This paper investigates the stability and bifurcation phenomena of a cholera transmission model in which individuals who have recovered from the disease may become susceptible again.The threshold for determining disea...This paper investigates the stability and bifurcation phenomena of a cholera transmission model in which individuals who have recovered from the disease may become susceptible again.The threshold for determining disease prevalence is established,and the parameter conditions for the existence of equilibria are discussed.The Routh-Hurwitz criterion is applied to demonstrate the local asymptotic stability of equilibria.By utilizing composite matrices and geometric techniques,the global dynamic behavior of the endemic equilibrium is investigated,and the sufficient conditions for its global asymptotic stability are derived.Furthermore,the disease-free equilibrium is a saddle-node when the basic reproductive number is 1,and tthe transcritical bifurcation in this case is discussed.展开更多
In the paper, under the framework of exploring the interaction between algae and bacteria, an algae-bacteria ecological model was established to analyze the interaction mechanism and growth coexistence mode between al...In the paper, under the framework of exploring the interaction between algae and bacteria, an algae-bacteria ecological model was established to analyze the interaction mechanism and growth coexistence mode between algicidal bacteria and algae. Firstly, mathematical work mainly provided some threshold conditions to ensure the occurrence of transcritical bifurcation and saddle-node bifurcation, which could provide certain theoretical support for selecting key ecological environmental factors and numerical simulations. Secondly, the numerical simulation work dynamically displayed the evolution process of the bifurcation dynamic behavior of the model (2.1) and the growth coexistence mode of algae and algicidal bacteria. Finally, it was worth summarizing that intrinsic growth rate and combined capture effort of algae population had a strong influence on the dynamic behavior of the model (2.1). Furthermore, it must also be noted that transcritical bifurcation and saddle-node bifurcation were the inherent driving forces behind the formation of steady-state growth coexistence mode between algicidal bacteria and algae. In summary, it was hoped that the results of this study would contribute to accelerating the study of the interaction mechanism between algicidal bacteria and algae.展开更多
A weakly nonlinear oscillator was modeled by a sort of differential equation, a saddle-node bifurcation was found in case of primary and secondary resonance. To control the jumping phenomena and the unstable region of...A weakly nonlinear oscillator was modeled by a sort of differential equation, a saddle-node bifurcation was found in case of primary and secondary resonance. To control the jumping phenomena and the unstable region of the nonlinear oscillator, feedback controllers were designed. Bifurcation control equations were obtained by using the multiple scales method. And through the numerical analysis, good controller could be obtained by changing the feedback control gain. Then a feasible way of further research of saddle-node bifurcation was provided. Finally, an example shows that the feedback control method applied to the hanging bridge system of gas turbine is doable.展开更多
The study of dynamical behavior of water or gas flows in broken rock is a basic research topic among a series of key projects about stability control of the surrounding rocks in mines and the prevention of some disast...The study of dynamical behavior of water or gas flows in broken rock is a basic research topic among a series of key projects about stability control of the surrounding rocks in mines and the prevention of some disasters such as water inrush or gas outburst and the protection of the groundwater resource. It is of great theoretical and engineering importance in respect of promo- tion of security in mine production and sustainable development of the coal industry. According to the non-Darcy property of seepage flow in broken rock dynamic equations of non-Darcy and non-steady flows in broken rock are established. By dimensionless transformation, the solution diagram of steady-states satisfying the given boundary conditions is obtained. By numerical analysis of low relaxation iteration, the dynamic responses corresponding to the different flow parameters have been obtained. The stability analysis of the steady-states indicate that a saddle-node bifurcaton exists in the seepage flow system of broken rock. Consequently, using catastrophe theory, the fold catastrophe model of seepage flow instability has been obtained. As a result, the bifurcation curves of the seepage flow systems with different control parameters are presented and the standard potential function is also given with respect to the generalized state variable for the fold catastrophe of a dynamic system of seepage flow in broken rock.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
