We study a chemostat system with two parameters, So-initial density and D-flow-speed of the solution. At first, a generalization of the traditional Hopf bifurcation theorem is given. Then, an existence theorem for the...We study a chemostat system with two parameters, So-initial density and D-flow-speed of the solution. At first, a generalization of the traditional Hopf bifurcation theorem is given. Then, an existence theorem for the Hopf bifurcation of the chemostat system is presented.展开更多
A dynamical model is constructed to depict the spatial-temporal evolution of malware in mobile wireless sensor networks(MWSNs). Based on such a model, we design a hybrid control scheme combining parameter perturbation...A dynamical model is constructed to depict the spatial-temporal evolution of malware in mobile wireless sensor networks(MWSNs). Based on such a model, we design a hybrid control scheme combining parameter perturbation and state feedback to effectively manipulate the spatiotemporal dynamics of malware propagation. The hybrid control can not only suppress the Turing instability caused by diffusion factor but can also adjust the occurrence of Hopf bifurcation induced by time delay. Numerical simulation results show that the hybrid control strategy can efficiently manipulate the transmission dynamics to achieve our expected desired properties, thus reducing the harm of malware propagation to MWSNs.展开更多
The lid-driven square cavity flow is investigated by numerical experiments.It is found that from Re=5,000 to Re=7,307.75 the solution is stationary,but at Re=7,308 the solution is time periodic.So the critical Reynold...The lid-driven square cavity flow is investigated by numerical experiments.It is found that from Re=5,000 to Re=7,307.75 the solution is stationary,but at Re=7,308 the solution is time periodic.So the critical Reynolds number for the first Hopf bifurcation localizes between Re=7,307.75 and Re=7,308.Time periodical behavior begins smoothly,imperceptibly at the bottom left corner at a tiny tertiary vortex;all other vortices stay still,and then it spreads to the three relevant corners of the square cavity so that all small vortices at all levels move periodically.The primary vortex stays still.At Re=13,393.5 the solution is time periodic;the long-term integration carried out past t_(∞)=126,562.5 and the fluctuations of the kinetic energy look periodic except slight defects.However at Re=13,393.75 the solution is not time periodic anymore:losing unambiguously,abruptly time periodicity,it becomes chaotic.So the critical Reynolds number for the second Hopf bifurcation localizes between Re=13,393.5 and Re=13,393.75.At high Reynolds numbers Re=20,000 until Re=30,000 the solution becomes chaotic.The long-term integration is carried out past the long time t_(∞)=150,000,expecting the time asymptotic regime of the flow has been reached.The distinctive feature of the flow is then the appearance of drops:tiny portions of fluid produced by splitting of a secondary vortex,becoming loose and then fading away or being absorbed by another secondary vortex promptly.At Re=30,000 another phenomenon arises—the abrupt appearance at the bottom left corner of a tiny secondary vortex,not produced by splitting of a secondary vortex.展开更多
In this paper, the Leslie predator-prey system with two delays is studied. The stability of the positive equilibrium is discussed by analyzing the associated characteristic transcendental equation. The direction and s...In this paper, the Leslie predator-prey system with two delays is studied. The stability of the positive equilibrium is discussed by analyzing the associated characteristic transcendental equation. The direction and stability of the bifurcating periodic solutions are determined by applying the center manifold theorem and normal form theory. The conditions to guarantee the global existence of periodic solutions are given.展开更多
Due to the increasing use of passive absorbers to control unwanted vibrations,many studies have been done on energy absorbers ideally,but the lack of studies of real environmental conditions on these absorbers is felt...Due to the increasing use of passive absorbers to control unwanted vibrations,many studies have been done on energy absorbers ideally,but the lack of studies of real environmental conditions on these absorbers is felt.The present work investigates the effect of viscoelasticity on the stability and bifurcations of a system attached to a nonlinear energy sink(NES).In this paper,the Burgers model is assumed for the viscoelasticity in an NES,and a linear oscillator system is considered for investigating the instabilities and bifurcations.The equations of motion of the coupled system are solved by using the harmonic balance and pseudo-arc-length continuation methods.The results show that the viscoelasticity affects the frequency intervals of the Hopf and saddle-node branches,and by increasing the stiffness parameters of the viscoelasticity,the conditions of these branches occur in larger ranges of the external force amplitudes,and also reduce the frequency range of the branches.In addition,increasing the viscoelastic damping parameter has the potential to completely eliminate the instability of the system and gradually reduce the amplitude of the jump phenomenon.展开更多
