The time-fractional modified Korteweg-de Vries(KdV)equation is committed to establish exact solutions by employing the bifurcation method.Firstly,the phase portraits and related qualitative analysis are comprehensivel...The time-fractional modified Korteweg-de Vries(KdV)equation is committed to establish exact solutions by employing the bifurcation method.Firstly,the phase portraits and related qualitative analysis are comprehensively provided.Then,we give parametric expressions of different types of solutions matching with the corresponding orbits.Finally,solution profiles,3D and density plots of some solutions are presented with proper parametric choices.展开更多
This paper presents a bifurcation study of a mRNA-protein network with negative feedback and time delay. The network is modeled as a coupled system of N ordinary differential equations (ODEs) and N delay differential ...This paper presents a bifurcation study of a mRNA-protein network with negative feedback and time delay. The network is modeled as a coupled system of N ordinary differential equations (ODEs) and N delay differential equations (DDEs). Linear analysis of the stable equilibria shows the existence of a critical time delay beyond which limit cycle oscillations are born in a Hopf bifurcation. The Poincaré-Lindstedt perturbation method is applied to the nonlinear system, resulting in closed form approximate expressions for the amplitude and frequency of oscillation. We confirm our perturbation analysis results by numerically simulating the full 2N-dimensional nonlinear system for N = 2, 7, 15, and 30. Our results show that for small perturbations the equilibrium undergoes a supercritical Hopf and the system exhibits stable periodic solutions. Furthermore, our closed form numerical expressions exhibit the importance of the network’s negative feedback and nonlinear effects.展开更多
In this paper,a discrete Lotka-Volterra predator-prey model is proposed that considers mixed functional responses of Holling types I and III.The equilibrium points of the model are obtained,and their stability is test...In this paper,a discrete Lotka-Volterra predator-prey model is proposed that considers mixed functional responses of Holling types I and III.The equilibrium points of the model are obtained,and their stability is tested.The dynamical behavior of this model is studied according to the change of the control parameters.We find that the complex dynamical behavior extends from a stable state to chaotic attractors.Finally,the analytical results are clarified by some numerical simulations.展开更多
In the light of the visual angle model(VAM),an improved car-following model considering driver's visual angle,anticipated time and stabilizing driving behavior is proposed so as to investigate how the driver's...In the light of the visual angle model(VAM),an improved car-following model considering driver's visual angle,anticipated time and stabilizing driving behavior is proposed so as to investigate how the driver's behavior factors affect the stability of the traffic flow.Based on the model,linear stability analysis is performed together with bifurcation analysis,whose corresponding stability condition is highly fit to the results of the linear analysis.Furthermore,the time-dependent Ginzburg–Landau(TDGL)equation and the modified Korteweg–de Vries(m Kd V)equation are derived by nonlinear analysis,and we obtain the relationship of the two equations through the comparison.Finally,parameter calibration and numerical simulation are conducted to verify the validity of the theoretical analysis,whose results are highly consistent with the theoretical analysis.展开更多
This paper is concerned with the bifurcation analysis for a free boundary problem modeling the growth of solid tumor with inhibitors.In this problem,surface tension coefficient plays the role of bifurcation parameter,...This paper is concerned with the bifurcation analysis for a free boundary problem modeling the growth of solid tumor with inhibitors.In this problem,surface tension coefficient plays the role of bifurcation parameter,it is proved that there exists a sequence of the nonradially stationary solutions bifurcate from the radially symmetric stationary solutions.Our results indicate that the tumor grown in vivo may have various shapes.In particular,a tumor with an inhibitor is associated with the growth of protrusions.展开更多
A loss of ground directional stability can trigger a high-speed Unmanned Aerial Vehicle(UAV)to veer off the runway.In order to investigate the combined effects of the key structural and operational parameters on the U...A loss of ground directional stability can trigger a high-speed Unmanned Aerial Vehicle(UAV)to veer off the runway.In order to investigate the combined effects of the key structural and operational parameters on the UAV ground directional stability from a global perspective,a fully parameterized mathematical high-speed UAV ground nonlinear dynamic model is developed considering several nonlinear factors.The bifurcation analysis procedure of a UAV ground steering system is introduced,following which the simulation efficiency is greatly improved comparing with the time-domain simulation method.Then the numerical continuation method is employed to investigate the influence of the nose wheel steering angle and the global stability region is obtained.The bifurcation parameter plane is divided into several parts with different stability properties by the saddle nodes and the Hopf bifurcation points.We find that the UAV motion states will never cross the bifurcation curve in the nonlinear system.Also,the dual-parameter bifurcation analyses are presented to give a complete description of the possible steering performance.It is also found that BT bifurcation appears when the UAV initial rectilinear velocity and the tire frictional coefficient vary.In addition,results indicate that the influence of tire frictional coefficient has an opposite trend to the influence of initial rectilinear velocity.Overall,using bifurcation analysis method to identify the parameter regions of a UAV nonlinear ground dynamic system helps to improve the development efficiency and quality during UAV designing phase.展开更多
