Spatiotemporal structures arising in two identical cells, which are governed by higher autocatalator kinetics and coupled via diffusive interchange of autocatalyst, are discussed. The stability of the unique homogeneo...Spatiotemporal structures arising in two identical cells, which are governed by higher autocatalator kinetics and coupled via diffusive interchange of autocatalyst, are discussed. The stability of the unique homogeneous steady state is obtained by the linearized theory. A necessary condition for bifurcations in spatially non-uniform solutions in uncoupled and coupled systems is given. Further information about Turing pattern solutions near bifurcation points is obtained by weakly nonlinear theory. Finally, the stability of equilibrium points of the amplitude equation is discussed by weakly nonlinear theory, with the bifurcation branches of the weakly coupled system.展开更多
This paper theoretically analyses and studies stationary patterns in diffusively coupled bistable elements. Since these stationary patterns consist of two types of stationary mode structure： kink and pulse, a mode an...This paper theoretically analyses and studies stationary patterns in diffusively coupled bistable elements. Since these stationary patterns consist of two types of stationary mode structure： kink and pulse, a mode analysis method is proposed to approximate the solutions of these localized basic modes and to analyse their stabilities. Using this method, it reconstructs the whole stationary patterns. The cellular mode structures （kink and pulse） in bistable media fundamentally differ from stationary patterns in monostable media showing spatial periodicity induced by a diffusive Taring bifurcation.展开更多
Turing demonstrated that spatially heterogeneous patterns can be self-organized, when the two substances interact locally and diffuse randomly. Turing systems have been applied not only to explain patterns observed wi...Turing demonstrated that spatially heterogeneous patterns can be self-organized, when the two substances interact locally and diffuse randomly. Turing systems have been applied not only to explain patterns observed within the biological and chemical fields, but also to develop image information processing tools. In a twin study, to evaluate the V-shaped bundle of the inner ear outer hair, we developed a method that utilizes a reaction-diffusion system with anisotropic diffusion that exhibited triangular patterns with the introduction of a certain anisotropy strength. In this study, we explored the parameter range over which these periodic triangular patterns were obtained. First, we defined an index for triangular clearness, TC. Triangular patterns can be obtained by introducing a large anisotropy δ, but the range of δ depends on the diffusion coefficient. We found an explanatory variable that can explain the change in TC based on a heuristic argument of the relative distance of the pitchfork bifurcation point between the maximum and minimum anisotropic diffusion function values. Clear periodic triangular patterns were obtained when the distance between the minimum anisotropic function value and pitchfork bifurcation point was over 2.5 times the distance to the anisotropic diffusion function maximum value. By changing the diffusion coefficients or the reaction terms, we further confirmed the accuracy of this condition using computer simulation. Its relevance to diffusion instability has also been discussed.展开更多
Differential method and homotopy analysis method are used for solving the two-dimensional reaction-diffusion model. And the structure of the solutions is analyzed. Finally, the homotopy series solutions are simulated ...Differential method and homotopy analysis method are used for solving the two-dimensional reaction-diffusion model. And the structure of the solutions is analyzed. Finally, the homotopy series solutions are simulated with the mathematical software Matlab, so the Turing patterns will be produced. Overall analysis and experimental simulation of the model show that the different parameters lead to different Turing pattern structures. As time goes on, the structure of Turing patterns changes, and the final solutions tend to stationary state.展开更多
Control of the spatiotemporal patterns near the codimension-three Turing–Hopf–Wave bifurcations is studied by using time-delayed feedback in a three-variable Brusselator model. Linear stability analysis of the syste...Control of the spatiotemporal patterns near the codimension-three Turing–Hopf–Wave bifurcations is studied by using time-delayed feedback in a three-variable Brusselator model. Linear stability analysis of the system shows that the competition among the Turing-, Hopf- and Wave-modes, the wavenumber, and the oscillation frequency of patterns can be controlled by changing the feedback parameters. The role of the feedback intensity Pu played on controlling the pattern competition is equivalent to that of Pw, but opposite to that of Pv. The role of the feedback intensity Pu played on controlling the wavenumber and oscillation frequency of patterns is equivalent to that of Pv, but opposite to that of Pw. When the intensities of feedback are applied equally, changing the delayed time could not alter the competition among these modes, however, it can control the oscillation frequency of patterns. The analytical results are verified by two-dimensional （2D） numerical simulations.展开更多