The singularly perturbed bifurcation subsystem is described, and the test conditions of subsystem persistence are deduced. By use of fast and slow reduced subsystem model, the result does not require performing nonlin...The singularly perturbed bifurcation subsystem is described, and the test conditions of subsystem persistence are deduced. By use of fast and slow reduced subsystem model, the result does not require performing nonlinear transformation. Moreover, it is shown and proved that the persistence of the periodic orbits for Hopf bifurcation in the reduced model through center manifold. Van der Pol oscillator circuit is given to illustrate the persistence of bifurcation subsystems with the full dynamic system.展开更多
In this paper n-dimensional flows （described by continuous-time system） with static bifurcations are considered with the aim of classification of different elementary bifurcations using the frequency domain formalis...In this paper n-dimensional flows （described by continuous-time system） with static bifurcations are considered with the aim of classification of different elementary bifurcations using the frequency domain formalism. Based on frequency domain approach, we prove some criterions for the saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation, and give an example to illustrate the efficiency of the result obtained.展开更多
In this paper, we examine a discrete-time Host-Parasitoid model which is a non-dimensionalized Nicholson and Bailey model. Phase portraits are drawn for different ranges of parameters and display the complicated dynam...In this paper, we examine a discrete-time Host-Parasitoid model which is a non-dimensionalized Nicholson and Bailey model. Phase portraits are drawn for different ranges of parameters and display the complicated dynamics of this system. We conduct the bifurcation analysis with respect to intrinsic growth rate <em>r</em> and searching efficiency <em>a</em>. Many forms of complex dynamics such as chaos, periodic windows are observed. Transition route to chaos dynamics is established via period-doubling bifurcations. Conditions of occurrence of the period-doubling, Neimark-Sacker and saddle-node bifurcations are analyzed for <em>b≠a</em> where <em>a,b</em> are searching efficiency. We study stable and unstable manifolds for different equilibrium points and coexistence of different attractors for this non-dimensionalize system. Without the parasitoid, the host population follows the dynamics of the Ricker model.展开更多
The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf...The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf bifurcation is given. Both the period-doubling bifurcation and saddle-node bifurcation of periodical solutions are computed since the observed floquet multiplier overpass the unit circle by DDE-Biftool software in Matlab. The continuation of saddle-node bifurcation line or period-doubling curve is carried out as varying free parameters and time delays. Two different transition modes of saddle-node bifurcation are discovered which is verified by numerical simulation work with aids of DDE-Biftool.展开更多
Due to the random search of species and from the economic point of view,combined harvesting is more suitable than selective harvesting.Thus,we have developed and analyzed a prey-predator model with the combined effect...Due to the random search of species and from the economic point of view,combined harvesting is more suitable than selective harvesting.Thus,we have developed and analyzed a prey-predator model with the combined effect of nonlinear harvesting in this research paper.Nonlinear harvesting possesses multiple predator-free and interior equilibrium points in the dynamical system.We have examined the local stability analysis of all the equilibrium points.Besides these various types,rich and complex dynamical behaviors such as backward,saddle-node,Hopf and Bogdanov-Takens(BT)bifurcations,homoclinic loop and limit cycles appear in this model.Furthermore,interesting phenomena like bi-stability and tri-stability occur in our model between the different equilibrium points.Also,we have derived different threshold values of predator harvesting parameters and prey environmental carrying capacity from these bifurcations to obtain the different harvesting strategies for both species.We have observed that the extinction of predator species may not happen due to backward bifurcation,although a stable predator-free equilibrium(PFE)exists.Finally,numerical simulations are discussed using MATLAB to verify all the theoretical results.展开更多
According to the H_(∞)principle,the dynamical performance optimization of a quasi-zero-stiffness(QZS)isolation system with an additional tuned viscous inerter damper(TVID)is studied by using analytical method.The app...According to the H_(∞)principle,the dynamical performance optimization of a quasi-zero-stiffness(QZS)isolation system with an additional tuned viscous inerter damper(TVID)is studied by using analytical method.The approximate analytical solutions of the QZS system coupled with TVID are solved by using the complexification-averaging method,and the expression of stability conditions for steady-state solutions is derived based on Lyapunov method and Routh-Hurwitz criterion.Based on the fixed-point theory,considering the nonlinear stiffness and weak damping of the primary system,the stiffness and damping ratios of TVID coupled to QZS system are optimized by using the equal-peak method.The detailed analysis is conducted on the impact of TVID parameters and their corresponding optimization parameters on the dynamic behavior of the QZS primary system,including saddle-node(SN)bifurcation,Hopf bifurcation,backbone curve of amplitude-frequency response,and force transmissibility.According to the analysis,it is found that the steady-state motion of the system can enter quasi-periodic motion or even chaotic motion after losing stability through Hopf bifurcation.By optimizing the parameters of TVID,the number of SN bifurcation regions of the QZS main system can be reduced from 2 to 1,the Hopf bifurcation region can be eliminated,and the number of branches of backbone curve can be reduced from 2 to 1,thereby improving the dynamical performance of the QZS system.展开更多