This paper considers a delayed human respiratory model. Firstly, the stability of the equilibrium of the model is investigated and the occurrence of a sequence of Hopf bifurcations of the model is proved. Secondly, th...This paper considers a delayed human respiratory model. Firstly, the stability of the equilibrium of the model is investigated and the occurrence of a sequence of Hopf bifurcations of the model is proved. Secondly, the explicit algorithms which determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived by applying the normal form method and the center manifold theory. Finally, the existence of the global periodic solutions is showed under some assumptions on the model.展开更多
A memristor-coupled heterogenous neural network consisting of two-dimensional(2D)FitzHugh–Nagumo(FHN)and Hindmarsh–Rose(HR)neurons with two time delays is established.Taking the time delays as the control parameters...A memristor-coupled heterogenous neural network consisting of two-dimensional(2D)FitzHugh–Nagumo(FHN)and Hindmarsh–Rose(HR)neurons with two time delays is established.Taking the time delays as the control parameters,the existence of Hopf bifurcation near the stable equilibrium point in four cases is derived theoretically,and the validity of the Hopf bifurcation condition is verified by numerical analysis.The results show that the two time delays can make the stable equilibrium point unstable,thus leading to periodic oscillations induced by Hopf bifurcation.Furthermore,the time delays in FHN and HR neurons have different effects on the firing activity of neural network.Complex firing patterns,such as quiescent state,chaotic spiking,and periodic spiking can be induced by the time delay in FHN neuron,while the neural network only exhibits quiescent state and periodic spiking with the change of the time delay in HR neuron.Especially,phase synchronization between the heterogeneous neurons is explored,and the results show that the time delay in HR neurons has a greater effect on blocking the synchronization than the time delay in FHN neuron.Finally,the theoretical analysis is verified by circuit simulations.展开更多
The article mainly explores the Hopf bifurcation of a kind of nonlinear system with Gaussian white noise excitation and bounded random parameter.Firstly,the nonlinear system with multisource stochastic fac-tors is red...The article mainly explores the Hopf bifurcation of a kind of nonlinear system with Gaussian white noise excitation and bounded random parameter.Firstly,the nonlinear system with multisource stochastic fac-tors is reduced to an equivalent deterministic nonlinear system by the sequential orthogonal decomposi-tion method and the Karhunen-Loeve(K-L)decomposition theory.Secondly,the critical conditions about the Hopf bifurcation of the equivalent deterministic system are obtained.At the same time the influence of multisource stochastic factors on the Hopf bifurcation for the proposed system is explored.Finally,the theorical results are verified by the numerical simulations.展开更多
In this paper,an eco-epidemiological model with time delay is studied.The local stability of the four equilibria,the existence of stability switches about the predationfree equilibrium and the coexistence equilibrium ...In this paper,an eco-epidemiological model with time delay is studied.The local stability of the four equilibria,the existence of stability switches about the predationfree equilibrium and the coexistence equilibrium are discussed.It is found that Hopf bifurcations occur when the delay passes through some critical values.Formulae are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem.Some numerical simulations are carried out to illustrate the theoretical results.展开更多
This paper constructed and studied a nonresident computer virus model with age structure and two delays effects. The non-negativity and boundedness of the solution of the model have been discussed, and then gave the b...This paper constructed and studied a nonresident computer virus model with age structure and two delays effects. The non-negativity and boundedness of the solution of the model have been discussed, and then gave the basic regeneration number, and obtained the conditions for the existence and the stability of the virus-free equilibrium and the computer virus equilibrium. Theoretical analysis shows the conditions under which the model undergoes Hopf bifurcation in three different cases. The numerical examples are provided to demonstrate the theoretical results.展开更多