By using micron α-SisN4, SiO2, Al2O3 and h-BN as starting materials, O＇ -SiAION-BN （ Si2-z AlzO1 ＋z N2-z, z= 0. 3） composite was synthesized by reaction sintering. According to theoretical proportion ratio： n（ ...By using micron α-SisN4, SiO2, Al2O3 and h-BN as starting materials, O＇ -SiAION-BN （ Si2-z AlzO1 ＋z N2-z, z= 0. 3） composite was synthesized by reaction sintering. According to theoretical proportion ratio： n（ SiO2）/n（ α-Si3N4） = 1, the effects of two sintering aid composites, Y2O3 ＋ B2O3 and Y2O3 ＋ TiO2 at 1700℃ for 2h, were studied. The results indicate that Y2 O3 ＋ TiO2 as sintering aid can accelerate reaction sintering of O＇ -SiAION-BN more effectively than Y2O3 ＋ B2O3, and the relative density of the composites declined with the increase of BN addition （10%, 20% and 30% respectively）; XRD analysis found that excessive β-Si3N4 existed in the O＇ -SiAION-BN composite. Therefore, in order to get more pure O＇ -SiAION and BN phases in the composites ore SiO2 is needed. When Y2O3 ＋ TiO2 was used as sintering aid and addition of BN was 10%, the result of cross experiment on condition of A- n（SiO2）/n（α-Si3N4） was 1.05, 1.1 and 1.2; B-- addition of sintering aid was 2%, 4% and 6% ; C-- firing temperature was 1600℃, 1650℃ and 1700℃ ; D--soaking time was 1h, 2h and 3h, shows that the sintering properties were influenced by factors of firing temperature, soaking time, addition of sintering aid and n（ SiO2 ）/n（α-SisN4） in order of importance. In addition, the technical parameter A s B 3 C s D3 can achieve the highest relative density. Besides, using Pattern Recognition method, the optimized parameter range to form pure O＇ -SiAION and BN without β-Si3N4 remained was determined as Y 〉 1024X^2 - 230. 400X ＋ 11.088 （ X = 0. 9999A -0. 0006C - 0. 0163D, Y = 0. 0163A ＋ 0. 009B -0. 0014C ＋0. 9999D）.展开更多
Pattern formations in an Oregonator model with superdiffusion are studied in two-dimensional(2D) numerical simulations. Stability analyses are performed by applying Fourier and Laplace transforms to the space fraction...Pattern formations in an Oregonator model with superdiffusion are studied in two-dimensional(2D) numerical simulations. Stability analyses are performed by applying Fourier and Laplace transforms to the space fractional reaction–diffusion systems. Antispiral, stable turing patterns, and travelling patterns are observed by changing the diffusion index of the activator. Analyses of Floquet multipliers show that the limit cycle solution loses stability at the wave number of the primitive vector of the travelling hexagonal pattern. We also observed a transition between antispiral and spiral by changing the diffusion index of the inhibitor.展开更多
Based on the principle given in nonlinear diffusion-reaction dynamics, a new dynamic model for dislocation patterning is proposed by introducing a relaxation time to the relation between dislocation density and disloc...Based on the principle given in nonlinear diffusion-reaction dynamics, a new dynamic model for dislocation patterning is proposed by introducing a relaxation time to the relation between dislocation density and dislocation flux. The so-called chemical potential like quantities, which appear in the model can be derived from variation principle for free energy functional of dislocated media, where the free energy density function is expressed in terms of not only the dislocation density itself but also their spatial gradients. The Linear stability analysis on the governing equations of a simple dislocation density shows that there exists an intrinsic wave number leading to bifurcation of space structure of dislocation density. At the same time, the numerical results also demonstrate the coexistence and transition between different dislocation patterns.展开更多
In this paper, a spatial tri-trophic food chain model with ratio-dependent Michaelis-Menten type functional response under homogeneous Neumann boundary conditions is studied. Conditions for Hopf and Turing bifurcation...In this paper, a spatial tri-trophic food chain model with ratio-dependent Michaelis-Menten type functional response under homogeneous Neumann boundary conditions is studied. Conditions for Hopf and Turing bifurcation are derived. Sufficient conditions for the emergence of spatial patterns are obtained. The results of numerical simulations reveal the formation of labyrinth patterns and the coexistence of spotted and stripe-like patterns.展开更多
基金the National Natural Science Foundation of China(No.60574075)
文摘Spatiotemporal structures arising in two identical cells, which are governed by higher autocatalator kinetics and coupled via diffusive interchange of autocatalyst, are discussed. The stability of the unique homogeneous steady state is obtained by the linearized theory. A necessary condition for bifurcations in spatially non-uniform solutions in uncoupled and coupled systems is given. Further information about Turing pattern solutions near bifurcation points is obtained by weakly nonlinear theory. Finally, the stability of equilibrium points of the amplitude equation is discussed by weakly nonlinear theory, with the bifurcation branches of the weakly coupled system.