In this paper,a new generalized non-monotonic and saturated incidence rate was introduced into a susceptible-infected-susceptible(SIS)epidemic model to account for inhibitory effect and crowding effect.The dynamic pro...In this paper,a new generalized non-monotonic and saturated incidence rate was introduced into a susceptible-infected-susceptible(SIS)epidemic model to account for inhibitory effect and crowding effect.The dynamic properties of the model were studied by qualitative theory and bifurcation theory.It is shown that when the infuence of psychological factors is large,the model has only disease-free equilibrium point,and this disease-free equilibrium point is globally asymptotically stable;when the influence of psychological factors is small,for some parameter conditions,the model has a unique endemic equilibrium point,which is a cusp point of co-dimension two,and for other parameter conditions the model has two endemic equilibrium points,one of which could be weak focus or center.In addition,the results of the model undergoing saddle-node bifurcation,Hopf bifurcation and Bogdanov-Takens bifurcation as the parameters vary were also proved.These results shed light on the impact of psychological behavior of susceptible people on the disease transmission.展开更多
A reduced three-degree-of-freedom model simulating the fluid-structure interactions （FSI） of the turbine blades and the on- coming air flows is proposed. The equations of motions consist of the coupling of bending a...A reduced three-degree-of-freedom model simulating the fluid-structure interactions （FSI） of the turbine blades and the on- coming air flows is proposed. The equations of motions consist of the coupling of bending and torsion of a blade as well as a van der Pol oscillation which represents the time-varying of the fluid. The 1：1 internal resonance of the system is analyzed with the multiple scale method, and the modulation equations are derived. The two-parameter bifurcation diagrams are computed. The effects of the system parameters, including the detuning parameter and the reduced frequency, on responses of the struc- ture and fluid are investigated. Bifurcation curves are computed and the stability is determined by examining the eigenvalues of the Jacobian matrix. The results indicate that rich dynamic phenomena of the steady-state solutions including the sad- dle-node and Hopf bifurcations can occur under certain parameter conditions. The parameter region where the unstable solu- tions occur should be avoided to keep the safe operation of the blades. The analytical solutions are verified by the direct nu- merical simulations.展开更多
This paper deals with bifurcations of subharmonic solutions and invariant tori generated from limit cycles in the fast dynamics for a nonautonomous singularly perturbed system. Based on Poincare map, a series of blow-...This paper deals with bifurcations of subharmonic solutions and invariant tori generated from limit cycles in the fast dynamics for a nonautonomous singularly perturbed system. Based on Poincare map, a series of blow-up transformations and the theory of integral manifold, the conditions for the existence of invariant tori are obtained, and the saddle-node bifurcations of subharmonic solutions are studied.展开更多
This paper uses the geometric singular perturbation theory to investigate dynamical behaviors and singularities in a fundamental power system presented in a single-machine infinite-bus formulation. The power system ca...This paper uses the geometric singular perturbation theory to investigate dynamical behaviors and singularities in a fundamental power system presented in a single-machine infinite-bus formulation. The power system can be approximated by two simplified systems S and F, which correspond respectively to slow and fast subsystems. The singularities, including Hopf bifurcation (HB), saddle-node bifurcation (SNB) and singularity induced bifurcation (SIB), are characterized. We show that SNB occurs at P Tc = 3.4382, SIB at P T0 = 2.8653 and HB at P Th = 2.802 for the singular perturbation system. It means that the power system will collapse near SIB which precedes SNB and that the power system will oscillate near HB which precedes SIB. In other words, the power system will lose its stability by means of oscillation near the HB which precedes SIB and SNB as P T is increasing to a critical value. The boundary of the stability region of the system can be described approximately by a combination of boundaries of the stability regions of the fast subsystem and slow subsystem.展开更多
