In the present paper, the existence of global attractor for dissipative Hamiltonian amplitude equation governing the modulated wave instabilities in E0 is considered. By a decomposition of solution operator, it is sho...In the present paper, the existence of global attractor for dissipative Hamiltonian amplitude equation governing the modulated wave instabilities in E0 is considered. By a decomposition of solution operator, it is shown that the global attractor in E0 is actually equal to a global attractor in E1.展开更多
We investigate the Turing instability and pattern formation mechanism of a plant-wrack model with both self-diffusion and cross-diffusion terms.We first study the effect of self-diffusion on the stability of equilibri...We investigate the Turing instability and pattern formation mechanism of a plant-wrack model with both self-diffusion and cross-diffusion terms.We first study the effect of self-diffusion on the stability of equilibrium.We then derive the conditions for the occurrence of the Turing patterns induced by cross-diffusion based on self-diffusion stability.Next,we analyze the pattern selection by using the amplitude equation and obtain the exact parameter ranges of different types of patterns,including stripe patterns,hexagonal patterns and mixed states.Finally,numerical simulations confirm the theoretical results.展开更多
We present Turing pattern selection in a reaction-diffusion epidemic model under zero-flux boundary conditions. The value of this study is twofold. First, it establishes the amplitude equations for the excited modes, ...We present Turing pattern selection in a reaction-diffusion epidemic model under zero-flux boundary conditions. The value of this study is twofold. First, it establishes the amplitude equations for the excited modes, which determines the stability of amplitudes towards uniform and inhomogeneous perturbations. Second, it illustrates all five categories of Turing patterns close to the onset of Turing bifurcation via numerical simulations which indicates that the model dynamics exhibits complex pattern replication: on increasing the control parameter v, the sequence "H0 hexagons → H0-hexagon-stripe mixtures →stripes → Hπ-hexagon-stripe mixtures → Hπ hexagons" is observed. This may enrich the pattern dynamics in a diffusive epidemic model.展开更多
In the cylindrical coordinate system,a singular perturbation theory of multiple-scale asymptotic expansions was developed to study single standing water wave mode by solving potential equations of water waves in a rig...In the cylindrical coordinate system,a singular perturbation theory of multiple-scale asymptotic expansions was developed to study single standing water wave mode by solving potential equations of water waves in a rigid circular cylinder, which is subject to a vertical oscillation.It is assumed that the fluid in the circular cylindrical vessel is inviscid,incompressible and the motion is irrotational, a nonlinear amplitude equation with cubic and vertically excited terms of the vessel was derived by expansion of two-time scales without considering the effect of surface tension.It is shown by numerical computation that different free surface standing wave patterns will be formed in different excited frequencies and amplitudes.The contours of free surface waves are agreed well with the experimental results which were carried out several years ago.展开更多
In this paper, we present the amplitude equations for the excited modes in a cross-diffusive predator-prey model with zero-flux boundary conditions. From these equations, the stability of patterns towards uniform and ...In this paper, we present the amplitude equations for the excited modes in a cross-diffusive predator-prey model with zero-flux boundary conditions. From these equations, the stability of patterns towards uniform and inhomogenous perturbations is determined. Furthermore, we present novel numerical evidence of six typical turing patterns, and find that the model dynamics exhibits complex pattern replications: for μ1 〈μ ≤μ2, the steady state is the only stable solution of the model; for μ2 〈 μ ≤ μ4, by increasing the control parameter μ, the sequence Hπ-hexagons→ Hπ- hexagon-stripe mixtures → stripes → H0-hexagon-stripe mixtures → H0-hexagons is observed; for μ 〉 μ4, the stripe pattern emerges. This may enrich the pattern formation in the cross-diffusive predatorprey model.展开更多
In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions was developed in inviscid fluids to investigate the motion of single free surface standing wave inc...In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions was developed in inviscid fluids to investigate the motion of single free surface standing wave including the effect of surface tension. A nonlinear slowly varying amplitude equation, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from potential flow equation. The results show that, when forced frequency is lower, the effect of surface ten- sion on mode selection of surface wave is not important. However, when forced frequency is higher, the surface tension can not be neglected. This proved that the surface tension causes free surface returning to equilibrium location. In addition, due to considering the effect of surface tension, the theoretical result approaches to experimental results much more than that of no surface tension.展开更多
The Turing instability and the phenomena of pattern formation for a nonlinear reaction-diffusion(RD) system of turbulence-shear flowinteraction are investigated.By the linear stability analysis,the essential condition...The Turing instability and the phenomena of pattern formation for a nonlinear reaction-diffusion(RD) system of turbulence-shear flowinteraction are investigated.By the linear stability analysis,the essential conditions for Turing instability are obtained.It indicates that the emergence of cross-diffusion terms leads to the destabilizing mechanism.Then the amplitude equations and the asymptotic solutions of the model closed to the onset of instability are derived by using the weakly nonlinear analysis.展开更多