In this paper, a system of Lorenz-type ordinary differential equations is considered and, under some assumptions about the parameter space, the presence of the supercritical non-degenerate Hopf bifurcation is demonstr...In this paper, a system of Lorenz-type ordinary differential equations is considered and, under some assumptions about the parameter space, the presence of the supercritical non-degenerate Hopf bifurcation is demonstrated. The technical tool used consists of the Central Manifold theorem, a well-known formula to calculate the Lyapunov coefficient and Hopf’s Theorem. For particular values of the parameters in the parameter space established in the main result of this work, a graph is presented that describes the evolution of the trajectories, obtained by means of numerical simulation.展开更多
The homogeneity-breaking instability of the periodic solutions triggered by Hopf bifurcations of a diffusive Gierer-Meinhart system is studied in this paper.Sufficient conditions on the diffusion coefficients and the ...The homogeneity-breaking instability of the periodic solutions triggered by Hopf bifurcations of a diffusive Gierer-Meinhart system is studied in this paper.Sufficient conditions on the diffusion coefficients and the cross diffusion coefficients were derived to guarantee the occurrence of the aforementioned homogeneity-breaking instability.展开更多
The dynamic model of a bistable laminated composite shell simply supported by four corners is further developed to investigate the resonance responses and chaotic behaviors.The existence of the 1:1 resonance relations...The dynamic model of a bistable laminated composite shell simply supported by four corners is further developed to investigate the resonance responses and chaotic behaviors.The existence of the 1:1 resonance relationship between two order vibration modes of the system is verified.The resonance response of this class of bistable structures in the dynamic snap-through mode is investigated,and the four-dimensional(4D)nonlinear modulation equations are derived based on the 1:1 internal resonance relationship by means of the multiple scales method.The Hopf bifurcation and instability interval of the amplitude frequency and force amplitude curves are analyzed.The discussion focuses on investigating the effects of key parameters,e.g.,excitation amplitude,damping coefficient,and detuning parameters,on the resonance responses.The numerical simulations show that the foundation excitation and the degree of coupling between the vibration modes exert a substantial effect on the chaotic dynamics of the system.Furthermore,the significant motions under particular excitation conditions are visualized by bifurcation diagrams,time histories,phase portraits,three-dimensional(3D)phase portraits,and Poincare maps.Finally,the vibration experiment is carried out to study the amplitude frequency responses and bifurcation characteristics for the bistable laminated composite shell,yielding results that are qualitatively consistent with the theoretical results.展开更多
In recent years, the traffic congestion problem has become more and more serious, and the research on traffic system control has become a new hot spot. Studying the bifurcation characteristics of traffic flow systems ...In recent years, the traffic congestion problem has become more and more serious, and the research on traffic system control has become a new hot spot. Studying the bifurcation characteristics of traffic flow systems and designing control schemes for unstable pivots can alleviate the traffic congestion problem from a new perspective. In this work, the full-speed differential model considering the vehicle network environment is improved in order to adjust the traffic flow from the perspective of bifurcation control, the existence conditions of Hopf bifurcation and saddle-node bifurcation in the model are proved theoretically, and the stability mutation point for the stability of the transportation system is found. For the unstable bifurcation point, a nonlinear system feedback controller is designed by using Chebyshev polynomial approximation and stochastic feedback control method. The advancement, postponement, and elimination of Hopf bifurcation are achieved without changing the system equilibrium point, and the mutation behavior of the transportation system is controlled so as to alleviate the traffic congestion. The changes in the stability of complex traffic systems are explained through the bifurcation analysis, which can better capture the characteristics of the traffic flow. By adjusting the control parameters in the feedback controllers, the influence of the boundary conditions on the stability of the traffic system is adequately described, and the effects of the unstable focuses and saddle points on the system are suppressed to slow down the traffic flow. In addition, the unstable bifurcation points can be eliminated and the Hopf bifurcation can be controlled to advance, delay, and disappear,so as to realize the control of the stability behavior of the traffic system, which can help to alleviate the traffic congestion and describe the actual traffic phenomena as well.展开更多
