In this paper, a novel four dimensional hyper-chaotic system is coined based on the Chen system, which contains two quadratic terms and five system parameters. The proposed system can generate a hyper-chaotic attracto...In this paper, a novel four dimensional hyper-chaotic system is coined based on the Chen system, which contains two quadratic terms and five system parameters. The proposed system can generate a hyper-chaotic attractor in wide parameters regions. By using the center manifold theorem and the local bifurcation theory, a pitchfork bifurcation is demonstrated to arise at the zero equilibrium point. Numerical analysis demonstrates that the hyper-cha^tic system can generate complex dynamical behaviors, e.g., a direct transition from quasi-periodic behavior to hyper-chaotic behavior. Finally, an electronic circuit is designed to implement the hyper-chaotic system, the experimental results are consist with the numerical simulations, which verifies the existence of the hyper-chaotic attractor. Due to the complex dynamic behaviors, this new hyper-cha^tic system is useful in the secure communication.展开更多
The clustering behavior of a mono-disperse granular gas is experimentally studied in an asymmetric two-compartment setup. Unlike the random clustering in either compartment in the case of symmetric configuration when ...The clustering behavior of a mono-disperse granular gas is experimentally studied in an asymmetric two-compartment setup. Unlike the random clustering in either compartment in the case of symmetric configuration when lowering the shaking strength to below a critical value, the directed clustering is observed, which corresponds to an imperfect pitchfork bifurcation. Numerical solutions of the flux equation using a modified simple flux function show qualitative agreements with the experimental results. The potential application of this asymmetric structure is discussed.展开更多
Based on the empirical rate law and the kinetic data reported in the literature, it is predicted that the iodate-arsenous acid reaction may exhibit pitchfork bifurcation phenomenon in CSTR with two inflows of the reac...Based on the empirical rate law and the kinetic data reported in the literature, it is predicted that the iodate-arsenous acid reaction may exhibit pitchfork bifurcation phenomenon in CSTR with two inflows of the reactants.展开更多
In this paper,we present a criterion for pitchfork bifurcations of smooth vector elds based on a topological argument.Our result expands Rajapakse and Smale's result[15]signi cantly.Based on our criterion,we prese...In this paper,we present a criterion for pitchfork bifurcations of smooth vector elds based on a topological argument.Our result expands Rajapakse and Smale's result[15]signi cantly.Based on our criterion,we present a class of families of non-symmetric vector elds undergoing a pitchfork bifurcation.展开更多
Local bifurcation phenomena in a four-dlmensional continuous hyperchaotic system, which has rich and complex dynamical behaviours, are analysed. The local bifurcations of the system are investigated by utilizing the b...Local bifurcation phenomena in a four-dlmensional continuous hyperchaotic system, which has rich and complex dynamical behaviours, are analysed. The local bifurcations of the system are investigated by utilizing the bifurcation theory and the centre manifold theorem, and thus the conditions of the existence of pitchfork bifurcation and Hopf bifurcation are derived in detail. Numerical simulations are presented to verify the theoretical analysis, and they show some interesting dynamics, including stable periodic orbits emerging from the new fixed points generated by pitchfork bifurcation, coexistence of a stable limit cycle and a chaotic attractor, as well as chaos within quite a wide parameter region.展开更多
Both the symmetric period n-2 motion and asymmetric one of a one-degree- of-freedom impact oscillator are considered. The theory of bifurcations of the fixed point is applied to such model, and it is proved that the s...Both the symmetric period n-2 motion and asymmetric one of a one-degree- of-freedom impact oscillator are considered. The theory of bifurcations of the fixed point is applied to such model, and it is proved that the symmetric periodic motion has only pitchfork bifurcation by the analysis of the symmetry of the Poincar6 map. The numerical simulation shows that one symmetric periodic orbit could bifurcate into two antisymmet- ric ones via pitchfork bifurcation. While the control parameter changes continuously, the two antisymmetric periodic orbits will give birth to two synchronous antisymmetric period-doubling sequences, and bring about two antisymmetric chaotic attractors subse- quently. If the symmetric system is transformed into asymmetric one, bifurcations of the asymmetric period n-2 motion can be described by a two-parameter unfolding of cusp, and the pitchfork changes into one unbifurcated branch and one fold branch.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