Stapes fracture causes hearing loss and instability in the middle ear hearing system(MEHS). The material used in the stapes reconstruction restores stapes, but the effects of the nonlinear material parameters on the s...Stapes fracture causes hearing loss and instability in the middle ear hearing system(MEHS). The material used in the stapes reconstruction restores stapes, but the effects of the nonlinear material parameters on the stability of the MEHS are still unknown. To address this challenge, the nonlinear dynamic response and stability of the stapes reconstruction are investigated using a multi-degree-of-freedom mechanical model. The material parameters of the implant are tentatively determined by analyzing the natural frequencies of the undamped system. The dynamical properties of the MEHS are characterized under different external excitations. The approximate solution of the MEHS near the resonant frequency is derived through the multiple-time-scale method(MTSM). The results show that the nonlinear stiffness of the material has little influence on the MEHS in the healthy state, but it causes resonant phenomena between the ossicle and the implant in the pathological state.展开更多
In this paper, we propose a discrete ratio-dependent predator-prey system. The stability of the fixed points of this model is studied. At the same time, it is shown that the discrete model undergoes fold bifurcation a...In this paper, we propose a discrete ratio-dependent predator-prey system. The stability of the fixed points of this model is studied. At the same time, it is shown that the discrete model undergoes fold bifurcation and flip bifurcation by using bifurcation theory and the method of approximation by a flow. Numerical simulations are presented not only to demonstrate the consistence with our theoretical analyses, but also to exhibit the complex dynamical behaviors, such as the cascade of period-doubling bifurcation in period-2 and the chaotic sets. The Maximum Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors. These results show that the direct discrete method has more rich dynamic behaviors than the discrete model obtained by Euler method.展开更多
This paper reports a new four-dimensional hyperchaotic system obtained by adding a controller to a threedimensional autonomous chaotic system. The new system has two parameters, and each equation of the system has one...This paper reports a new four-dimensional hyperchaotic system obtained by adding a controller to a threedimensional autonomous chaotic system. The new system has two parameters, and each equation of the system has one quadratic cross-product term. Some basic properties of the new system are analysed. The different dynamic behaviours of the new system are studied when the system parameter a or b is varied. The system is hyperchaotic in several different regions of the parameter b. Especially, the two positive Lyapunov exponents are both larger, and the hyperchaotic region is also larger when this system is hyperchaotic in the case of varying a. The hyperchaotic system is analysed by Lyapunov-exponents spectrum, bifurcation diagrams and Poincaré sections.展开更多
This paper presents a new 3D quadratic autonomous chaotic system which contains five system parameters and three quadratic cross-product terms,and the system can generate a single four-wing chaotic attractor with wide...This paper presents a new 3D quadratic autonomous chaotic system which contains five system parameters and three quadratic cross-product terms,and the system can generate a single four-wing chaotic attractor with wide parameter ranges. Through theoretical analysis,the Hopf bifurcation processes are proved to arise at certain equilibrium points.Numerical bifurcation analysis shows that the system has many interesting complex dynamical behaviours;the system trajectory can evolve to a chaotic attractor from a periodic orbit or a fixed point as the proper parameter varies. Finally,an analog electronic circuit is designed to physically realize the chaotic system;the existence of four-wing chaotic attractor is verified by the analog circuit realization.展开更多
In view of engineering application, it is practicable to decompose the aerodynamics into three components: the static aerodynamics, the aerodynamic increment due to steady rotations, and the aerodynamic increment due...In view of engineering application, it is practicable to decompose the aerodynamics into three components: the static aerodynamics, the aerodynamic increment due to steady rotations, and the aerodynamic increment due to unsteady separated and vortical flow. The first and the second components can be presented in conventional forms, while the third is described using a one-order differential equation and a radial-basis-function (RBF) network. For an aircraft configuration, the mathematical models of 6- component aerodynamic coefficients are set up from the wind tunnel test data of pitch, yaw, roll, and coupled yawroll large-amplitude oscillations. The flight dynamics of an aircraft is studied by the bifurcation analysis technique in the case of quasi-steady aerodynamics and unsteady aerodynam- ics, respectively. The results show that: (1) unsteady aerodynamics has no effect upon the existence of trim points, but affects their stability; (2) unsteady aerodynamics has great effects upon the existence, stability, and amplitudes of periodic solutions; and (3) unsteady aerodynamics changes the stable regions of trim points obviously. Furthermore, the dynamic responses of the aircraft to elevator deflections are inspected. It is shown that the unsteady aerodynamics is beneficial to dynamic stability for the present aircraft. Finally, the effects of unsteady aerodynamics on the post-stall maneuverability展开更多
