International Journal of Bifurcation and Chaos

A feedforward network is a unidirectionally coupled chain of dynamical systems in which the first cell may be coupled to itself, and each successive cell is coupled to the next one. Previous work has shown that a feedforward network of nonlinear oscillators can lead, in principle, to a novel beamforming system, in which the collective dynamics of the network can produce, simultaneously, signal amplification and beam steering capabilities through phase-locking solutions. These two features can find applicability in future design and fabrication of modern antenna and radar systems. We now investigate the response of the feedforward network of oscillators, and the beamforming pattern, to an incoming signal. The study shows, in particular, that synchronization with an incoming signal can yield a beamforming pattern with a narrow mainlobe and lower sidelobes. These are desirable features because they can lead to improved resolution and less susceptibility to interference by strenuous signals.

In this paper, a new vibro-impact bistable oscillator by installing an adjustable rigid wall to one side of the smooth–discontinuous (SD) oscillator, is constructed to analytically and numerically reveal its unique global dynamic characteristics. In the unperturbed case, there exist complex unilateral homoclinic structures transversal intersecting, grazing and nonintersecting with the impact boundary due to the adjustability of the rigid constraint. When weak vicious damping, weak periodic excitation and impact energy dissipation are considered, the influences of the adjustable rigid constraint and the evolutions from coexisting periodic orbits to chaos are analyzed through bifurcation diagrams, phase diagrams and Poincaré map and basins of attraction. As the position of the impact boundary moves, the orbits graze multiple times with the boundary, causing period-doubling bifurcations and presenting some new impacting periodic orbits and disconnected chaotic attractors. The vibro-impact SD oscillator also gradually loses stability due to the rapid migration of attractors among coexistence orbits.

Studies have shown that inhibitory interneurons include inhibitory interneurons on fast time scales (I1) and inhibitory interneurons on slow time scales (I2), both of which can modulate epileptic activities. Based on this, we improve the corticothalamic model by introducing self-inhibition of I1 and I2 to explore the onset and transitions of epileptic activities from the dynamic perspective. First, we investigate the case that the excitatory loop between the excitatory pyramidal neurons (PY) and the thalamic relay neurons (TC) induces epileptic activities in the absence of self-inhibition. The emergence of multistability drives the system to produce a variety of epileptic activities, such as m-spike-wave discharges, tonic and clonic oscillations. Subsequently, we introduce self-inhibition of inhibitory interneurons and find that the advancement or disappearance of the Hopf bifurcation leads to a reduction in the area of epileptic discharges. Finally, we explore epileptic activities induced by the self-inhibition of I1 and the excitability of TC on I1. The strong self-inhibition can effectively modulate seizures, but excessive exchange of information between TC and I1 induces absence seizures. In addition, the paper focuses on revealing the loop mechanisms and related bifurcation characteristics in various situations. Besides Hopf and fold limit cycle bifurcations, period-doubling and homoclinic bifurcations are also key factors leading to the state transitions. These may provide some theoretical guidance for the generation and evolution of seizures.

A Filippov–Gause predator–prey model is proposed, which suggests that cooperation and competition among predators depend on prey abundance. In this case, when the prey abundance is high, predators cooperate in hunting, but when the prey abundance decreases and falls below a critical level, predators compete for scarce food. The proposed model is analyzed in terms of the existence and stability of its equilibria and its pseudo-equilibrium, which is located on the differentiable curve separating the two vector fields. Given the conditions found which determine the existence of cycles and limit cycles, contrasted with a bifurcation diagram, we find that the proposed model can have an asymptotically stable limit cycle around the pseudo-equilibrium and two positive inner equilibria, together with an unstable limit cycle around the pseudo-equilibrium. Similarly, if the model has a pseudo-equilibrium and no positive inner equilibria, then we could have at least two limit cycles, one stable and one unstable, surrounding the pseudo-equilibrium.

This paper focuses on the dense uniform Li–Yorke chaos for linear operators on a Banach space. Some sufficient conditions and equivalent conditions are established under which the dynamical system is densely uniformly Li–Yorke chaotic. It is shown that there are plenty of densely uniformly Li–Yorke chaotic operators. For unilateral backward weighted shifts and bilateral backward weighted shifts on ${\ell }^p$, it is shown that Li–Yorke chaos is equivalent to dense uniform Li–Yorke chaos.