This paper mainly focus on the research of a predator⁃prey system with Gompertz growth of prey.When the system does not contain diffusion,the stability conditions of positive equilibrium and the occurring condition of...This paper mainly focus on the research of a predator⁃prey system with Gompertz growth of prey.When the system does not contain diffusion,the stability conditions of positive equilibrium and the occurring condition of the Hopf bifurcation are obtained.When the diffusion term of the system appears,the stable conditions of positive equilibrium and the Turing instability condition are also obtained.Turing instability is induced by the diffusion term through theoretical analysis.Thus,the region of parameters in which Turing instability occurs is presented.Then the amplitude equations are derived by the multiple scale method.The results will enrich the pattern dynamics in predator⁃prey systems.展开更多
Cross-diffusion is a ubiquitous phenomenon in complex networks, but it is often neglected in the study of reaction–diffusion networks. In fact, network connections are often random. In this paper, we investigate patt...Cross-diffusion is a ubiquitous phenomenon in complex networks, but it is often neglected in the study of reaction–diffusion networks. In fact, network connections are often random. In this paper, we investigate pattern dynamics of random networks with cross-diffusion by using the method of network analysis and obtain a condition under which the network loses stability and Turing bifurcation occurs. In addition, we also derive the amplitude equation for the network and prove the stability of the amplitude equation which is also an effective tool to investigate pattern dynamics of the random network with cross diffusion. In the meantime, the pattern formation consistently matches the stability of the system and the amplitude equation is verified by simulations. A novel approach to the investigation of specific real systems was presented in this paper. Finally, the example and simulation used in this paper validate our theoretical results.展开更多
The study of rumor propagation dynamics is of great significance to reduce.false news and ensure the authenticity of news information.In this paper,a SI reaction-diffusion rumor propagation model with nonlinear satura...The study of rumor propagation dynamics is of great significance to reduce.false news and ensure the authenticity of news information.In this paper,a SI reaction-diffusion rumor propagation model with nonlinear saturation incidence is studied.First,through stability analysis,we obtain the conditions for the existence and local stability of the positive equilibrium point.By selecting suitable variable as the control parameter,the critical value of Turing bifurcation and the existence theorem of Turing bifurcation are obtained.Then,using the above theorem and multi-scale standard analysis,the expression of amplitude equation around Turing bifurcation point is obtained.By analyzing the amplitude equation,different types of Turing pattern are divided such as uniform steady-state mode,hexagonal mode,stripe mode and mixed structure mode.Further,in the numerical simulation part,by observing different patterns corresponding to different values of control variable,the correctness of the theory is verified.Finally,the effects of different network structures on patterns are investigated.The results show that there are significant differences in the distribution of users on different network structures.展开更多
By the method of dynamical system, we construct the exact travelling wave solu- tions of a new Hamiltonian amplitude equation and the Ostrovsky equation. Based on this method, the new exact travelling wave solutions o...By the method of dynamical system, we construct the exact travelling wave solu- tions of a new Hamiltonian amplitude equation and the Ostrovsky equation. Based on this method, the new exact travelling wave solutions of the new Hamiltonian am- plitude equation and the Ostrovsky equation, such as solitary wave solutions, kink and anti-kink wave solutions and periodic travelling wave solutions, are obtained, respectively.展开更多
In this paper,several new constant-amplitude and variable-amplitude wave solutions(namely,traveling wave solutions) of a generalized nonlinear Schrdinger equation are investigated by using the extended homogeneous b...In this paper,several new constant-amplitude and variable-amplitude wave solutions(namely,traveling wave solutions) of a generalized nonlinear Schrdinger equation are investigated by using the extended homogeneous balance method,where the balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation,respectively.In addition,stability analysis of those solutions are also conducted by regular phase plane technique.展开更多
The natural frequency of surface wave, which has been derived from avertically oscillating circular cylindrical vessel in inviscid fluid, was modified by consideringthe influence of surface tension and weak viscosity....The natural frequency of surface wave, which has been derived from avertically oscillating circular cylindrical vessel in inviscid fluid, was modified by consideringthe influence of surface tension and weak viscosity. Many flow patterns were found at differentforced frequencies by numerical computation. In addition, the nonlinear amplitude equation derivedin inviscid fluid was modified by adding viscous damping and the unstable regions were determined bystability analysis.展开更多
基金Supported by the National Natural Science Foundation of China (No.19861004)
文摘In the present paper, the existence of global attractor for dissipative Hamiltonian amplitude equation governing the modulated wave instabilities in E0 is considered. By a decomposition of solution operator, it is shown that the global attractor in E0 is actually equal to a global attractor in E1.