A kind of 2-dimensional neural network model with delay is considered. By analyzing the distribution of the roots of the characteristic equation associated with the model, a bifurcation diagram was drawn in an appropr...A kind of 2-dimensional neural network model with delay is considered. By analyzing the distribution of the roots of the characteristic equation associated with the model, a bifurcation diagram was drawn in an appropriate parameter plane. It is found that a line is a pitchfork bifurcation curve. Further more, the stability of each fixed point and existence of Hopf bifurcation were obtained. Finally, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions were determined by using the normal form method and centre manifold theory.展开更多
From the point of view of dynamics, the phenomenon of mode jumping in the imperfect pitchfork problem is discussed. The dynamical mechanism of model jumping of structures, such as plate and shell, that is brought abou...From the point of view of dynamics, the phenomenon of mode jumping in the imperfect pitchfork problem is discussed. The dynamical mechanism of model jumping of structures, such as plate and shell, that is brought about by the extremum instability, is explained. Finally, we give numerical simulation to show the validity of our results.展开更多
This study explores the nonlinear dynamics of a quasi-zero stiffness(QZS)vibration isolator coupled with a piezoelectric energy harvester connected to an RL-resonant circuit.The model of the system is formulated with ...This study explores the nonlinear dynamics of a quasi-zero stiffness(QZS)vibration isolator coupled with a piezoelectric energy harvester connected to an RL-resonant circuit.The model of the system is formulated with the Lagrangian mechanics,representing a two-degree-of-freedom nonlinear electromechanical system subject to harmonic base excitation under a 1:1 internal resonance condition.The model is normalized,and the conditions dictating monostable and bistable oscillation modes are identified.The bifurcation characteristics of the coupled system are analyzed in both oscillation modes by means of harmonic balance and continuation methods.The vibration isolation performance,with and without the coupled harvester,is evaluated in terms of displacement transmissibility to assess its dual functionalities for vibration isolation and energy harvesting.Analytical results demonstrate that integrating a piezoelectric harvester into a monostable QZS isolator under 1:1 internal resonance does not compromise its vibration isolation capability while enabling efficient energy harvesting at extremely low-frequency base excitation.Furthermore,the system's response under strong base excitation is investigated exclusively for energy harvesting in both monostable and bistable modes,leading to optimal structural parameter design.The conditions for intra-well and inter-well periodic oscillation modes,as well as chaotic responses,are analyzed analytically and validated numerically through stability charts,basins of attraction,bifurcation diagrams,time histories,and Poincarémaps.This work provides a comprehensive understanding of the oscillation dynamics of QZS isolators and offers valuable insights for optimizing their geometric parameters to function as high-performance vibration isolators and/or energy harvesters.展开更多
This paper introduces a four-dimensional （4D） segmented disc dynamo which possesses coexisting hidden attractors with one stable equilibrium or a line equilibrium when parameters vary. In addition, by choosing an ap...This paper introduces a four-dimensional （4D） segmented disc dynamo which possesses coexisting hidden attractors with one stable equilibrium or a line equilibrium when parameters vary. In addition, by choosing an appropriate bifurcation parameter, the paper proves that Hopf bifurcation and pitchfork bifurcation occur in the system. The ultimate bound is also estimated. Some numerical investigations are also exploited to demonstrate and visualize the corresponding theoretical results.展开更多
In this paper, the complex dynamical behavior of a fractional-order Lorenz-like system with two quadratic terms is investigated. The existence and uniqueness of solutions for this system are proved, and the stabilitie...In this paper, the complex dynamical behavior of a fractional-order Lorenz-like system with two quadratic terms is investigated. The existence and uniqueness of solutions for this system are proved, and the stabilities of the equilibrium points are analyzed as one of the system parameters changes. The pitchfork bifurcation is discussed for the first time, and the necessary conditions for the commensurate and incommensurate fractional-order systems to remain in chaos are derived. The largest Lyapunov exponents and phase portraits are given to check the existence of chaos. Finally, the sliding mode control law is provided to make the states of the Lorenz-like system asymptotically stable. Numerical simulation results show that the presented approach can effectively guide chaotic trajectories to the unstable equilibrium points.展开更多