Fractals and chaotic systems play an important role in science and engineering. However, few scholars have focused on further deep integration of multiple fractals and multi-level cascaded chaotic systems to create new discrete chaotic maps. In this paper, we propose an innovative method to construct a series of new fractal chaotic maps by borrowing the concept of Julia fractal. Unlike previous fractal discrete map designs, this method enriches the dynamical behavior of the new maps by introducing the topological complexity of multiple fractal transformations into chaotic systems in terms of system structure. More importantly, it ensures that the new system dynamical structure satisfies the necessary condition for generating chaos through an internal feedback mechanism, i.e. the initial error is continuously transmitted in each state of the system iteration. The paper applies the method to eight types of discrete chaotic maps, generating various fractal chaotic maps, revealing new chaotic attractors as a result, and discussing the dynamical characteristics of discrete fractal chaotic maps through Lyapunov exponent, information entropy, spectral entropy, C₀ complexity and bifurcation diagram analysis. The dynamical analysis indicates that the proposed fractal chaotic maps have higher complexity and stronger randomness. Consequently, the method proposed in this paper for generating new discrete chaotic maps through Julia fractal offers a novel approach to the design of chaotic systems, with promising potential for engineering and scientific applications.

We establish a long-term dynamic Stackelberg game model based on dual-channel retailing, taking into account the manufacturer’s dynamic quality decision-making. The model involves a manufacturer who has a self-built online platform and a retailer, and the complex characteristics of the system when the manufacturer makes dynamic quality decision-making are explored. We further analyze the supply chain system by exploring the stable region, global bifurcation and multistability of the dynamic system, the bifurcation diagrams, 3D-profit diagrams, and consumer utility. The conclusions are as follows: (1) There are local steady and multisteady states in the dynamic system, and the adjustment parameters of all decision variables should be kept within an appropriate range to figure out the dynamic evolutionary orbits and reduce the uncertainty. (2) The development of the supply chain is hampered by excessive channel rivalry. (3) The market-dominant manufacturer must strike a balance between product quality and the self-built online platform’s business focus. (4) While a chaotic condition of the system might lead to greater efficiency for supply chain participants, major disruptions to supply chain operations can also result from market uncertainty. (5) The chaotic behavior can be effectively controlled using the delay feedback control method.

In this paper, we scrutinize the dynamics of a temporal and spatiotemporal prey–predator model incorporating the fear effect on prey and team hunting by the predator. Additionally, we explore the influence of delayed anti-predation response. The analysis includes discussions on well-posedness, local stability, and various bifurcations such as saddle-node, transcritical, Hopf and Bogdanov–Takens bifurcations. The impact of fear cost and delay parameters on model dynamics is investigated by considering them as bifurcation parameters. We investigate how bifurcation values change with varying parameters by exploring different bi-parameter planes. It is observed that the system transitions into chaotic behavior through Hopf bifurcation for significant anti-predation response delay. The positivity of the maximal Lyapunov exponent indicates the confirmed characteristics of chaotic behavior. Furthermore, within the spatiotemporal model framework, a thorough analysis of local and global stability is provided, including the establishment of criteria for identifying Turing instability in cases of self- and cross-diffusion. Various stationary and dynamic patterns are elucidated as diffusion coefficients vary, showcasing the diverse dynamics of the spatiotemporal model. In order to illustrate the dynamic characteristics of the system, a series of comprehensive numerical simulations are conducted. The discoveries outlined in this paper could prove advantageous for understanding the biological implications resulting from the examination of predator–prey relationships.

A fractional-order proportional-derivative controller is designed to address bifurcation issues in a dual-time-delay fractional-order predator–prey model, achieving anti-control of Hopf bifurcation at equilibrium of the system. By selecting different delays as bifurcation parameters, the stability and Hopf bifurcation conditions of the controlled system are derived. The results show that the fractional-order, delays and control parameters play an important role in determining the stability and Hopf bifurcation of the system. By selecting reasonable system parameters (fractional-order, delays, and control parameters), effective system control strategies can be devised. Finally, the key findings of this study are verified through numerical examples. The research in this paper has potential application value for the management of natural ecosystems.