基金the National Natural Science Foundation of China(Grant Nos.10971009,11771033,and12201046)Fundamental Research Funds for the Central Universities(Grant No.BLX201925)China Postdoctoral Science Foundation(Grant No.2020M670175)。
文摘We investigate the Turing instability and pattern formation mechanism of a plant-wrack model with both self-diffusion and cross-diffusion terms.We first study the effect of self-diffusion on the stability of equilibrium.We then derive the conditions for the occurrence of the Turing patterns induced by cross-diffusion based on self-diffusion stability.Next,we analyze the pattern selection by using the amplitude equation and obtain the exact parameter ranges of different types of patterns,including stripe patterns,hexagonal patterns and mixed states.Finally,numerical simulations confirm the theoretical results.
基金Project supported by the Natural Science Foundation of Zhejiang Province of China (Grant No.Y7080041)
文摘We present Turing pattern selection in a reaction-diffusion epidemic model under zero-flux boundary conditions. The value of this study is twofold. First, it establishes the amplitude equations for the excited modes, which determines the stability of amplitudes towards uniform and inhomogeneous perturbations. Second, it illustrates all five categories of Turing patterns close to the onset of Turing bifurcation via numerical simulations which indicates that the model dynamics exhibits complex pattern replication: on increasing the control parameter v, the sequence "H0 hexagons → H0-hexagon-stripe mixtures →stripes → Hπ-hexagon-stripe mixtures → Hπ hexagons" is observed. This may enrich the pattern dynamics in a diffusive epidemic model.
文摘In the cylindrical coordinate system,a singular perturbation theory of multiple-scale asymptotic expansions was developed to study single standing water wave mode by solving potential equations of water waves in a rigid circular cylinder, which is subject to a vertical oscillation.It is assumed that the fluid in the circular cylindrical vessel is inviscid,incompressible and the motion is irrotational, a nonlinear amplitude equation with cubic and vertically excited terms of the vessel was derived by expansion of two-time scales without considering the effect of surface tension.It is shown by numerical computation that different free surface standing wave patterns will be formed in different excited frequencies and amplitudes.The contours of free surface waves are agreed well with the experimental results which were carried out several years ago.
基金supported by the Natural Science Foundation of Zhejiang Province,China (Grant No. Y7080041)the Shanghai Postdoctoral Scientific Program,China (Grant No. 09R21410700)
文摘In this paper, we present the amplitude equations for the excited modes in a cross-diffusive predator-prey model with zero-flux boundary conditions. From these equations, the stability of patterns towards uniform and inhomogenous perturbations is determined. Furthermore, we present novel numerical evidence of six typical turing patterns, and find that the model dynamics exhibits complex pattern replications: for μ1 〈μ ≤μ2, the steady state is the only stable solution of the model; for μ2 〈 μ ≤ μ4, by increasing the control parameter μ, the sequence Hπ-hexagons→ Hπ- hexagon-stripe mixtures → stripes → H0-hexagon-stripe mixtures → H0-hexagons is observed; for μ 〉 μ4, the stripe pattern emerges. This may enrich the pattern formation in the cross-diffusive predatorprey model.
基金Project supported by the National Natural Science Foundation of China (Nos.19772063 and 19772068) and the Doctoral Research Fund of the Ministry of Education (No.20010141024)
文摘In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions was developed in inviscid fluids to investigate the motion of single free surface standing wave including the effect of surface tension. A nonlinear slowly varying amplitude equation, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from potential flow equation. The results show that, when forced frequency is lower, the effect of surface ten- sion on mode selection of surface wave is not important. However, when forced frequency is higher, the surface tension can not be neglected. This proved that the surface tension causes free surface returning to equilibrium location. In addition, due to considering the effect of surface tension, the theoretical result approaches to experimental results much more than that of no surface tension.