Stem cell regeneration is an essential biological process in the maintenance of tissue homeostasis;dysregulation of stem cell regeneration may result in dynamic diseases that show oscillations in cell numbers.Cell het...Stem cell regeneration is an essential biological process in the maintenance of tissue homeostasis;dysregulation of stem cell regeneration may result in dynamic diseases that show oscillations in cell numbers.Cell heterogeneity and plasticity are necessary for the dynamic equilibrium of tissue homeostasis;however,how these features may affect the oscillatory dynamics of the stem cell regeneration process remains poorly understood.Here,based on a mathematical model of heterogeneous stem cell regeneration that includes cell heterogeneity and random transition of epigenetic states,we study the conditions to have oscillation solutions through bifurcation analysis and numerical simulations.Our results show various model system dynamics with changes in different parameters associated with kinetic rates,cellular heterogeneity,and plasticity.We show that introducing heterogeneity and plasticity to cells can result in oscillation dynamics,as we have seen in the homogeneous stem cell regeneration system.However,increasing the cell heterogeneity and plasticity intends to maintain tissue homeostasis under certain conditions.The current study is an initiatory investigation of how cell heterogeneity and plasticity may affect stem cell regeneration dynamics,and many questions remain to be further studied both biologically and mathematically.展开更多
In this paper, we study a modified Leslie-Gower predator-prey model with Smith growth subject to homogeneous Neumann boundary condition, in which the functional response is the Crowley-Martin functional response term....In this paper, we study a modified Leslie-Gower predator-prey model with Smith growth subject to homogeneous Neumann boundary condition, in which the functional response is the Crowley-Martin functional response term. Firstly, for ODE model, the local stability of equilibrium point is given. And by using bifurcation theory and selecting suitable bifurcation parameters, we find many kinds of bifurcation phenomena, including Transcritical bifurcation and Hopf bifurcation. For the reaction-diffusion model, we find that Turing instability occurs. Besides, it is proved that Hopf bifurcation exists in the model. Finally, numerical simulations are presented to verify and illustrate the theoretical results.展开更多
Since the last century, various predator-prey systems have garnered widespread attention. In particular, the predator-prey systems have sparked significant interest among applied mathematicians and ecologists. From th...Since the last century, various predator-prey systems have garnered widespread attention. In particular, the predator-prey systems have sparked significant interest among applied mathematicians and ecologists. From the perspectives of both mathematics and biology, a predator-prey system with the Allee effect and featuring the Bazykin functional response has been established. For this model, analyses have been conducted on its boundedness, the properties of its solutions, the existence of equilibrium points, as well as its local stability and Hopf bifurcations.展开更多
In the paper, a novel four-wing hyper-chaotic system is proposed and analyzed. A rare dynamic phenomenon is found that this new system with one equilibrium generates a four-wing-hyper-chaotic attractor as parameter va...In the paper, a novel four-wing hyper-chaotic system is proposed and analyzed. A rare dynamic phenomenon is found that this new system with one equilibrium generates a four-wing-hyper-chaotic attractor as parameter varies. The system has rich and complex dynamical behaviors, and it is investigated in terms of Lyapunov exponents, bifurcation diagrams, Poincare maps, frequency spectrum, and numerical simulations. In addition, the theoretical analysis shows that the system undergoes a Hopf bifurcation as one parameter varies, which is illustrated by the numerical simulation. Finally, an analog circuit is designed to implement this hyper-chaotic system.展开更多
This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms,and with linear delayed velocity feedback.The analysis indicates that for a sufficiently large velocity feedback gain,th...This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms,and with linear delayed velocity feedback.The analysis indicates that for a sufficiently large velocity feedback gain,the equilibrium of the system may undergo a number of stability switches with an increase of time delay,and then becomes unstable forever.At each critical value of time delay for which the system changes its stability,a generic Hopf bifurcation occurs and a periodic motion emerges in a one-sided neighbourhood of the critical time delay.The method of Fredholm alternative is applied to determine the bifurcating periodic motions and their stability.It stresses on the effect of the system parameters on the stable regions and the amplitudes of the bifurcating periodic solutions.展开更多
基金The NSF (10171010) of China Major Project of Education Ministry (01061) of China.