We study the swelling of a gel annulus attached to a rigid core when it is immersed in a solvent.For equilibrium states,the free-energy function of the gel can be converted into a strain energy function,and as a resul...We study the swelling of a gel annulus attached to a rigid core when it is immersed in a solvent.For equilibrium states,the free-energy function of the gel can be converted into a strain energy function,and as a result the gel can be treated as a compressible hyperelastic material.Asymptotic methods are used to study the inhomogeneous swelling in order to obtain the leading-order solution.Some analytical insights are then deduced.Because of the compressive hoop stress in this state,at some stage instability can occur,leading to wrinkles in the gel.An incremental deformation theory in nonlinear elasticity is used to conduct a linear bifurcation analysis for understanding such instability.More specifically,the critical loading for the onset of a wrinkled state is obtained.Detailed discussions on the behaviors of various physical quantities in this critical state are given.It is found that the critical mode number,while insensitive to the material parameters,is greatly influenced by the ratio of the inner and outer radii of the gel.Also,an interesting finding is that the critical swelling ratio is an increasing function of this geometrical parameter,which implies that a thin annulus is more likely to be unstable than a thick one.展开更多
Inositol 1,4,5-trisphosphate receptors(IP_(3)R)-mediated calcium ion(Ca^(2+))release plays a central role in the regulation of cell survival and death.Bcl-2 limits the Ca^(2+)release function of the IP3R through a dir...Inositol 1,4,5-trisphosphate receptors(IP_(3)R)-mediated calcium ion(Ca^(2+))release plays a central role in the regulation of cell survival and death.Bcl-2 limits the Ca^(2+)release function of the IP3R through a direct or indirect mechanism.However,the two mechanisms are overwhelmingly complex and not completely understood.Here,we convert the mechanisms into a set of ordinary differential equations.We firstly simulate the time evolution of Ca^(2+)concentration under two different levels of Bcl-2 for the direct and indirect mechanism models and compare them with experimental results available in the literature.Secondly,we employ one-and two-parameter bifurcation analysis to demonstrate that Bcl-2 can suppress Ca^(2+)signal from a global point of view both in the direct and indirect mechanism models.We then use mathematical analysis to clarify that the indirect mechanism is more efficient than the direct mechanism in repressing Ca^(2+)signal.Lastly,we predict that the two mechanisms restrict Ca^(2+)signal synergistically.Together,our study provides theoretical insights into Bcl-2 regulation in IP_(3)R-mediated Ca^(2+)release,which may be instrumental for the successful development of therapies to target Bcl-2 for cancer treatment.展开更多
Based on the principle given in nonlinear diffusion-reaction dynamics, a new dynamic model for dislocation patterning is proposed by introducing a relaxation time to the relation between dislocation density and disloc...Based on the principle given in nonlinear diffusion-reaction dynamics, a new dynamic model for dislocation patterning is proposed by introducing a relaxation time to the relation between dislocation density and dislocation flux. The so-called chemical potential like quantities, which appear in the model can be derived from variation principle for free energy functional of dislocated media, where the free energy density function is expressed in terms of not only the dislocation density itself but also their spatial gradients. The Linear stability analysis on the governing equations of a simple dislocation density shows that there exists an intrinsic wave number leading to bifurcation of space structure of dislocation density. At the same time, the numerical results also demonstrate the coexistence and transition between different dislocation patterns.展开更多
We present a class of two-dimensional memristive maps with a cosine memristor. The memristive maps do not have any fixed points, so they belong to the category of nonlinear maps with hidden attractors. The rich dynami...We present a class of two-dimensional memristive maps with a cosine memristor. The memristive maps do not have any fixed points, so they belong to the category of nonlinear maps with hidden attractors. The rich dynamical behaviors of these maps are studied and investigated using different numerical tools, including phase portrait, basins of attraction,bifurcation diagram, and Lyapunov exponents. The two-parameter bifurcation analysis of the memristive map is carried out to reveal the bifurcation mechanism of its dynamical behaviors. Based on our extensive simulation studies, the proposed memristive maps can produce hidden periodic, chaotic, and hyper-chaotic attractors, exhibiting extremely hidden multistability, namely the coexistence of infinite hidden attractors, which was rarely observed in memristive maps. Potentially,this work can be used for some real applications in secure communication, such as data and image encryptions.展开更多
An extended car-following model with multiple delays is constructed to describe driver's driving behavior.Through stability analysis,the stability condition of this uncontrolled model is given.To dampen the negati...An extended car-following model with multiple delays is constructed to describe driver's driving behavior.Through stability analysis,the stability condition of this uncontrolled model is given.To dampen the negative impact of the driver's multiple delays(i.e.,stability condition is not satisfied),a novel control strategy is proposed to assist the driver in adjusting vehicle operation.The control strategy consists of two parts:the design of control term as well as the design of the parameters in the term.Bifurcation analysis is performed to illustrate the necessity of the design of parameters in control terms.In the course of the design of parameters in the control term,we improve the definite integral stability method to reduce the iterations by incorporating the characteristics of bifurcation,which can determine the appropriate parameters in the control terms more quickly.Finally,in the case study,we validate the control strategy by utilizing measured data and configuring scenario,which is closer to the actual traffic.The results of validation show that the control strategy can effectively stabilize the unstable traffic flow caused by driver's delays.展开更多