A bounded confidence model of opinion dynamics in multi-group projects is presented in which each group＇s opinion evolution is driven by two types of forces：（i） the group＇s cohesive force which tends to restore t...A bounded confidence model of opinion dynamics in multi-group projects is presented in which each group＇s opinion evolution is driven by two types of forces：（i） the group＇s cohesive force which tends to restore the opinion back towards the initial status because of its company culture;and（ii） nonlinear coupling forces with other groups which attempt to bring opinions closer due to collaboration willingness.Bifurcation analysis for the case of a two-group project shows a cusp catastrophe phenomenon and three distinctive evolutionary regimes,i.e.,a deadlock regime,a convergence regime,and a bifurcation regime in opinion dynamics.The critical value of initial discord between the two groups is derived to discriminate which regime the opinion evolution belongs to.In the case of a three-group project with a symmetric social network,both bifurcation analysis and simulation results demonstrate that if each pair has a high initial discord,instead of symmetrically converging to consensus with the increase of coupling scale as expected by Gabbay＇s result（Physica A 378（2007） p.125 Fig.5）,project organization（PO） may be split into two distinct clusters because of the symmetry breaking phenomenon caused by pitchfork bifurcations,which urges that apart from divergence in participants＇ interests,nonlinear interaction can also make conflict inevitable in the PO.The effects of two asymmetric level parameters are tested in order to explore the ways of inducing dominant opinion in the whole PO.It is found that the strong influence imposed by a leader group with firm faith on the flexible and open minded follower groups can promote the formation of a positive dominant opinion in the PO.展开更多
In this paper, the catastrophe of a spherical cavity and the cavitation of a spherical cavity for Hooke's material with 1/2 Poisson's ratio are studied. A nonlinear problem, which is a moving boundary problem ...In this paper, the catastrophe of a spherical cavity and the cavitation of a spherical cavity for Hooke's material with 1/2 Poisson's ratio are studied. A nonlinear problem, which is a moving boundary problem for the geometrically nonlinear elasticity in radial symmetric, is solved analytically. The governing equations are written on the deformed region or on the present configuration. And the conditions are described on moving boundary. A closed form solution is found. Furthermore, a bifurcation solution in closed form is given from the trivial homogeneous solution of a solid sphere. The results indicate that there is a tangent bifurcation on the displacement_load curve for a sphere with a cavity. On the tangent bifurcation point, the cavity grows up suddenly, which is a kind of catastrophe. And there is a pitchfork bifurcation on the displacement_load curve for a solid sphere. On the pitchfork bifurcation point, there is a cavitation in the solid sphere.展开更多
The local dynamical behaviors of a four-dimensional hyperchaotic Lorenz system, including stability and bifurcations, are investigated in this paper by analytical and numerical methods. The equilibriums and their stab...The local dynamical behaviors of a four-dimensional hyperchaotic Lorenz system, including stability and bifurcations, are investigated in this paper by analytical and numerical methods. The equilibriums and their stability under different parameter conditions are analyzed by applying Routh-Hurwitz criterion. The results indicate that the system may exist one, three and five equilibrium points for different system parameters. Based on the central manifold theorem and normal form theorem, the pitchfork bifurcation and Hopf bifurcation are studied respectively. By using the Hopf bifurcation theorem and calculating the first Lyapunov coefficient, the Hopf bifurcation of this system is obtained as supercritical for certain parameters. Finally, the results of theoretical parts are verified by some numerical simulations.展开更多
A cross-diffusion prey-predator system exhibiting the prey group defense under homogeneous Neumann boundary conditions is studied.By considering the diffusive rate of the prey as a bifurcation parameter,we investigate...A cross-diffusion prey-predator system exhibiting the prey group defense under homogeneous Neumann boundary conditions is studied.By considering the diffusive rate of the prey as a bifurcation parameter,we investigate sudden changes in the population dynamics of the prey and predator which can have a substantial effect on population size of the species.First a priori estimate for positive steady states is obtained.Next we prove the existence of a pitchfork bifurcation of positive steady states at a simple eigenvalue.The structure of the global steady-state bifurcation is discussed.We also investigate the stability of the trivial solution line and nontrivial steady-state solutions via the eigenvalue perturbation theory.To illustrate our theoretical results some numerical simulations are given.Numerical examples contain a supercritical and a subcritical pitchfork bifurcation.展开更多