文摘We study a chemostat system with two parameters, So-initial density and D-flow-speed of the solution. At first, a generalization of the traditional Hopf bifurcation theorem is given. Then, an existence theorem for the Hopf bifurcation of the chemostat system is presented.
基金Project supported by the National Natural Science Foundation of China (Grant No. 62073172)the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20221329)。
文摘A dynamical model is constructed to depict the spatial-temporal evolution of malware in mobile wireless sensor networks(MWSNs). Based on such a model, we design a hybrid control scheme combining parameter perturbation and state feedback to effectively manipulate the spatiotemporal dynamics of malware propagation. The hybrid control can not only suppress the Turing instability caused by diffusion factor but can also adjust the occurrence of Hopf bifurcation induced by time delay. Numerical simulation results show that the hybrid control strategy can efficiently manipulate the transmission dynamics to achieve our expected desired properties, thus reducing the harm of malware propagation to MWSNs.
基金supported in part by the National Science Foundation Grant No.DMS-0604235.
文摘The lid-driven square cavity flow is investigated by numerical experiments.It is found that from Re=5,000 to Re=7,307.75 the solution is stationary,but at Re=7,308 the solution is time periodic.So the critical Reynolds number for the first Hopf bifurcation localizes between Re=7,307.75 and Re=7,308.Time periodical behavior begins smoothly,imperceptibly at the bottom left corner at a tiny tertiary vortex;all other vortices stay still,and then it spreads to the three relevant corners of the square cavity so that all small vortices at all levels move periodically.The primary vortex stays still.At Re=13,393.5 the solution is time periodic;the long-term integration carried out past t_(∞)=126,562.5 and the fluctuations of the kinetic energy look periodic except slight defects.However at Re=13,393.75 the solution is not time periodic anymore:losing unambiguously,abruptly time periodicity,it becomes chaotic.So the critical Reynolds number for the second Hopf bifurcation localizes between Re=13,393.5 and Re=13,393.75.At high Reynolds numbers Re=20,000 until Re=30,000 the solution becomes chaotic.The long-term integration is carried out past the long time t_(∞)=150,000,expecting the time asymptotic regime of the flow has been reached.The distinctive feature of the flow is then the appearance of drops:tiny portions of fluid produced by splitting of a secondary vortex,becoming loose and then fading away or being absorbed by another secondary vortex promptly.At Re=30,000 another phenomenon arises—the abrupt appearance at the bottom left corner of a tiny secondary vortex,not produced by splitting of a secondary vortex.
基金This work is supported by the National Natural Sciences Foundation of China (No.10571064) the Natural Sciences Foundation of Guangdong Province(No.04010364).
文摘In this paper, the Leslie predator-prey system with two delays is studied. The stability of the positive equilibrium is discussed by analyzing the associated characteristic transcendental equation. The direction and stability of the bifurcating periodic solutions are determined by applying the center manifold theorem and normal form theory. The conditions to guarantee the global existence of periodic solutions are given.
