bifurcation

摘要： In this paper, we investigated stability and bifurcation behaviors of a predator-prey model with Michaelis-Menten type prey harvesting. Sufficient conditions for local and global asymptotically stability of the interior equilibrium point were established. Some critical threshold conditions for transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation were explored analytically. Furthermore, It should be stressed that the fear factor could not only reduce the predator density, but also affect the prey growth rate. Finally, these theoretical results revealed that nonlinear Michaelis-Menten type prey harvesting has played an important role in the dynamic relationship, which also in turn proved the validity of theoretical derivation.

摘要： Vibration energy harvesting has emerged as a promising method to harvest energy for small-scale applications.Enhancing the performance of a vibration energy harvester(VEH)incorporating nonlinear techniques,for example,the snap-through VEH with geometric non-linearity,has gained attention in recent years.A conventional snap-through VEH is a bi-stable system with a time-invariant potential function,which was investigated extensively in the past.In this work,a modified snap-through VEH with a time-varying potential function subject to harmonic and random base excitations is investigated.Modified snap-through VEHs,such as the one considered in this study,are used in wave energy harvesters.However,the studies on their dynamics and energy harvesting under harmonic and random excitations are limited.The dynamics of the modified snap-through VEH is represented by a system of differential algebraic equations(DAEs),and the numerical schemes are proposed for its solutions.Under a harmonic excitation,the system exhibits periodic and chaotic motions,and the energy harvesting is superior compared with the conventional counterpart.The dynamics under a random excitation is investigated by the moment differential method and the numerical scheme based on the modified Euler-Maruyama method.The Fokker-Planck equation representing the dynamics is derived,and the marginal and joint probability density functions(PDFs)are obtained by the Monte Carlo simulation.The study shows that the modified snap-through oscillator based VEH performs better under both harmonic and random excitations.The dynamics of the system under stochastic resonance(SR)is investigated,and performance enhancement is observed.The results from this study will help in the development of adaptive VEH techniques in the future.

摘要： This review article aims to give a comprehensive review of periodic orbits in the circular restricted three-body problem(CRTBP),which is a standard ideal model for the Earth-Moon system and is closest to the practical mechanical model.It focuses the attention on periodic orbits in the Earth-Moon system.This work is primarily motivated by a series of missions and plans that take advantages of the three-body periodic orbits near the libration points or around two gravitational celestial bodies.Firstly,simple periodic orbits and their classification that is usually considered to be early work before 1970 are summarized,and periodic orbits around Lagrange points,either planar or three-dimensional,are intensively studied during past decades.Subsequently,stability index of a periodic orbit and bifurcation analysis are presented,which demonstrate a guideline to find more periodic orbits inspired by bifurcation signals.Then,the practical techniques for computing a wide range of periodic orbits and associated quasi-periodic orbits,as well as constructing database of periodic orbits by numerical searching techniques are also presented.For those unstable periodic orbits,the station keeping maneuvers are reviewed.Finally,the applications of periodic orbits are presented,including those in practical missions,under consideration,and still in conceptual design stage.This review article has the function of bridging between engineers and researchers,so as to make it more convenient and faster for engineers to understand the complex restricted three-body problem(RTBP).At the same time,it can also provide some technical thinking for general researchers.

摘要： By bifurcation and topological methods,we study the global structure of a radial nodal solutions set of the mean curvature equation in a standard static spacetime div {a∇u√1−a^(2)|∇u|^(2)+g(∇u,∇a)/√1−a^(2)|∇u|^(2)=λNH,with a 0-Dirichlet boundary condition on the unit ball.According to the behavior of H near 0,we obtain the global structure of sign-changing radial spacelike graphs for this problem.

摘要： In this paper, based on the dynamic relationship between algae and protozoa, an aquatic ecological model with Allee effect was established to investigate how some ecological environment factors affect coexistence mode of algae and protozoa. Mathematical derivation works mainly gave some key conditions to ensure the existence and stability of all possible equilibrium points, and to induce the occurrence of transcritical bifurcation and Hopf bifurcation. The numerical simulation works mainly revealed ecological relationship change characteristics of algae and protozoa with the help of bifurcation dynamics evolution process. Furthermore, it was also worth emphasizing that Allee effect had a strong influence on the dynamic relationship between algae and protozoa. In a word, it was hoped that the research results could provide some theoretical support for algal bloom control, and also be conducive to the rapid development of aquatic ecological models.