The Whitham-Broer-Kaup model is widely used to study the tsunami waves.The classical Whitham-Broer-Kaup equations are re-investigated in detail by the generalized projective Riccati-equation method.20 sets of solution...The Whitham-Broer-Kaup model is widely used to study the tsunami waves.The classical Whitham-Broer-Kaup equations are re-investigated in detail by the generalized projective Riccati-equation method.20 sets of solutions are obtained of which,to the best of the authors’knowledge,some have not been reported in literature.Bifurcation analysis of the planar dynamical systems is then used to show different phase portraits of the traveling wave solutions under various parametric conditions.展开更多
We provide an overview of current techniques and typical applications of numerical bifurcation analysis in fluid dynamical problems.Many of these problems are characterized by high-dimensional dynamical systems which ...We provide an overview of current techniques and typical applications of numerical bifurcation analysis in fluid dynamical problems.Many of these problems are characterized by high-dimensional dynamical systems which undergo transitions as parameters are changed.The computation of the critical conditions associated with these transitions,popularly referred to as‘tipping points’,is important for understanding the transition mechanisms.We describe the two basic classes of methods of numerical bifurcation analysis,which differ in the explicit or implicit use of the Jacobian matrix of the dynamical system.The numerical challenges involved in both methods are mentioned and possible solutions to current bottlenecks are given.To demonstrate that numerical bifurcation techniques are not restricted to relatively low-dimensional dynamical systems,we provide several examples of the application of the modern techniques to a diverse set of fluid mechanical problems.展开更多
For food chain system with three populations,direct predation is the basic interaction between species.Different species often have different predation functional responses,so a food chain system with Holling-II respo...For food chain system with three populations,direct predation is the basic interaction between species.Different species often have different predation functional responses,so a food chain system with Holling-II response for middle predator and Beddinton-DeAngelis response for top predator is proposed.Apart from direct predation,predator population can significantly impact the survival of prey population by inducing the prey's fear,but the impact often possesses a time delay.This paper is concentrated to explore how the fear and time delay affect the system stability and the species persistence.By use of Lyapunov functional method and bifurcation theory,the positiveness and boundedness of solutions,local and global behavior of species,the system stability around the equilibrium states and various kinds of bifurcation are investigated.Numerically,some simulations are carried out to validate the main findings and the critical values of the bifurcation parameters of fear and conversion rate are obtained.It is observed that fear and delay can not only stabilize,but also destabilize the system,which depends on the magnitude of the fear and delay.The system varies from unstable to stable due to the continuous increase of the prey's fear by middle predator.Small fear induced by top predator or small delay of the prey's fear can stabilize the system,while they are sufficiently large,the system stability is to be destroyed.Simultaneously,the conversion rate can also change the stability and even make the species to be extinct.Some rich dynamics like multiple stabilities and various types of bistability behaviors are also exhibited,which results in the convergence of the species from one stable equilibrium to another.展开更多
Meningococcal meningitis(MCM)is one of the serious public health threats in the tropical and sub-tropical regions.In this paper,we propose an epidemic model to study the transmission dynamics of MCM with high-and low-...Meningococcal meningitis(MCM)is one of the serious public health threats in the tropical and sub-tropical regions.In this paper,we propose an epidemic model to study the transmission dynamics of MCM with high-and low-risk susceptible populations.The model considers two different groups of susceptible individuals depending on the availability of medical resources(MR,including hospitals,health workers,etc.),which varies the infection risk.We find that the model exhibits the phenomenon of backward bifurcation(BB),which increases the difficulty of MCM control since the dynamics are not merely relying on the basic reproduction number,TZo.This study explores the effects of MR on the MCM epidemics by mathematical analysis and shows the existence of BB on MCM disease.Our findings suggest that providing adequate MR in a community is crucial in mitigating MCM incidences and deaths,especially,in the MCM endemic regions.展开更多
基金Project supported by the Natural Science Foundation of Shandong Province (Grant No.ZR2021MA084)the Natural Science Foundation of Liaocheng University (Grant No.318012025)Discipline with Strong Characteristics of Liaocheng University–Intelligent Science and Technology (Grant No.319462208)。
文摘The time-fractional modified Korteweg-de Vries(KdV)equation is committed to establish exact solutions by employing the bifurcation method.Firstly,the phase portraits and related qualitative analysis are comprehensively provided.Then,we give parametric expressions of different types of solutions matching with the corresponding orbits.Finally,solution profiles,3D and density plots of some solutions are presented with proper parametric choices.