The power-law fluid flow past a row of uniform placed square cylinders is investigated using the Lattice Boltzmann method (LBM).The flow is assumed to be two-dimensional and incompressible.The relaxation time is assum...The power-law fluid flow past a row of uniform placed square cylinders is investigated using the Lattice Boltzmann method (LBM).The flow is assumed to be two-dimensional and incompressible.The relaxation time is assumed to be shear-dependent and determined by using a variable parameter related to the local shear rate.The effects of both shear-thinning/shear-thickening property and the cylinder spacing on the confluence of the jets are mainly concerned.The bifurcation diagrams of the flow are obtained,which include confluences of double and quadruple jets.The results show that both the first and second pitchfork bifurcations are advanced due to the effect of the shear-thinning property,and postponed due to the shear-thickening property.展开更多
基金Project supported by the Natural Science Foundation of China (Grant Nos.61174094, 50977063, and 60904063)the Foundation of the Application Base and Frontier Technology Research Project of Tianjin, China (Grant No.10JCZDJC23100)the Development of Science and Technology Foundation of the Higher Education Institutions of Tianjin, China (Grant No.20080826)
文摘In this paper, a novel four dimensional hyper-chaotic system is coined based on the Chen system, which contains two quadratic terms and five system parameters. The proposed system can generate a hyper-chaotic attractor in wide parameters regions. By using the center manifold theorem and the local bifurcation theory, a pitchfork bifurcation is demonstrated to arise at the zero equilibrium point. Numerical analysis demonstrates that the hyper-cha^tic system can generate complex dynamical behaviors, e.g., a direct transition from quasi-periodic behavior to hyper-chaotic behavior. Finally, an electronic circuit is designed to implement the hyper-chaotic system, the experimental results are consist with the numerical simulations, which verifies the existence of the hyper-chaotic attractor. Due to the complex dynamic behaviors, this new hyper-cha^tic system is useful in the secure communication.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11034010 and 11274354)the Chinese Academy of Sciences "Strategic Priority Research Program - SJ-10" (Grant No. XDA04020200)the Special Fund for Earthquake Research of China (Grant No. 201208011)
文摘The clustering behavior of a mono-disperse granular gas is experimentally studied in an asymmetric two-compartment setup. Unlike the random clustering in either compartment in the case of symmetric configuration when lowering the shaking strength to below a critical value, the directed clustering is observed, which corresponds to an imperfect pitchfork bifurcation. Numerical solutions of the flux equation using a modified simple flux function show qualitative agreements with the experimental results. The potential application of this asymmetric structure is discussed.
文摘Based on the empirical rate law and the kinetic data reported in the literature, it is predicted that the iodate-arsenous acid reaction may exhibit pitchfork bifurcation phenomenon in CSTR with two inflows of the reactants.
基金The second author was supported by the Smale Institute.This work was finished during the third author's stay in Graduate Center of City University of New York.
文摘In this paper,we present a criterion for pitchfork bifurcations of smooth vector elds based on a topological argument.Our result expands Rajapakse and Smale's result[15]signi cantly.Based on our criterion,we present a class of families of non-symmetric vector elds undergoing a pitchfork bifurcation.
基金supported by the National Natural Science Foundation of China (Grant Nos 60774088,10772135 and 60574036)the Research Foundation from the Ministry of Education of China (Grant Nos 107024 and 207005)+1 种基金the Program for New Century Excellent Talents in University of China (NCET)the Application Base and Frontier Technology Project of Tianjin,China(Grant No 08JCZDJC21900)
文摘Local bifurcation phenomena in a four-dlmensional continuous hyperchaotic system, which has rich and complex dynamical behaviours, are analysed. The local bifurcations of the system are investigated by utilizing the bifurcation theory and the centre manifold theorem, and thus the conditions of the existence of pitchfork bifurcation and Hopf bifurcation are derived in detail. Numerical simulations are presented to verify the theoretical analysis, and they show some interesting dynamics, including stable periodic orbits emerging from the new fixed points generated by pitchfork bifurcation, coexistence of a stable limit cycle and a chaotic attractor, as well as chaos within quite a wide parameter region.