摘要： Under the control framework of algae bloom in eutrophic lakes and reservoirs based on biological manipulation, the temperature variable is introduced into ecological modeling to show that it is a necessary condition for the rapid occurrence of algal blooms, and an aquatic ecological model with temperature effect is proposed to describe dynamic relationship between algae and biological manipulation predator. The mathematical theory work mainly investigates the existence and stability of some equilibrium points and some critical conditions for the occurrence of transcritical bifurcation and Hopf bifurcation. The numerical simulation mainly shows the dynamic evolution process of bifurcation dynamics, which can not only verify the validity and feasibility of these theoretical works but also analyze the influence of some key parameters on dynamic behavior evolution. Furthermore, It is worth emphasizing that temperature plays an important role in the coexistence of algae and biological manipulation predators. Moreover, the coexistence mode of algae and biological manipulation predators is discovered by means of dynamic bifurcation evolution. Finally, it is hoped that these research results can provide some reference for the study of aquatic ecosystems.

摘要： In the traditional viewpoint,inhibitory and excitatory effects always induce opposite responses.In the present study,the enhanced bursting activities induced by excitatory autapses,which are consistent with the recent experimental observations,and those induced by inhibitory autapses,which is a paradoxical phenomenon,were simulated using the Chay model.The same bifurcations and different ionic currents for the same responses were acquired with fast-slow variable dissection and current decomposition,respectively.As the inhibitory or excitatory autaptic conductance increased,the ending phase of the burst related to a homoclinic bifurcation of the fast subsystem changed to widen the burst duration to contain more spikes,which was induced by an elevated minimal potential(V_(min)) of spiking of the fast subsystem.Larger inhibitory and excitatory autaptic conductances induced smaller and larger maximal potentials(V_(max)) of spiking,respectively.During the downstroke,a weaker potassium current induced by the smaller V_(max) played a dominant role for the inhibitory autapse,and the stronger potassium current induced by the larger V_(max) became weaker due to the opposite autaptic current of the excitatory autapse,which induced the V_(min) elevated.The results present the nonlinear and biophysical mechanisms of the same responses to opposite effects,which extends nonlinear dynamics knowledge and provides potential modulation measures for the nervous system.

摘要： This paper introduces a hyperchaotic system from the Lü system with a sinusoïdal perturbation. This hyperchaotic system has more complex dynamical behaviors, and can generate 2-scroll hyperchaotic attractor and 2-scroll chaotic attractor under different control parameters. Theoretical analyses and simulation are conducted to investigate the dynamical behaviors of the proposed hyperchaotic system by means of Lyapunov exponents, analysis of the bifurcation diagram and phase portraits.

摘要： This work presents the complexity that emerges in a Bertrand duopoly between two companies in the Greek oil market, one of which is semi-public and the other is private. The game uses linear demand functions for differentiated products from the existing literature and asymmetric cost functions that arose after approaches using the published financial reports of the two oil companies (Hellenic Petroleum and Motor Oil). The game is based on the assumption of homogeneous players who are characterized by bounded rationality and follow an adjustment mechanism. The players’ decisions for each time period are expressed by two difference equations. A dynamical analysis of the game’s discrete dynamical system is made by finding the equilibrium positions and studying their stability. Numerical simulations include bifurcation diagrams and strange attractors. Lyapunov numbers’ graphs and sensitivity analysis in initial conditions prove the algebraic results and reveal the complexity and chaotic behavior of the system focusing on the two parameters k1 and k2 (speed of adjustment for each player). The d-Backtest method is applied through which an attempt is made to control the chaos that appears outside the stability space in order to return to the locally asymptotically stable Nash equilibrium for the system.

摘要： Avariety of nonlinear circuits can be tamed to reproduce the main dynamical properties in neural activities for biological neurons while designing reliable artificial synapses for connecting these neural circuits becomes a challenge.In this paper,a Josephson junction is used to build coupling channel for connecting two FitzHugh-Nagumo neural circuits driven by periodical voltage source.The hybrid synapse is designed by using a linear resistor paralleled with a Josephson junction,which can estimate the effect of external magnetic field by generating additive phase error between the junction.Indeed,it activates nonlinear coupling due to phase diversity in the Josephson junction.The coupled circuits are estimated in dimensionless dynamical systems by applying scale transformation on the variables and parameters in neural circuits.In fact,the intrinsic parameters of coupling channel are adjusted to detect the occurrence of synchronization.Bifurcation analysis is calculated to predict and confirm the occurrence of synchronization between two neural circuits.It is found that synchronization can be stabilized between two FHN neural circuits by selecting appropriate values for parameters in the Josephson junction involved in the coupling channel.It gives useful guidance for implementing artificial synapse for signal processing in neural circuits.