The predation process plays a significant role in advancing life evolution and the maintenance of ecological balance and biodiversity.Hunting cooperation in predators is one of the most remarkable features of the pred...The predation process plays a significant role in advancing life evolution and the maintenance of ecological balance and biodiversity.Hunting cooperation in predators is one of the most remarkable features of the predation process,which benefits the predators by developing fear upon their prey.This study investigates the dynamical behavior of a modified LV-type predator-prey system with Michaelis-Menten-type harvesting of predators where predators adopt cooperation strategy during hunting.The ecologically feasible steady states of the system and their asymptotic stabilities are explored.The local codimension one bifurcations,viz.transcritical,saddle-node and Hopf bifurcations,that emerge in the system are investigated.Sotomayors approach is utilized to show the appearance of transcritical bifurcation and saddle-node bifurcation.A backward Hopfbifurcation is detected when the harvesting effort is increased,which destabilizes the system by generating periodic solutions.The stability nature of the Hopf-bifurcating periodic orbits is determined by computing the first Lyapunov coefficient.Our analyses revealed that above a threshold value of the harvesting effort promotes the coexistence of both populations.Similar periodic solutions of the system are also observed when the conversion efficiency rate or the hunting cooperation rate is increased.We have also explored codimension two bifurcations viz.the generalized Hopf and the Bogdanov-Takens bifurcation exhibit by the system.To visualize the dynamical behavior of the system,numerical simulations are conducted using an ecologically plausible parameter set.The existence of the bionomic equilibrium of the model is analyzed.Moreover,an optimal harvesting policy for the proposed model is derived by considering harvesting effort as a control parameter with the help of Pontryagins maximum principle.展开更多
This paper deals with a three-dimensional nonlinear mathematical model to analyze an epidemic's future course when the public healthcare facilities,specifically the number of hospital beds,are limited.The feasibil...This paper deals with a three-dimensional nonlinear mathematical model to analyze an epidemic's future course when the public healthcare facilities,specifically the number of hospital beds,are limited.The feasibility and stability of the obtained equilibria are analyzed,and the basic reproduction number(Ro)is obtained.We show that the system exhibits transcritical bifurcation.To show the existence of Bogdanov-Takens bifurcation,we have derived the normal form.We have also discussed a generalized Hopf(or Bautin)bifurcation at which the first Lyapunov coefficient evanescences.To show the existence of saddle-node bifurcation,we used Sotomayor's theorem.Furthermore,we have identified an optimal layout of hospital beds in order to control the disease with minimum possible expenditure.An optimal control setting is studied analytically using optimal control theory,and numerical simulations of the optimal regimen are presented as well.展开更多
In this paper,we construct a mathematical model to investigate the interaction between the tumor cells,the immune cells and the helper T cells(HTCs).We perform math-ematical analysis to reveal the stability of the equ...In this paper,we construct a mathematical model to investigate the interaction between the tumor cells,the immune cells and the helper T cells(HTCs).We perform math-ematical analysis to reveal the stability of the equilibria of the model.In our model,the HTCs are stimulated by the identification of the presence of tumor antigens.Our investigation implies that the presence of tumor antigens may inhibit the existence of high steady state of tumor cells,which leads to the elimination of the bistable behavior of the tumor-immune system,i.e.the equilibrium corresponding to the high steady state of tumor cells is destabilized.Choosing immune intensity c as bifurcation parameter,there exists saddle-node bifurcation.Besides,there exists a critical value C*,at which a Hopf bifurcation occurs.The stability and direction of Hopf bifurcation are discussed.展开更多
基金supported by the National Natural Science Foundation of China(No.12171337)the Central Government Guided Local Science and Technology Development Projects(No.2024ZYD0059)+1 种基金the Natural Science Foundation of Sichuan Province(No.2022NSFSC0529)the Open Research Fund Program of Data Recovery Key Laboratory of Sichuan Province(No.DRN2405)。
文摘This paper investigates the stability and bifurcation phenomena of a cholera transmission model in which individuals who have recovered from the disease may become susceptible again.The threshold for determining disease prevalence is established,and the parameter conditions for the existence of equilibria are discussed.The Routh-Hurwitz criterion is applied to demonstrate the local asymptotic stability of equilibria.By utilizing composite matrices and geometric techniques,the global dynamic behavior of the endemic equilibrium is investigated,and the sufficient conditions for its global asymptotic stability are derived.Furthermore,the disease-free equilibrium is a saddle-node when the basic reproductive number is 1,and tthe transcritical bifurcation in this case is discussed.
