of complex dynamics including two coexisting chaotic or periodic attractors. Moreover, some chaotic features of the attractor are justified numerically. Finally, 0-1 test is used to analyze and describe the complex chaotic dynamic behavior of the new system.

Global analysis is often necessary for exploiting various applications or understanding the mechanisms of many dynamical phenomena in engineering practice where the underlying system model is too complex to analyze or even unavailable. Without a mathematical model, however, it is very difficult to apply cell mapping for global analysis. This paper for the first time proposes a data-driven generalized cell mapping to investigate the global properties of nonlinear systems from a sequence of measurement data, without prior knowledge of the underlying system. The proposed method includes the estimation of the state dimension of the system and time step for creating a mapping from the data. With the knowledge of the estimated state dimension and proper mapping time step, the one-step transition probability matrix can be computed from a statistical approach. The global properties of the underlying system can be uncovered with the one-step transition probability matrix. Three examples from applications are presented to illustrate a quality global analysis with the proposed data-driven generalized cell mapping method.

In this paper, we provide a new method for constructing chaotic Hopfield neural networks. Our approach is based on structuring the domain to form a special set through the discrete evolution of the network state variables. In the chaotic regime, the formed set is invariant under the system governing the dynamics of the neural network. The approach can be viewed as an extension of the unimodality technique for one-dimensional map, thereby generating chaos from higher-dimensional systems. We show that the discrete Hopfield neural network considered is chaotic in the sense of Devaney, Li–Yorke, and Poincaré. Mathematical analysis and numerical simulation are provided to confirm the presence of chaos in the network.

In this paper, a new 3D discrete hyperchaotic system is constructed, and its Lyapunov exponent and approximate entropy are calculated. We adopt the drive-response method and self-adaptive method to make the hyperchaotic system to reach synchronization, and then design an encrypted transmission system based on the synchronization of this discrete hyperchaotic system. In the synchronous hyperchaotic system, the initial value related to the hash value of the speech signal of the chaotic system is designed. Some security analyses are studied in detail.

Conditional symmetry is known as a new regime for providing coexisting duplicate oscillations with opposite polarity. Polarity balance can be obtained from a function for hosting conditional symmetry. In this paper, new cases of chaotic systems with conditional symmetry are coined from 1D and 2D offset boosting based on a suitable polarity adjustment. Conditional symmetric attractors are controlled effectively by a simple absolute value function, where the distance between two coexisting attractors of conditional symmetry is modified linearly by the offset boosting constant, meanwhile the size of the coexisting attractors gets controlled by the slope. Coexisting attractors of conditional symmetry are thereafter implemented based on the develop kit of STM32.