基金financial support from K.N.Toosi University of Technology,Tehran,Iran。
文摘Due to the increasing use of passive absorbers to control unwanted vibrations,many studies have been done on energy absorbers ideally,but the lack of studies of real environmental conditions on these absorbers is felt.The present work investigates the effect of viscoelasticity on the stability and bifurcations of a system attached to a nonlinear energy sink(NES).In this paper,the Burgers model is assumed for the viscoelasticity in an NES,and a linear oscillator system is considered for investigating the instabilities and bifurcations.The equations of motion of the coupled system are solved by using the harmonic balance and pseudo-arc-length continuation methods.The results show that the viscoelasticity affects the frequency intervals of the Hopf and saddle-node branches,and by increasing the stiffness parameters of the viscoelasticity,the conditions of these branches occur in larger ranges of the external force amplitudes,and also reduce the frequency range of the branches.In addition,increasing the viscoelastic damping parameter has the potential to completely eliminate the instability of the system and gradually reduce the amplitude of the jump phenomenon.
基金This research is supported by the National Natural Science Foundation of China under Grant No.10571064 the Natural Science Foundation of Guangdong Province under Grant No.04010364.
文摘This paper considers a delayed human respiratory model. Firstly, the stability of the equilibrium of the model is investigated and the occurrence of a sequence of Hopf bifurcations of the model is proved. Secondly, the explicit algorithms which determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived by applying the normal form method and the center manifold theory. Finally, the existence of the global periodic solutions is showed under some assumptions on the model.
基金the National Natural Science Foundations of China(Grant Nos.62171401 and 62071411).
文摘A memristor-coupled heterogenous neural network consisting of two-dimensional(2D)FitzHugh–Nagumo(FHN)and Hindmarsh–Rose(HR)neurons with two time delays is established.Taking the time delays as the control parameters,the existence of Hopf bifurcation near the stable equilibrium point in four cases is derived theoretically,and the validity of the Hopf bifurcation condition is verified by numerical analysis.The results show that the two time delays can make the stable equilibrium point unstable,thus leading to periodic oscillations induced by Hopf bifurcation.Furthermore,the time delays in FHN and HR neurons have different effects on the firing activity of neural network.Complex firing patterns,such as quiescent state,chaotic spiking,and periodic spiking can be induced by the time delay in FHN neuron,while the neural network only exhibits quiescent state and periodic spiking with the change of the time delay in HR neuron.Especially,phase synchronization between the heterogeneous neurons is explored,and the results show that the time delay in HR neurons has a greater effect on blocking the synchronization than the time delay in FHN neuron.Finally,the theoretical analysis is verified by circuit simulations.
基金This work was supported by the grants from the National Nat-ural Science Foundation of China(No.11772002)Ningxia higher education first-class discipline construction funding project(No.NXYLXK2017B09)+2 种基金Major Special project of North Minzu University(No.ZDZX201902)Open project of The Key Laboratory of In-telligent Information and Big Data Processing of NingXia Province(No.2019KLBD008)Postgraduate Innovation Project of North Minzu University(No.YCX22099).
文摘The article mainly explores the Hopf bifurcation of a kind of nonlinear system with Gaussian white noise excitation and bounded random parameter.Firstly,the nonlinear system with multisource stochastic fac-tors is reduced to an equivalent deterministic nonlinear system by the sequential orthogonal decomposi-tion method and the Karhunen-Loeve(K-L)decomposition theory.Secondly,the critical conditions about the Hopf bifurcation of the equivalent deterministic system are obtained.At the same time the influence of multisource stochastic factors on the Hopf bifurcation for the proposed system is explored.Finally,the theorical results are verified by the numerical simulations.
文摘In this paper,an eco-epidemiological model with time delay is studied.The local stability of the four equilibria,the existence of stability switches about the predationfree equilibrium and the coexistence equilibrium are discussed.It is found that Hopf bifurcations occur when the delay passes through some critical values.Formulae are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem.Some numerical simulations are carried out to illustrate the theoretical results.