文摘This paper presents a bifurcation study of a mRNA-protein network with negative feedback and time delay. The network is modeled as a coupled system of N ordinary differential equations (ODEs) and N delay differential equations (DDEs). Linear analysis of the stable equilibria shows the existence of a critical time delay beyond which limit cycle oscillations are born in a Hopf bifurcation. The Poincaré-Lindstedt perturbation method is applied to the nonlinear system, resulting in closed form approximate expressions for the amplitude and frequency of oscillation. We confirm our perturbation analysis results by numerically simulating the full 2N-dimensional nonlinear system for N = 2, 7, 15, and 30. Our results show that for small perturbations the equilibrium undergoes a supercritical Hopf and the system exhibits stable periodic solutions. Furthermore, our closed form numerical expressions exhibit the importance of the network’s negative feedback and nonlinear effects.
基金the Deanship of Scientific Research at King Khalid University for funding this work through the Big Research Group Project under grant number(R.G.P2/16/40).
文摘In this paper,a discrete Lotka-Volterra predator-prey model is proposed that considers mixed functional responses of Holling types I and III.The equilibrium points of the model are obtained,and their stability is tested.The dynamical behavior of this model is studied according to the change of the control parameters.We find that the complex dynamical behavior extends from a stable state to chaotic attractors.Finally,the analytical results are clarified by some numerical simulations.
基金the Natural Science Foundation of Zhejiang Province,China(Grant Nos.LY22G010001,LY20G010004)the Program of Humanities and Social Science of Education Ministry of China(Grant No.20YJA630008)+1 种基金the National Key Research and Development Program of China-Traffic Modeling,Surveillance and Control with Connected&Automated Vehicles(Grant No.2017YFE9134700)the K.C.Wong Magna Fund in Ningbo University,China。
文摘In the light of the visual angle model(VAM),an improved car-following model considering driver's visual angle,anticipated time and stabilizing driving behavior is proposed so as to investigate how the driver's behavior factors affect the stability of the traffic flow.Based on the model,linear stability analysis is performed together with bifurcation analysis,whose corresponding stability condition is highly fit to the results of the linear analysis.Furthermore,the time-dependent Ginzburg–Landau(TDGL)equation and the modified Korteweg–de Vries(m Kd V)equation are derived by nonlinear analysis,and we obtain the relationship of the two equations through the comparison.Finally,parameter calibration and numerical simulation are conducted to verify the validity of the theoretical analysis,whose results are highly consistent with the theoretical analysis.
基金The NSF(11361029)of Chinathe NSF(20142BAB211001)of Jiangxi Province
文摘This paper is concerned with the bifurcation analysis for a free boundary problem modeling the growth of solid tumor with inhibitors.In this problem,surface tension coefficient plays the role of bifurcation parameter,it is proved that there exists a sequence of the nonradially stationary solutions bifurcate from the radially symmetric stationary solutions.Our results indicate that the tumor grown in vivo may have various shapes.In particular,a tumor with an inhibitor is associated with the growth of protrusions.
基金supported by the National Natural Science Foundation of China(Nos.51905264 and 12002157)the China Postdoctoral Science Foundation Funded Project,China(Nos.2019M650115,2019M661818 and 2020T130298)+3 种基金the Science&Technology Innovation Project for Overseas Scholars in Nanjing,China(No.YQR20046)the National Defense Outstanding Youth Science Foundation,China(No.2018-JCJQ-ZQ-053)the Fundamental Research Funds for the Central Universities,China(No.NF2018001)the Priority Academic Program Development of Jiangsu Higher Education Institutions,China。
文摘A loss of ground directional stability can trigger a high-speed Unmanned Aerial Vehicle(UAV)to veer off the runway.In order to investigate the combined effects of the key structural and operational parameters on the UAV ground directional stability from a global perspective,a fully parameterized mathematical high-speed UAV ground nonlinear dynamic model is developed considering several nonlinear factors.The bifurcation analysis procedure of a UAV ground steering system is introduced,following which the simulation efficiency is greatly improved comparing with the time-domain simulation method.Then the numerical continuation method is employed to investigate the influence of the nose wheel steering angle and the global stability region is obtained.The bifurcation parameter plane is divided into several parts with different stability properties by the saddle nodes and the Hopf bifurcation points.We find that the UAV motion states will never cross the bifurcation curve in the nonlinear system.Also,the dual-parameter bifurcation analyses are presented to give a complete description of the possible steering performance.It is also found that BT bifurcation appears when the UAV initial rectilinear velocity and the tire frictional coefficient vary.In addition,results indicate that the influence of tire frictional coefficient has an opposite trend to the influence of initial rectilinear velocity.Overall,using bifurcation analysis method to identify the parameter regions of a UAV nonlinear ground dynamic system helps to improve the development efficiency and quality during UAV designing phase.