基金National Natural Science Foundation of China(No.11371087)
文摘The Turing instability and the phenomena of pattern formation for a nonlinear reaction-diffusion(RD) system of turbulence-shear flowinteraction are investigated.By the linear stability analysis,the essential conditions for Turing instability are obtained.It indicates that the emergence of cross-diffusion terms leads to the destabilizing mechanism.Then the amplitude equations and the asymptotic solutions of the model closed to the onset of instability are derived by using the weakly nonlinear analysis.
基金National Natural Science Foundation of China(No.11971143)。
文摘This paper mainly focus on the research of a predator⁃prey system with Gompertz growth of prey.When the system does not contain diffusion,the stability conditions of positive equilibrium and the occurring condition of the Hopf bifurcation are obtained.When the diffusion term of the system appears,the stable conditions of positive equilibrium and the Turing instability condition are also obtained.Turing instability is induced by the diffusion term through theoretical analysis.Thus,the region of parameters in which Turing instability occurs is presented.Then the amplitude equations are derived by the multiple scale method.The results will enrich the pattern dynamics in predator⁃prey systems.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11272277,11572278,and 11572084)the Innovation Scientists and Technicians Troop Construction Projects of Henan Province,China(Grant No.2017JR0013)
文摘Cross-diffusion is a ubiquitous phenomenon in complex networks, but it is often neglected in the study of reaction–diffusion networks. In fact, network connections are often random. In this paper, we investigate pattern dynamics of random networks with cross-diffusion by using the method of network analysis and obtain a condition under which the network loses stability and Turing bifurcation occurs. In addition, we also derive the amplitude equation for the network and prove the stability of the amplitude equation which is also an effective tool to investigate pattern dynamics of the random network with cross diffusion. In the meantime, the pattern formation consistently matches the stability of the system and the amplitude equation is verified by simulations. A novel approach to the investigation of specific real systems was presented in this paper. Finally, the example and simulation used in this paper validate our theoretical results.
基金supported by the National Natural Science Foundation of China(Grant No.12002135)Young Science and Technology Talents Lifting Project of Jiangsu Association for Science and Technology.
文摘The study of rumor propagation dynamics is of great significance to reduce.false news and ensure the authenticity of news information.In this paper,a SI reaction-diffusion rumor propagation model with nonlinear saturation incidence is studied.First,through stability analysis,we obtain the conditions for the existence and local stability of the positive equilibrium point.By selecting suitable variable as the control parameter,the critical value of Turing bifurcation and the existence theorem of Turing bifurcation are obtained.Then,using the above theorem and multi-scale standard analysis,the expression of amplitude equation around Turing bifurcation point is obtained.By analyzing the amplitude equation,different types of Turing pattern are divided such as uniform steady-state mode,hexagonal mode,stripe mode and mixed structure mode.Further,in the numerical simulation part,by observing different patterns corresponding to different values of control variable,the correctness of the theory is verified.Finally,the effects of different network structures on patterns are investigated.The results show that there are significant differences in the distribution of users on different network structures.
文摘By the method of dynamical system, we construct the exact travelling wave solu- tions of a new Hamiltonian amplitude equation and the Ostrovsky equation. Based on this method, the new exact travelling wave solutions of the new Hamiltonian am- plitude equation and the Ostrovsky equation, such as solitary wave solutions, kink and anti-kink wave solutions and periodic travelling wave solutions, are obtained, respectively.
基金supported by the National NSF of China(11571088)NSF of Zhejiang Province(LY13A010020)Program(HNUEYT2013)
文摘In this paper,several new constant-amplitude and variable-amplitude wave solutions(namely,traveling wave solutions) of a generalized nonlinear Schrdinger equation are investigated by using the extended homogeneous balance method,where the balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation,respectively.In addition,stability analysis of those solutions are also conducted by regular phase plane technique.
文摘The natural frequency of surface wave, which has been derived from avertically oscillating circular cylindrical vessel in inviscid fluid, was modified by consideringthe influence of surface tension and weak viscosity. Many flow patterns were found at differentforced frequencies by numerical computation. In addition, the nonlinear amplitude equation derivedin inviscid fluid was modified by adding viscous damping and the unstable regions were determined bystability analysis.