文摘This paper constructed and studied a nonresident computer virus model with age structure and two delays effects. The non-negativity and boundedness of the solution of the model have been discussed, and then gave the basic regeneration number, and obtained the conditions for the existence and the stability of the virus-free equilibrium and the computer virus equilibrium. Theoretical analysis shows the conditions under which the model undergoes Hopf bifurcation in three different cases. The numerical examples are provided to demonstrate the theoretical results.
文摘In this paper, a system of Lorenz-type ordinary differential equations is considered and, under some assumptions about the parameter space, the presence of the supercritical non-degenerate Hopf bifurcation is demonstrated. The technical tool used consists of the Central Manifold theorem, a well-known formula to calculate the Lyapunov coefficient and Hopf’s Theorem. For particular values of the parameters in the parameter space established in the main result of this work, a graph is presented that describes the evolution of the trajectories, obtained by means of numerical simulation.
基金Sponsored by the National Natural Science Foundation of China(Grant Nos.12061033,2020GG0130,2020MS04007,2020BS11,and NJZZ22286).
文摘The homogeneity-breaking instability of the periodic solutions triggered by Hopf bifurcations of a diffusive Gierer-Meinhart system is studied in this paper.Sufficient conditions on the diffusion coefficients and the cross diffusion coefficients were derived to guarantee the occurrence of the aforementioned homogeneity-breaking instability.
基金Project supported by the National Natural Science Foundation of China(Nos.12293000,12293001,11988102,12172006,and 12202011)。
文摘The dynamic model of a bistable laminated composite shell simply supported by four corners is further developed to investigate the resonance responses and chaotic behaviors.The existence of the 1:1 resonance relationship between two order vibration modes of the system is verified.The resonance response of this class of bistable structures in the dynamic snap-through mode is investigated,and the four-dimensional(4D)nonlinear modulation equations are derived based on the 1:1 internal resonance relationship by means of the multiple scales method.The Hopf bifurcation and instability interval of the amplitude frequency and force amplitude curves are analyzed.The discussion focuses on investigating the effects of key parameters,e.g.,excitation amplitude,damping coefficient,and detuning parameters,on the resonance responses.The numerical simulations show that the foundation excitation and the degree of coupling between the vibration modes exert a substantial effect on the chaotic dynamics of the system.Furthermore,the significant motions under particular excitation conditions are visualized by bifurcation diagrams,time histories,phase portraits,three-dimensional(3D)phase portraits,and Poincare maps.Finally,the vibration experiment is carried out to study the amplitude frequency responses and bifurcation characteristics for the bistable laminated composite shell,yielding results that are qualitatively consistent with the theoretical results.
基金Project supported by the National Natural Science Foundation of China(Grant No.72361031)the Gansu Province University Youth Doctoral Support Project(Grant No.2023QB-049)。
文摘In recent years, the traffic congestion problem has become more and more serious, and the research on traffic system control has become a new hot spot. Studying the bifurcation characteristics of traffic flow systems and designing control schemes for unstable pivots can alleviate the traffic congestion problem from a new perspective. In this work, the full-speed differential model considering the vehicle network environment is improved in order to adjust the traffic flow from the perspective of bifurcation control, the existence conditions of Hopf bifurcation and saddle-node bifurcation in the model are proved theoretically, and the stability mutation point for the stability of the transportation system is found. For the unstable bifurcation point, a nonlinear system feedback controller is designed by using Chebyshev polynomial approximation and stochastic feedback control method. The advancement, postponement, and elimination of Hopf bifurcation are achieved without changing the system equilibrium point, and the mutation behavior of the transportation system is controlled so as to alleviate the traffic congestion. The changes in the stability of complex traffic systems are explained through the bifurcation analysis, which can better capture the characteristics of the traffic flow. By adjusting the control parameters in the feedback controllers, the influence of the boundary conditions on the stability of the traffic system is adequately described, and the effects of the unstable focuses and saddle points on the system are suppressed to slow down the traffic flow. In addition, the unstable bifurcation points can be eliminated and the Hopf bifurcation can be controlled to advance, delay, and disappear,so as to realize the control of the stability behavior of the traffic system, which can help to alleviate the traffic congestion and describe the actual traffic phenomena as well.
