International Journal of Bifurcation and Chaos最新文献

To exemplify the existence of hidden strange nonchaotic attractors (HSNAs) and transition mechanism, we consider a rational memristive map with additional force. We find that the four-torus bifurcates into the eight-torus through torus doubling as a function of the control parameter. Following that, the formation of strange nonchaotic attractors occurs when increasing the control parameter. As a result, it is clear that the suggested rational maps reach hidden chaotic attractors via HSNAs through the route of torus doubling to torus breakdown. The obtained dynamical transitions are validated further using bifurcation analysis and the largest Lyapunov exponents. In particular, the obtained HSNAs are confirmed through distinct statistical measures including 0–1 test and singular continuous spectrum.

The renowned 2D invertible Hénon map turns into 1D noninvertible quadratic map when its leading parameter $b$ becomes zero. This well-known link was studied by Mira who demonstrated that the bifurcation set of Hénon diffeomorphism is similar to his “box-within-a-box” bifurcation structure of 1D endomorphism. In general, such similarity has not been strictly established, especially in multidimensional cases. In this paper, we proved that the Mira bifurcation structure of a quadratic noninvertible map persists when the parameter increases from zero and the map turns into an invertible multidimensional generalized Hénon map. The changes of periodic and homoclinic orbits and chaotic attractors at this transition are described. We proved the existence of Newhouse regions is different from those Mira boxes that accumulate to the homoclinic bifurcations.

Our paper is a continuation of a previous work referenced as [Katsanikas & Wiggins, 2024b]. In this new paper, we present a second method for computing three-dimensional generating surfaces in Hamiltonian systems with three degrees of freedom. These 3D generating surfaces are distinct from the Normally Hyperbolic Invariant Manifold (NHIM) and have the unique property of producing dividing surfaces with no-recrossing characteristics, as explained in our previous work [Katsanikas & Wiggins, 2024b]. This second method for computing 3D generating surfaces is valuable, especially in cases where the first method is unable to achieve the desired results. This research aims to provide alternative techniques and solutions for addressing specific challenges in Hamiltonian systems with three degrees of freedom and improving the accuracy and reliability of generating surfaces. This research may find applications in the broader field of dynamical systems and attract the attention of researchers and scholars interested in these areas.

In this paper, a tripartite synapse network is constructed to examine external and internal triggering factors of epilepsy transition and propagation in neurons with the Epileptor-2 model. We first explore the external stimuli in the environment that induce epileptic activities and transition behaviors among Ictal Discharges (IDs) and Interictal Discharges (IIDs) states. The higher the strength and abruptness of the stimuli, the more severe is the occurrence of epilepsy within a reasonable range of parameters. Then for the internal triggering factors, the results of the tripartite synapse network, which is improved by combining the Epileptor-2 model with astrocyte by means of ion exchange and new connections, show that astrocytes can transmit normal physiological activity information and filter out abnormal discharge information of neurons. One of the causes for epileptic seizures is the abnormal release of glial neurotransmitters in astrocytes. The excessive release of glutamate causes the discharge state of neurons to transit from nonepileptic to IIDs, IDs and tonic, while adenosine triphosphate can alleviate epilepsy. Meanwhile, the synapse dysfunction of an astrocyte-free network can also lead to seizures, and the epilepsy propagation ability of a tripartite synapse network becomes weaker than that of an astrocyte-free network. Our research is expected to provide some theoretical basis for the therapeutic approach to curing epilepsy in the intracellular and extracellular contexts.

In this paper, we consider a food chain model with ratio-dependent functional response and prey-taxis. We first investigate the global existence and boundedness of the unique positive classical solutions of the system in a bounded domain with smooth boundary and Neumann boundary conditions. Then, we analyze the local stability of the system and the existence of Hopf bifurcation. In addition, we prove the global asymptotic stability of steady states under some conditions by constructing a Lyapunov functional, and investigate convergence rates. Finally, we present several numerical simulations to illustrate the results.

In this work, a dengue transmission model with logistic growth and time delay $(\tau )$ is investigated. Through detailed mathematical analysis, the local stability of a disease-free equilibrium and an endemic equilibrium is discussed, the existence of Hopf bifurcation and stability switch is established, and it is proved that the system is permanent if the basic reproduction number is greater than 1. On the basis of Lyapunov functional and LaSalle’s invariance principle, sufficient conditions are derived for the global stability of the endemic equilibrium. The primary theoretical results are simulated numerically. In addition, when $\tau =0$, relevant properties of the Hopf bifurcation are analyzed. Finally, sensitivity analysis is given and data fitting is carried out to predict the epidemic development trend of dengue fever in Singapore in 2020.

Because of their applications, the study of piecewise-linear differential systems has become increasingly important in recent years. This type of system already exists to model many different natural phenomena in physics, biology, economics, etc. As is well known, the study of the qualitative theory of piecewise differential systems focuses mainly on limit cycles. Most papers studying the problem of existence and the maximum number of limit cycles of piecewise differential systems have precisely considered planar systems. However, few papers have examined this problem in ${ℝ}^3$. In this paper, our main goal is to examine a class of discontinuous piecewise differential systems in ${ℝ}^3$, where we consider the unit sphere as the separation surface that divides the entire space into two regions, each one has a linear vector field analogous to planar center. In general, it is hard to determine an exact upper bound for the number of limit cycles that a class of differential systems can exhibit. We prove that this class of differential systems can have at most four limit cycles. We show that there are examples of such differential systems with exactly 1, 2, 3 and 4 limit cycles.

In this paper, two prey–predator models with distributed delays are presented based on the growth and loss rates of the predator, which are much smaller than that of the prey, leading to a singular perturbation problem. It is obtained that Hopf bifurcation can occur, where the coexistence equilibrium becomes unstable leading to a stable limit cycle. Subsequently, considering the perturbation parameter $0<𝜀\ll 1$, the fact that the solution crossing the transcritical point converges to a stable equilibrium is discussed for the model with Holling type I using the linear chain criterion, center-manifold reduction, the geometric singular perturbation theory and entry–exit function. The existence and uniqueness of relaxation oscillation cycle for the model with Holling type II are obtained. In addition, numerical simulations are provided to verify the analytical results.

Gas detection plays different roles in different environments. Traditional algorithms implemented on electronic nose for gas detection and recognition have high complexity and cannot resist device drift. In response to the above issues, we propose a convolutional neural network based on memristive Stochastic Computing (SC), which combines the characteristics of small devices and low power consumption of memristor devices, as well as the fast and fault-tolerant random calculation speed. It can effectively utilize hardware advantages, recognizing gases by electronic nose. The experimental results show that for two different gas sensor array datasets, the accuracy of the proposed method can achieve the level of 99%. When using memristive SC for deduction, the accuracy decreases by less than 1%, but in drift data, the accuracy can be improved by about 3%. Finally, the improvement in area, power, and energy compared to inference in GPU (NVIDIA Geforce RTX 3060 Laptop) is 1104X, 48X, and 9X, respectively.

For the cantilever beam vibration model without damping and forced terms, the corresponding differential system is a planar dynamical system with some singular straight lines. In this paper, by using the techniques from dynamical systems and singular traveling wave theory developed by [Li & Chen, 2007] to analyze its corresponding differential system, the bifurcations and the dynamical behaviors of the corresponding phase portraits are identified and analyzed. Under different parameter conditions, exact homoclinic and heteroclinic solutions, periodic solutions, compacton solutions, as well as peakons and periodic peakons, are all found explicitly.