基金Project supported by the National Natural Science Foundation of China (Nos. 12072222, 12132010,12021002, 11991032, and 12372019)the State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures of China (No. SKLTESKF1901)the Aeronautical Science Foundation of China (No. ASFC-201915048001)。
文摘Stapes fracture causes hearing loss and instability in the middle ear hearing system(MEHS). The material used in the stapes reconstruction restores stapes, but the effects of the nonlinear material parameters on the stability of the MEHS are still unknown. To address this challenge, the nonlinear dynamic response and stability of the stapes reconstruction are investigated using a multi-degree-of-freedom mechanical model. The material parameters of the implant are tentatively determined by analyzing the natural frequencies of the undamped system. The dynamical properties of the MEHS are characterized under different external excitations. The approximate solution of the MEHS near the resonant frequency is derived through the multiple-time-scale method(MTSM). The results show that the nonlinear stiffness of the material has little influence on the MEHS in the healthy state, but it causes resonant phenomena between the ossicle and the implant in the pathological state.
文摘In this paper, we propose a discrete ratio-dependent predator-prey system. The stability of the fixed points of this model is studied. At the same time, it is shown that the discrete model undergoes fold bifurcation and flip bifurcation by using bifurcation theory and the method of approximation by a flow. Numerical simulations are presented not only to demonstrate the consistence with our theoretical analyses, but also to exhibit the complex dynamical behaviors, such as the cascade of period-doubling bifurcation in period-2 and the chaotic sets. The Maximum Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors. These results show that the direct discrete method has more rich dynamic behaviors than the discrete model obtained by Euler method.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 60374037 and 60574036), the Specialized Research Fund for the Doctoral Program of China (Grant No 20050055013) and the Program for New Century Excellent Talents in University of China (NCET).
文摘This paper reports a new four-dimensional hyperchaotic system obtained by adding a controller to a threedimensional autonomous chaotic system. The new system has two parameters, and each equation of the system has one quadratic cross-product term. Some basic properties of the new system are analysed. The different dynamic behaviours of the new system are studied when the system parameter a or b is varied. The system is hyperchaotic in several different regions of the parameter b. Especially, the two positive Lyapunov exponents are both larger, and the hyperchaotic region is also larger when this system is hyperchaotic in the case of varying a. The hyperchaotic system is analysed by Lyapunov-exponents spectrum, bifurcation diagrams and Poincaré sections.
基金Project supported by the National Natural Science Foundation of China(Grant Nos 60774088 and 10772135)the Foundation of the Application Base and Frontier Technology Research Project of Tianjin,China (Grant Nos 07JCZDJC09600,08JCZDJC21900 and 08JCZDJC18600)the Tianjin Key Laboratory for Control Theory & Applications in Complicated Industry Systems of China
文摘This paper presents a new 3D quadratic autonomous chaotic system which contains five system parameters and three quadratic cross-product terms,and the system can generate a single four-wing chaotic attractor with wide parameter ranges. Through theoretical analysis,the Hopf bifurcation processes are proved to arise at certain equilibrium points.Numerical bifurcation analysis shows that the system has many interesting complex dynamical behaviours;the system trajectory can evolve to a chaotic attractor from a periodic orbit or a fixed point as the proper parameter varies. Finally,an analog electronic circuit is designed to physically realize the chaotic system;the existence of four-wing chaotic attractor is verified by the analog circuit realization.