基金funded by the Scientific Research Project of Tianjin Education Commission(Grant No.2019KJ026).
文摘Stem cell regeneration is an essential biological process in the maintenance of tissue homeostasis;dysregulation of stem cell regeneration may result in dynamic diseases that show oscillations in cell numbers.Cell heterogeneity and plasticity are necessary for the dynamic equilibrium of tissue homeostasis;however,how these features may affect the oscillatory dynamics of the stem cell regeneration process remains poorly understood.Here,based on a mathematical model of heterogeneous stem cell regeneration that includes cell heterogeneity and random transition of epigenetic states,we study the conditions to have oscillation solutions through bifurcation analysis and numerical simulations.Our results show various model system dynamics with changes in different parameters associated with kinetic rates,cellular heterogeneity,and plasticity.We show that introducing heterogeneity and plasticity to cells can result in oscillation dynamics,as we have seen in the homogeneous stem cell regeneration system.However,increasing the cell heterogeneity and plasticity intends to maintain tissue homeostasis under certain conditions.The current study is an initiatory investigation of how cell heterogeneity and plasticity may affect stem cell regeneration dynamics,and many questions remain to be further studied both biologically and mathematically.
文摘In this paper, we study a modified Leslie-Gower predator-prey model with Smith growth subject to homogeneous Neumann boundary condition, in which the functional response is the Crowley-Martin functional response term. Firstly, for ODE model, the local stability of equilibrium point is given. And by using bifurcation theory and selecting suitable bifurcation parameters, we find many kinds of bifurcation phenomena, including Transcritical bifurcation and Hopf bifurcation. For the reaction-diffusion model, we find that Turing instability occurs. Besides, it is proved that Hopf bifurcation exists in the model. Finally, numerical simulations are presented to verify and illustrate the theoretical results.
文摘Since the last century, various predator-prey systems have garnered widespread attention. In particular, the predator-prey systems have sparked significant interest among applied mathematicians and ecologists. From the perspectives of both mathematics and biology, a predator-prey system with the Allee effect and featuring the Bazykin functional response has been established. For this model, analyses have been conducted on its boundedness, the properties of its solutions, the existence of equilibrium points, as well as its local stability and Hopf bifurcations.
基金supported by the National Natural Science Foundation of China(Grant Nos.10772135 and 60874028)the Young Scientists Fund of the National Natural Science Foundation of China(Grant No.11202148)+2 种基金the Incentive Funding of the National Research Foundation of South Africa(GrantNo.IFR2009090800049)the Eskom Tertiary Education Support Programme of South Africathe Research Foundation of Tianjin University of Science and Technology
文摘In the paper, a novel four-wing hyper-chaotic system is proposed and analyzed. A rare dynamic phenomenon is found that this new system with one equilibrium generates a four-wing-hyper-chaotic attractor as parameter varies. The system has rich and complex dynamical behaviors, and it is investigated in terms of Lyapunov exponents, bifurcation diagrams, Poincare maps, frequency spectrum, and numerical simulations. In addition, the theoretical analysis shows that the system undergoes a Hopf bifurcation as one parameter varies, which is illustrated by the numerical simulation. Finally, an analog circuit is designed to implement this hyper-chaotic system.
基金The project supported by the National Natural Science Foundation of China (19972025)
文摘This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms,and with linear delayed velocity feedback.The analysis indicates that for a sufficiently large velocity feedback gain,the equilibrium of the system may undergo a number of stability switches with an increase of time delay,and then becomes unstable forever.At each critical value of time delay for which the system changes its stability,a generic Hopf bifurcation occurs and a periodic motion emerges in a one-sided neighbourhood of the critical time delay.The method of Fredholm alternative is applied to determine the bifurcating periodic motions and their stability.It stresses on the effect of the system parameters on the stable regions and the amplitudes of the bifurcating periodic solutions.