文摘In the paper, under the framework of exploring the interaction between algae and bacteria, an algae-bacteria ecological model was established to analyze the interaction mechanism and growth coexistence mode between algicidal bacteria and algae. Firstly, mathematical work mainly provided some threshold conditions to ensure the occurrence of transcritical bifurcation and saddle-node bifurcation, which could provide certain theoretical support for selecting key ecological environmental factors and numerical simulations. Secondly, the numerical simulation work dynamically displayed the evolution process of the bifurcation dynamic behavior of the model (2.1) and the growth coexistence mode of algae and algicidal bacteria. Finally, it was worth summarizing that intrinsic growth rate and combined capture effort of algae population had a strong influence on the dynamic behavior of the model (2.1). Furthermore, it must also be noted that transcritical bifurcation and saddle-node bifurcation were the inherent driving forces behind the formation of steady-state growth coexistence mode between algicidal bacteria and algae. In summary, it was hoped that the results of this study would contribute to accelerating the study of the interaction mechanism between algicidal bacteria and algae.
基金Project(10672053) supported by the National Natural Science Foundation of ChinaProject(2002AA503010) supported by the National High-Tech Research and Development Program of China
文摘A weakly nonlinear oscillator was modeled by a sort of differential equation, a saddle-node bifurcation was found in case of primary and secondary resonance. To control the jumping phenomena and the unstable region of the nonlinear oscillator, feedback controllers were designed. Bifurcation control equations were obtained by using the multiple scales method. And through the numerical analysis, good controller could be obtained by changing the feedback control gain. Then a feasible way of further research of saddle-node bifurcation was provided. Finally, an example shows that the feedback control method applied to the hanging bridge system of gas turbine is doable.
基金Projects 50490273 and 50674087 supported by the National Natural Science Foundation of ChinaBK2007029 by the Natural Science Foundation of Jiangsu Province
文摘The study of dynamical behavior of water or gas flows in broken rock is a basic research topic among a series of key projects about stability control of the surrounding rocks in mines and the prevention of some disasters such as water inrush or gas outburst and the protection of the groundwater resource. It is of great theoretical and engineering importance in respect of promo- tion of security in mine production and sustainable development of the coal industry. According to the non-Darcy property of seepage flow in broken rock dynamic equations of non-Darcy and non-steady flows in broken rock are established. By dimensionless transformation, the solution diagram of steady-states satisfying the given boundary conditions is obtained. By numerical analysis of low relaxation iteration, the dynamic responses corresponding to the different flow parameters have been obtained. The stability analysis of the steady-states indicate that a saddle-node bifurcaton exists in the seepage flow system of broken rock. Consequently, using catastrophe theory, the fold catastrophe model of seepage flow instability has been obtained. As a result, the bifurcation curves of the seepage flow systems with different control parameters are presented and the standard potential function is also given with respect to the generalized state variable for the fold catastrophe of a dynamic system of seepage flow in broken rock.