文摘In view of engineering application, it is practicable to decompose the aerodynamics into three components: the static aerodynamics, the aerodynamic increment due to steady rotations, and the aerodynamic increment due to unsteady separated and vortical flow. The first and the second components can be presented in conventional forms, while the third is described using a one-order differential equation and a radial-basis-function (RBF) network. For an aircraft configuration, the mathematical models of 6- component aerodynamic coefficients are set up from the wind tunnel test data of pitch, yaw, roll, and coupled yawroll large-amplitude oscillations. The flight dynamics of an aircraft is studied by the bifurcation analysis technique in the case of quasi-steady aerodynamics and unsteady aerodynam- ics, respectively. The results show that: (1) unsteady aerodynamics has no effect upon the existence of trim points, but affects their stability; (2) unsteady aerodynamics has great effects upon the existence, stability, and amplitudes of periodic solutions; and (3) unsteady aerodynamics changes the stable regions of trim points obviously. Furthermore, the dynamic responses of the aircraft to elevator deflections are inspected. It is shown that the unsteady aerodynamics is beneficial to dynamic stability for the present aircraft. Finally, the effects of unsteady aerodynamics on the post-stall maneuverability
文摘We study the swelling of a gel annulus attached to a rigid core when it is immersed in a solvent.For equilibrium states,the free-energy function of the gel can be converted into a strain energy function,and as a result the gel can be treated as a compressible hyperelastic material.Asymptotic methods are used to study the inhomogeneous swelling in order to obtain the leading-order solution.Some analytical insights are then deduced.Because of the compressive hoop stress in this state,at some stage instability can occur,leading to wrinkles in the gel.An incremental deformation theory in nonlinear elasticity is used to conduct a linear bifurcation analysis for understanding such instability.More specifically,the critical loading for the onset of a wrinkled state is obtained.Detailed discussions on the behaviors of various physical quantities in this critical state are given.It is found that the critical mode number,while insensitive to the material parameters,is greatly influenced by the ratio of the inner and outer radii of the gel.Also,an interesting finding is that the critical swelling ratio is an increasing function of this geometrical parameter,which implies that a thin annulus is more likely to be unstable than a thick one.
基金supported by Shanxi Province Science Foundation for Youths(Grant No.201901D211159)the National Natural Science Foundation of China(Grant Nos.11504214,11874310,and 12090052).
文摘Inositol 1,4,5-trisphosphate receptors(IP_(3)R)-mediated calcium ion(Ca^(2+))release plays a central role in the regulation of cell survival and death.Bcl-2 limits the Ca^(2+)release function of the IP3R through a direct or indirect mechanism.However,the two mechanisms are overwhelmingly complex and not completely understood.Here,we convert the mechanisms into a set of ordinary differential equations.We firstly simulate the time evolution of Ca^(2+)concentration under two different levels of Bcl-2 for the direct and indirect mechanism models and compare them with experimental results available in the literature.Secondly,we employ one-and two-parameter bifurcation analysis to demonstrate that Bcl-2 can suppress Ca^(2+)signal from a global point of view both in the direct and indirect mechanism models.We then use mathematical analysis to clarify that the indirect mechanism is more efficient than the direct mechanism in repressing Ca^(2+)signal.Lastly,we predict that the two mechanisms restrict Ca^(2+)signal synergistically.Together,our study provides theoretical insights into Bcl-2 regulation in IP_(3)R-mediated Ca^(2+)release,which may be instrumental for the successful development of therapies to target Bcl-2 for cancer treatment.
基金The National Natural Science Foundation of China,Grant No.19392300
文摘Based on the principle given in nonlinear diffusion-reaction dynamics, a new dynamic model for dislocation patterning is proposed by introducing a relaxation time to the relation between dislocation density and dislocation flux. The so-called chemical potential like quantities, which appear in the model can be derived from variation principle for free energy functional of dislocated media, where the free energy density function is expressed in terms of not only the dislocation density itself but also their spatial gradients. The Linear stability analysis on the governing equations of a simple dislocation density shows that there exists an intrinsic wave number leading to bifurcation of space structure of dislocation density. At the same time, the numerical results also demonstrate the coexistence and transition between different dislocation patterns.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11972173 and 12172340)。
文摘We present a class of two-dimensional memristive maps with a cosine memristor. The memristive maps do not have any fixed points, so they belong to the category of nonlinear maps with hidden attractors. The rich dynamical behaviors of these maps are studied and investigated using different numerical tools, including phase portrait, basins of attraction,bifurcation diagram, and Lyapunov exponents. The two-parameter bifurcation analysis of the memristive map is carried out to reveal the bifurcation mechanism of its dynamical behaviors. Based on our extensive simulation studies, the proposed memristive maps can produce hidden periodic, chaotic, and hyper-chaotic attractors, exhibiting extremely hidden multistability, namely the coexistence of infinite hidden attractors, which was rarely observed in memristive maps. Potentially,this work can be used for some real applications in secure communication, such as data and image encryptions.