基金Project supported by the National Natural Science Foundation of China (No.10472096)the Fund for Doctoral Innovation of Southwest Jiaotong University
文摘Both the symmetric period n-2 motion and asymmetric one of a one-degree- of-freedom impact oscillator are considered. The theory of bifurcations of the fixed point is applied to such model, and it is proved that the symmetric periodic motion has only pitchfork bifurcation by the analysis of the symmetry of the Poincar6 map. The numerical simulation shows that one symmetric periodic orbit could bifurcate into two antisymmet- ric ones via pitchfork bifurcation. While the control parameter changes continuously, the two antisymmetric periodic orbits will give birth to two synchronous antisymmetric period-doubling sequences, and bring about two antisymmetric chaotic attractors subse- quently. If the symmetric system is transformed into asymmetric one, bifurcations of the asymmetric period n-2 motion can be described by a two-parameter unfolding of cusp, and the pitchfork changes into one unbifurcated branch and one fold branch.
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A kind of 2-dimensional neural network model with delay is considered. By analyzing the distribution of the roots of the characteristic equation associated with the model, a bifurcation diagram was drawn in an appropriate parameter plane. It is found that a line is a pitchfork bifurcation curve. Further more, the stability of each fixed point and existence of Hopf bifurcation were obtained. Finally, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions were determined by using the normal form method and centre manifold theory.
基金This work is supported by the Foundation of the National Educational Committeeand the National Natural Sciences Foundation of China.
文摘From the point of view of dynamics, the phenomenon of mode jumping in the imperfect pitchfork problem is discussed. The dynamical mechanism of model jumping of structures, such as plate and shell, that is brought about by the extremum instability, is explained. Finally, we give numerical simulation to show the validity of our results.
基金Project supported by the National Key R&D Program of China(No.2023YFE0125900)。
文摘This study explores the nonlinear dynamics of a quasi-zero stiffness(QZS)vibration isolator coupled with a piezoelectric energy harvester connected to an RL-resonant circuit.The model of the system is formulated with the Lagrangian mechanics,representing a two-degree-of-freedom nonlinear electromechanical system subject to harmonic base excitation under a 1:1 internal resonance condition.The model is normalized,and the conditions dictating monostable and bistable oscillation modes are identified.The bifurcation characteristics of the coupled system are analyzed in both oscillation modes by means of harmonic balance and continuation methods.The vibration isolation performance,with and without the coupled harvester,is evaluated in terms of displacement transmissibility to assess its dual functionalities for vibration isolation and energy harvesting.Analytical results demonstrate that integrating a piezoelectric harvester into a monostable QZS isolator under 1:1 internal resonance does not compromise its vibration isolation capability while enabling efficient energy harvesting at extremely low-frequency base excitation.Furthermore,the system's response under strong base excitation is investigated exclusively for energy harvesting in both monostable and bistable modes,leading to optimal structural parameter design.The conditions for intra-well and inter-well periodic oscillation modes,as well as chaotic responses,are analyzed analytically and validated numerically through stability charts,basins of attraction,bifurcation diagrams,time histories,and Poincarémaps.This work provides a comprehensive understanding of the oscillation dynamics of QZS isolators and offers valuable insights for optimizing their geometric parameters to function as high-performance vibration isolators and/or energy harvesters.
基金supported by the National Natural Science Foundation of China(Grant No.11671149)
文摘This paper introduces a four-dimensional （4D） segmented disc dynamo which possesses coexisting hidden attractors with one stable equilibrium or a line equilibrium when parameters vary. In addition, by choosing an appropriate bifurcation parameter, the paper proves that Hopf bifurcation and pitchfork bifurcation occur in the system. The ultimate bound is also estimated. Some numerical investigations are also exploited to demonstrate and visualize the corresponding theoretical results.
基金Projected supported by the National Natural Science Foundation of China (Grant No. 11202155)the Fundamental Research Funds for the Central Universities, China (Grant No. K50511700001)
文摘In this paper, the complex dynamical behavior of a fractional-order Lorenz-like system with two quadratic terms is investigated. The existence and uniqueness of solutions for this system are proved, and the stabilities of the equilibrium points are analyzed as one of the system parameters changes. The pitchfork bifurcation is discussed for the first time, and the necessary conditions for the commensurate and incommensurate fractional-order systems to remain in chaos are derived. The largest Lyapunov exponents and phase portraits are given to check the existence of chaos. Finally, the sliding mode control law is provided to make the states of the Lorenz-like system asymptotically stable. Numerical simulation results show that the presented approach can effectively guide chaotic trajectories to the unstable equilibrium points.