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the National Natural Science Foundation of China (60574011)Department of Science and Technology of Liaoning Province (2001401041).
文摘The singularly perturbed bifurcation subsystem is described, and the test conditions of subsystem persistence are deduced. By use of fast and slow reduced subsystem model, the result does not require performing nonlinear transformation. Moreover, it is shown and proved that the persistence of the periodic orbits for Hopf bifurcation in the reduced model through center manifold. Van der Pol oscillator circuit is given to illustrate the persistence of bifurcation subsystems with the full dynamic system.
基金This work was supported by the National Natural Science Foundation of China (No. 10371136).
文摘In this paper n-dimensional flows （described by continuous-time system） with static bifurcations are considered with the aim of classification of different elementary bifurcations using the frequency domain formalism. Based on frequency domain approach, we prove some criterions for the saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation, and give an example to illustrate the efficiency of the result obtained.
文摘In this paper, we examine a discrete-time Host-Parasitoid model which is a non-dimensionalized Nicholson and Bailey model. Phase portraits are drawn for different ranges of parameters and display the complicated dynamics of this system. We conduct the bifurcation analysis with respect to intrinsic growth rate <em>r</em> and searching efficiency <em>a</em>. Many forms of complex dynamics such as chaos, periodic windows are observed. Transition route to chaos dynamics is established via period-doubling bifurcations. Conditions of occurrence of the period-doubling, Neimark-Sacker and saddle-node bifurcations are analyzed for <em>b≠a</em> where <em>a,b</em> are searching efficiency. We study stable and unstable manifolds for different equilibrium points and coexistence of different attractors for this non-dimensionalize system. Without the parasitoid, the host population follows the dynamics of the Ricker model.
文摘The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf bifurcation is given. Both the period-doubling bifurcation and saddle-node bifurcation of periodical solutions are computed since the observed floquet multiplier overpass the unit circle by DDE-Biftool software in Matlab. The continuation of saddle-node bifurcation line or period-doubling curve is carried out as varying free parameters and time delays. Two different transition modes of saddle-node bifurcation are discovered which is verified by numerical simulation work with aids of DDE-Biftool.
文摘Due to the random search of species and from the economic point of view,combined harvesting is more suitable than selective harvesting.Thus,we have developed and analyzed a prey-predator model with the combined effect of nonlinear harvesting in this research paper.Nonlinear harvesting possesses multiple predator-free and interior equilibrium points in the dynamical system.We have examined the local stability analysis of all the equilibrium points.Besides these various types,rich and complex dynamical behaviors such as backward,saddle-node,Hopf and Bogdanov-Takens(BT)bifurcations,homoclinic loop and limit cycles appear in this model.Furthermore,interesting phenomena like bi-stability and tri-stability occur in our model between the different equilibrium points.Also,we have derived different threshold values of predator harvesting parameters and prey environmental carrying capacity from these bifurcations to obtain the different harvesting strategies for both species.We have observed that the extinction of predator species may not happen due to backward bifurcation,although a stable predator-free equilibrium(PFE)exists.Finally,numerical simulations are discussed using MATLAB to verify all the theoretical results.
基金supported by the National Natural Science Foundation of China(Grant Nos.12272241,12202286,and U1934201)Natural Science Foundation of Hebei Province(Grant Nos.A2021210012 and A2024210041).
文摘According to the H_(∞)principle,the dynamical performance optimization of a quasi-zero-stiffness(QZS)isolation system with an additional tuned viscous inerter damper(TVID)is studied by using analytical method.The approximate analytical solutions of the QZS system coupled with TVID are solved by using the complexification-averaging method,and the expression of stability conditions for steady-state solutions is derived based on Lyapunov method and Routh-Hurwitz criterion.Based on the fixed-point theory,considering the nonlinear stiffness and weak damping of the primary system,the stiffness and damping ratios of TVID coupled to QZS system are optimized by using the equal-peak method.The detailed analysis is conducted on the impact of TVID parameters and their corresponding optimization parameters on the dynamic behavior of the QZS primary system,including saddle-node(SN)bifurcation,Hopf bifurcation,backbone curve of amplitude-frequency response,and force transmissibility.According to the analysis,it is found that the steady-state motion of the system can enter quasi-periodic motion or even chaotic motion after losing stability through Hopf bifurcation.By optimizing the parameters of TVID,the number of SN bifurcation regions of the QZS main system can be reduced from 2 to 1,the Hopf bifurcation region can be eliminated,and the number of branches of backbone curve can be reduced from 2 to 1,thereby improving the dynamical performance of the QZS system.