基金Project supported by the Natural Science Foundation of Zhejiang Province,China(Grant No.LY20G010004)the Program of Humanities and Social Science of Education Ministry of China(Grant No.20YJA630008)+1 种基金the National Key Research and Development Program of China–Traffic Modeling,Surveillance and Control with Connected&Automated Vehicles(Grant No.2017YFE9134700)the K.C.Wong Magna Fund in Ningbo University,China。
文摘An extended car-following model with multiple delays is constructed to describe driver's driving behavior.Through stability analysis,the stability condition of this uncontrolled model is given.To dampen the negative impact of the driver's multiple delays(i.e.,stability condition is not satisfied),a novel control strategy is proposed to assist the driver in adjusting vehicle operation.The control strategy consists of two parts:the design of control term as well as the design of the parameters in the term.Bifurcation analysis is performed to illustrate the necessity of the design of parameters in control terms.In the course of the design of parameters in the control term,we improve the definite integral stability method to reduce the iterations by incorporating the characteristics of bifurcation,which can determine the appropriate parameters in the control terms more quickly.Finally,in the case study,we validate the control strategy by utilizing measured data and configuring scenario,which is closer to the actual traffic.The results of validation show that the control strategy can effectively stabilize the unstable traffic flow caused by driver's delays.
基金Project supported by the National Natural Science Foundation of China(No.11872241)the Discovery Early Career Researcher Award(No.DE150100169)the Centre of Excellence Grant funded by the Australian Research Council(No.CE140100003)。
文摘The Whitham-Broer-Kaup model is widely used to study the tsunami waves.The classical Whitham-Broer-Kaup equations are re-investigated in detail by the generalized projective Riccati-equation method.20 sets of solutions are obtained of which,to the best of the authors’knowledge,some have not been reported in literature.Bifurcation analysis of the planar dynamical systems is then used to show different phase portraits of the traveling wave solutions under various parametric conditions.
基金The workshop and the work of F.W.Wubs and H.A.Dijkstra was partially sponsored by the Netherlands Organization of Scientific Research(NWO)through the NWOCOMPLEXITY project PreKursThe participation of F.I.Dragomirescu to the workshop was partially supported by a Grant of the Romanian National Authority for Scientific Research,CNCS-UEFISCDI,project number PN-II-RU-PD-2011-3-0153,31/5.10.2011.Sandia National Laboratory is a multiprogram laboratory operated by Sandia Corporation,a Lockheed Martin Company,for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000The work of J.Sanchez-Umbria was supported by projects MTM2010-16930 and 2009-SGR-67.
文摘We provide an overview of current techniques and typical applications of numerical bifurcation analysis in fluid dynamical problems.Many of these problems are characterized by high-dimensional dynamical systems which undergo transitions as parameters are changed.The computation of the critical conditions associated with these transitions,popularly referred to as‘tipping points’,is important for understanding the transition mechanisms.We describe the two basic classes of methods of numerical bifurcation analysis,which differ in the explicit or implicit use of the Jacobian matrix of the dynamical system.The numerical challenges involved in both methods are mentioned and possible solutions to current bottlenecks are given.To demonstrate that numerical bifurcation techniques are not restricted to relatively low-dimensional dynamical systems,we provide several examples of the application of the modern techniques to a diverse set of fluid mechanical problems.
基金funded by the National Natural Science Foundation of China with number 11861027.
文摘For food chain system with three populations,direct predation is the basic interaction between species.Different species often have different predation functional responses,so a food chain system with Holling-II response for middle predator and Beddinton-DeAngelis response for top predator is proposed.Apart from direct predation,predator population can significantly impact the survival of prey population by inducing the prey's fear,but the impact often possesses a time delay.This paper is concentrated to explore how the fear and time delay affect the system stability and the species persistence.By use of Lyapunov functional method and bifurcation theory,the positiveness and boundedness of solutions,local and global behavior of species,the system stability around the equilibrium states and various kinds of bifurcation are investigated.Numerically,some simulations are carried out to validate the main findings and the critical values of the bifurcation parameters of fear and conversion rate are obtained.It is observed that fear and delay can not only stabilize,but also destabilize the system,which depends on the magnitude of the fear and delay.The system varies from unstable to stable due to the continuous increase of the prey's fear by middle predator.Small fear induced by top predator or small delay of the prey's fear can stabilize the system,while they are sufficiently large,the system stability is to be destroyed.Simultaneously,the conversion rate can also change the stability and even make the species to be extinct.Some rich dynamics like multiple stabilities and various types of bistability behaviors are also exhibited,which results in the convergence of the species from one stable equilibrium to another.
文摘Meningococcal meningitis(MCM)is one of the serious public health threats in the tropical and sub-tropical regions.In this paper,we propose an epidemic model to study the transmission dynamics of MCM with high-and low-risk susceptible populations.The model considers two different groups of susceptible individuals depending on the availability of medical resources(MR,including hospitals,health workers,etc.),which varies the infection risk.We find that the model exhibits the phenomenon of backward bifurcation(BB),which increases the difficulty of MCM control since the dynamics are not merely relying on the basic reproduction number,TZo.This study explores the effects of MR on the MCM epidemics by mathematical analysis and shows the existence of BB on MCM disease.Our findings suggest that providing adequate MR in a community is crucial in mitigating MCM incidences and deaths,especially,in the MCM endemic regions.