For linear reaction–diffusion equations, a general geometric singular perturbation framework was developed, to study the impact of strong, spatially localized, smooth nonlinear impurities on the existence, stability, and bifurcation of localized structure, in the paper [, ]. The multiscale nature enables deriving algebraic conditions determining the existence of pinned single- and multi-pulses. Moreover, linearity enables treating the spectral stability issue for pinned pulses similarly to the problem of existence. In this paper, we move one step further to treat a special type of nonlinear reaction–diffusion equation with the same type of impurity. The additional nonlinear term generates richer and more complex dynamics. We derive algebraic conditions for determining the existence and stability of pinned pulses in terms of Legendre functions.

In this work, switching of vibrations between different modes of a class of multistable systems with parametric excitations is studied. An intermittent control method without altering the underlying dynamics of the system or its coexisting attractors is proposed. When an orbit meets a proximity constraint with the desired orbit, the control action is applied intermittently in the time domain. This control input is operated by perturbing one of the attractors with an impulsive force, thereby switching the system response to the other attractor. The stability of the method is analyzed theoretically, and its effectiveness is verified by numerical results. Additionally, the influences of neighborhood boundary on the amplitude and duration of control inputs in unconstrained and constrained intermittent control methods are also analyzed.

This paper deals with the dynamic behaviors of a discrete-time fractional-order predator–prey model in the presence of both the Allee effect and immigration on the prey population and in the presence of harvesting on the predator population. The existence and uniqueness and parametric conditions for local asymptotic stability of fixed points of the discrete-time fractional-order model are studied. Moreover, using the center manifold theorem and bifurcation theory, it is shown that the considered model undergoes flip and Neimark–Sacker bifurcations in a small neighborhood of the interior fixed point. Then, the direction of bifurcation is calculated. A feedback controller is implemented in the proposed model to control chaos thanks to the emergence of the Neimark–Sacker bifurcation. Furthermore, numerical analysis confirms the theoretical analysis with the help of Matlab software.

This paper investigates a Leslie–Gower predator–prey model with simplified Holling type IV functional response and constant-yield prey harvesting. We analyze conditions for the existence of positive equilibria and prove that the system has at most four positive equilibria. The results show that the double positive equilibrium is a cusp of codimension at most 4, the triple positive equilibrium is a degenerate saddle or nilpotent focus of codimension-3, and the quadruple positive equilibrium is a nilpotent cusp of codimension-5. In addition, as the parameters vary, the system can undergo a cusp-type (or focus-type) degenerate Bogdanov–Takens bifurcation of codimension-4 (or codimension-3). Furthermore, the positive equilibrium is a weak focus of order at most 4, and the model can undergo a degenerate Hopf bifurcation of codimension-4. Finally, our main results are verified by some numerical simulations, which also reveal that there exist three limit cycles containing one positive equilibrium, or one (or two) limit cycles containing three positive equilibria, or a limit cycle as well as a homoclinic loop.

Many physical systems are represented by Partial Differential Equations (PDEs), and the study of chaotic dynamics in these systems is interesting and challenging. In this paper, the Li–Yorke chaos of PDEs is studied, and the Li–Yorke chaos is observable in several classes of PDEs, including systems with or without energy injection. For the PDEs without energy injection, three kinds of PDEs are investigated revealing the existence of Li–Yorke chaos, including a type of transport equations, a class of wave equations, and a kind of Navier–Stokes equations. For the PDEs with energy injection, only dissipative type of PDEs are studied, including a model of sound variation of the drum with damping, a kind of reaction–diffusion equations, and a class of two-dimensional Navier–Stokes equations. It is shown that Li–Yorke chaos is well defined for the characterization of the complicated dynamics of these systems. In particular, a physical explanation about chaos in such PDEs is provided, which gives an interesting explanation of acoustic chaos.

For the purpose of interpreting and controlling the irregular yaw motion of very large ships in waves, both analytical and numerical methods are employed in this study. A nonlinear dynamic model considering the behaviors of the human navigator for the yawing system of the ships in regular longitudinal waves is constructed and analyzed based on the bifurcation and chaos theory. The results of analysis indicate the existence of chaotic regions for the system, interpreting the irregular yaw motion of very large ships. Furthermore, to mitigate the chaotic yaw phenomenon of the subject ship, a concise robust integrator backstepping algorithm, incorporating the close-loop gain shaping algorithm is proposed based on the nonlinear dynamic model. It possesses a concise form and is easily implementable in marine practice. Comparative numerical experiments are conducted to validate the effectiveness and superiority of the proposed algorithm.