基金supported by the NSF of China[Grant No.11961021]the NSF of Guangdong province[Grant Nos.2022A1515010964 and 2022A1515010193]+1 种基金the Innovation and Developing School Project of Guangdong Province[Grant No.2019KzDXM032]the Special Fund of Science and Technology Innovation Strategy of Guangdong Province[Grant Nos.pdjh2022b0320 and pdjh2023b0325].
文摘In this paper,a new generalized non-monotonic and saturated incidence rate was introduced into a susceptible-infected-susceptible(SIS)epidemic model to account for inhibitory effect and crowding effect.The dynamic properties of the model were studied by qualitative theory and bifurcation theory.It is shown that when the infuence of psychological factors is large,the model has only disease-free equilibrium point,and this disease-free equilibrium point is globally asymptotically stable;when the influence of psychological factors is small,for some parameter conditions,the model has a unique endemic equilibrium point,which is a cusp point of co-dimension two,and for other parameter conditions the model has two endemic equilibrium points,one of which could be weak focus or center.In addition,the results of the model undergoing saddle-node bifurcation,Hopf bifurcation and Bogdanov-Takens bifurcation as the parameters vary were also proved.These results shed light on the impact of psychological behavior of susceptible people on the disease transmission.
基金supported by the National Basic Research Program of China(“973” Project)(Grant No.2015CB057405)the National Natural Science Foundation of China(Grant No.11372082)the State Scholarship Fund of CSC
文摘A reduced three-degree-of-freedom model simulating the fluid-structure interactions （FSI） of the turbine blades and the on- coming air flows is proposed. The equations of motions consist of the coupling of bending and torsion of a blade as well as a van der Pol oscillation which represents the time-varying of the fluid. The 1：1 internal resonance of the system is analyzed with the multiple scale method, and the modulation equations are derived. The two-parameter bifurcation diagrams are computed. The effects of the system parameters, including the detuning parameter and the reduced frequency, on responses of the struc- ture and fluid are investigated. Bifurcation curves are computed and the stability is determined by examining the eigenvalues of the Jacobian matrix. The results indicate that rich dynamic phenomena of the steady-state solutions including the sad- dle-node and Hopf bifurcations can occur under certain parameter conditions. The parameter region where the unstable solu- tions occur should be avoided to keep the safe operation of the blades. The analytical solutions are verified by the direct nu- merical simulations.
基金Project supported by the National Natural Science Foundation of China (No. 10671214)the Chongqing Natural Science Foundation of China (No. 2005cc14)Shanghai Shuguang Genzong Project (No.04SGG05)
文摘This paper deals with bifurcations of subharmonic solutions and invariant tori generated from limit cycles in the fast dynamics for a nonautonomous singularly perturbed system. Based on Poincare map, a series of blow-up transformations and the theory of integral manifold, the conditions for the existence of invariant tori are obtained, and the saddle-node bifurcations of subharmonic solutions are studied.