基金Project supported by the National Natural Science Foundation of China （Grant No. 70831002） Humanity and Social Science Youth Foundation of Ministry of Education of China （Grant No. 12YJCZH017）
文摘A bounded confidence model of opinion dynamics in multi-group projects is presented in which each group＇s opinion evolution is driven by two types of forces：（i） the group＇s cohesive force which tends to restore the opinion back towards the initial status because of its company culture;and（ii） nonlinear coupling forces with other groups which attempt to bring opinions closer due to collaboration willingness.Bifurcation analysis for the case of a two-group project shows a cusp catastrophe phenomenon and three distinctive evolutionary regimes,i.e.,a deadlock regime,a convergence regime,and a bifurcation regime in opinion dynamics.The critical value of initial discord between the two groups is derived to discriminate which regime the opinion evolution belongs to.In the case of a three-group project with a symmetric social network,both bifurcation analysis and simulation results demonstrate that if each pair has a high initial discord,instead of symmetrically converging to consensus with the increase of coupling scale as expected by Gabbay＇s result（Physica A 378（2007） p.125 Fig.5）,project organization（PO） may be split into two distinct clusters because of the symmetry breaking phenomenon caused by pitchfork bifurcations,which urges that apart from divergence in participants＇ interests,nonlinear interaction can also make conflict inevitable in the PO.The effects of two asymmetric level parameters are tested in order to explore the ways of inducing dominant opinion in the whole PO.It is found that the strong influence imposed by a leader group with firm faith on the flexible and open minded follower groups can promote the formation of a positive dominant opinion in the PO.
文摘In this paper, the catastrophe of a spherical cavity and the cavitation of a spherical cavity for Hooke's material with 1/2 Poisson's ratio are studied. A nonlinear problem, which is a moving boundary problem for the geometrically nonlinear elasticity in radial symmetric, is solved analytically. The governing equations are written on the deformed region or on the present configuration. And the conditions are described on moving boundary. A closed form solution is found. Furthermore, a bifurcation solution in closed form is given from the trivial homogeneous solution of a solid sphere. The results indicate that there is a tangent bifurcation on the displacement_load curve for a sphere with a cavity. On the tangent bifurcation point, the cavity grows up suddenly, which is a kind of catastrophe. And there is a pitchfork bifurcation on the displacement_load curve for a solid sphere. On the pitchfork bifurcation point, there is a cavitation in the solid sphere.
文摘The local dynamical behaviors of a four-dimensional hyperchaotic Lorenz system, including stability and bifurcations, are investigated in this paper by analytical and numerical methods. The equilibriums and their stability under different parameter conditions are analyzed by applying Routh-Hurwitz criterion. The results indicate that the system may exist one, three and five equilibrium points for different system parameters. Based on the central manifold theorem and normal form theorem, the pitchfork bifurcation and Hopf bifurcation are studied respectively. By using the Hopf bifurcation theorem and calculating the first Lyapunov coefficient, the Hopf bifurcation of this system is obtained as supercritical for certain parameters. Finally, the results of theoretical parts are verified by some numerical simulations.
文摘A cross-diffusion prey-predator system exhibiting the prey group defense under homogeneous Neumann boundary conditions is studied.By considering the diffusive rate of the prey as a bifurcation parameter,we investigate sudden changes in the population dynamics of the prey and predator which can have a substantial effect on population size of the species.First a priori estimate for positive steady states is obtained.Next we prove the existence of a pitchfork bifurcation of positive steady states at a simple eigenvalue.The structure of the global steady-state bifurcation is discussed.We also investigate the stability of the trivial solution line and nontrivial steady-state solutions via the eigenvalue perturbation theory.To illustrate our theoretical results some numerical simulations are given.Numerical examples contain a supercritical and a subcritical pitchfork bifurcation.
基金assistance from the Natural Science Foundation of China(GrantNo. 10972115)
文摘The power-law fluid flow past a row of uniform placed square cylinders is investigated using the Lattice Boltzmann method (LBM).The flow is assumed to be two-dimensional and incompressible.The relaxation time is assumed to be shear-dependent and determined by using a variable parameter related to the local shear rate.The effects of both shear-thinning/shear-thickening property and the cylinder spacing on the confluence of the jets are mainly concerned.The bifurcation diagrams of the flow are obtained,which include confluences of double and quadruple jets.The results show that both the first and second pitchfork bifurcations are advanced due to the effect of the shear-thinning property,and postponed due to the shear-thickening property.