基金Supported by the National Natural Science Fundation of China (No.50377018)a research grant from Research Office of the Hong Kong Polytechnic University(G.63.37.T494)
文摘This paper uses the geometric singular perturbation theory to investigate dynamical behaviors and singularities in a fundamental power system presented in a single-machine infinite-bus formulation. The power system can be approximated by two simplified systems S and F, which correspond respectively to slow and fast subsystems. The singularities, including Hopf bifurcation (HB), saddle-node bifurcation (SNB) and singularity induced bifurcation (SIB), are characterized. We show that SNB occurs at P Tc = 3.4382, SIB at P T0 = 2.8653 and HB at P Th = 2.802 for the singular perturbation system. It means that the power system will collapse near SIB which precedes SNB and that the power system will oscillate near HB which precedes SIB. In other words, the power system will lose its stability by means of oscillation near the HB which precedes SIB and SNB as P T is increasing to a critical value. The boundary of the stability region of the system can be described approximately by a combination of boundaries of the stability regions of the fast subsystem and slow subsystem.
基金jointly supported by the National Natural Science Foundation of China(62173139)the Science and Technology Innovation Program of Hunan Province(2021RC4030).
文摘The predation process plays a significant role in advancing life evolution and the maintenance of ecological balance and biodiversity.Hunting cooperation in predators is one of the most remarkable features of the predation process,which benefits the predators by developing fear upon their prey.This study investigates the dynamical behavior of a modified LV-type predator-prey system with Michaelis-Menten-type harvesting of predators where predators adopt cooperation strategy during hunting.The ecologically feasible steady states of the system and their asymptotic stabilities are explored.The local codimension one bifurcations,viz.transcritical,saddle-node and Hopf bifurcations,that emerge in the system are investigated.Sotomayors approach is utilized to show the appearance of transcritical bifurcation and saddle-node bifurcation.A backward Hopfbifurcation is detected when the harvesting effort is increased,which destabilizes the system by generating periodic solutions.The stability nature of the Hopf-bifurcating periodic orbits is determined by computing the first Lyapunov coefficient.Our analyses revealed that above a threshold value of the harvesting effort promotes the coexistence of both populations.Similar periodic solutions of the system are also observed when the conversion efficiency rate or the hunting cooperation rate is increased.We have also explored codimension two bifurcations viz.the generalized Hopf and the Bogdanov-Takens bifurcation exhibit by the system.To visualize the dynamical behavior of the system,numerical simulations are conducted using an ecologically plausible parameter set.The existence of the bionomic equilibrium of the model is analyzed.Moreover,an optimal harvesting policy for the proposed model is derived by considering harvesting effort as a control parameter with the help of Pontryagins maximum principle.
基金The authors also thankfully acknowledge financial support from Council of Scientific and Industrial Research,India through a research fellowship(File No.09/013(0841)/2018-EMR-I)Jyoti Maurya and DST-Science and Engineering Research Board,MATRICS Expert committee(File No.MTR/2021/000819)A.K.Misra to carry out this research work.
文摘This paper deals with a three-dimensional nonlinear mathematical model to analyze an epidemic's future course when the public healthcare facilities,specifically the number of hospital beds,are limited.The feasibility and stability of the obtained equilibria are analyzed,and the basic reproduction number(Ro)is obtained.We show that the system exhibits transcritical bifurcation.To show the existence of Bogdanov-Takens bifurcation,we have derived the normal form.We have also discussed a generalized Hopf(or Bautin)bifurcation at which the first Lyapunov coefficient evanescences.To show the existence of saddle-node bifurcation,we used Sotomayor's theorem.Furthermore,we have identified an optimal layout of hospital beds in order to control the disease with minimum possible expenditure.An optimal control setting is studied analytically using optimal control theory,and numerical simulations of the optimal regimen are presented as well.
文摘In this paper,we construct a mathematical model to investigate the interaction between the tumor cells,the immune cells and the helper T cells(HTCs).We perform math-ematical analysis to reveal the stability of the equilibria of the model.In our model,the HTCs are stimulated by the identification of the presence of tumor antigens.Our investigation implies that the presence of tumor antigens may inhibit the existence of high steady state of tumor cells,which leads to the elimination of the bistable behavior of the tumor-immune system,i.e.the equilibrium corresponding to the high steady state of tumor cells is destabilized.Choosing immune intensity c as bifurcation parameter,there exists saddle-node bifurcation.Besides,there exists a critical value C*,at which a Hopf bifurcation occurs.The stability and direction of Hopf bifurcation are discussed.