基金Project partially supported by the Outstanding Oversea Scholar Foundation of the Chinese Academy of Sciences (Bairenjihua)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry
文摘This paper theoretically analyses and studies stationary patterns in diffusively coupled bistable elements. Since these stationary patterns consist of two types of stationary mode structure： kink and pulse, a mode analysis method is proposed to approximate the solutions of these localized basic modes and to analyse their stabilities. Using this method, it reconstructs the whole stationary patterns. The cellular mode structures （kink and pulse） in bistable media fundamentally differ from stationary patterns in monostable media showing spatial periodicity induced by a diffusive Taring bifurcation.
文摘Turing demonstrated that spatially heterogeneous patterns can be self-organized, when the two substances interact locally and diffuse randomly. Turing systems have been applied not only to explain patterns observed within the biological and chemical fields, but also to develop image information processing tools. In a twin study, to evaluate the V-shaped bundle of the inner ear outer hair, we developed a method that utilizes a reaction-diffusion system with anisotropic diffusion that exhibited triangular patterns with the introduction of a certain anisotropy strength. In this study, we explored the parameter range over which these periodic triangular patterns were obtained. First, we defined an index for triangular clearness, TC. Triangular patterns can be obtained by introducing a large anisotropy δ, but the range of δ depends on the diffusion coefficient. We found an explanatory variable that can explain the change in TC based on a heuristic argument of the relative distance of the pitchfork bifurcation point between the maximum and minimum anisotropic diffusion function values. Clear periodic triangular patterns were obtained when the distance between the minimum anisotropic function value and pitchfork bifurcation point was over 2.5 times the distance to the anisotropic diffusion function maximum value. By changing the diffusion coefficients or the reaction terms, we further confirmed the accuracy of this condition using computer simulation. Its relevance to diffusion instability has also been discussed.
文摘Differential method and homotopy analysis method are used for solving the two-dimensional reaction-diffusion model. And the structure of the solutions is analyzed. Finally, the homotopy series solutions are simulated with the mathematical software Matlab, so the Turing patterns will be produced. Overall analysis and experimental simulation of the model show that the different parameters lead to different Turing pattern structures. As time goes on, the structure of Turing patterns changes, and the final solutions tend to stationary state.
基金Project supported by the National Nature Science Foundation of China(Grant No.11205044)the Fundamental Research Funds for the Central Universities(Grant No.10ML40)
文摘Control of the spatiotemporal patterns near the codimension-three Turing–Hopf–Wave bifurcations is studied by using time-delayed feedback in a three-variable Brusselator model. Linear stability analysis of the system shows that the competition among the Turing-, Hopf- and Wave-modes, the wavenumber, and the oscillation frequency of patterns can be controlled by changing the feedback parameters. The role of the feedback intensity Pu played on controlling the pattern competition is equivalent to that of Pw, but opposite to that of Pv. The role of the feedback intensity Pu played on controlling the wavenumber and oscillation frequency of patterns is equivalent to that of Pv, but opposite to that of Pw. When the intensities of feedback are applied equally, changing the delayed time could not alter the competition among these modes, however, it can control the oscillation frequency of patterns. The analytical results are verified by two-dimensional （2D） numerical simulations.
文摘By using micron α-SisN4, SiO2, Al2O3 and h-BN as starting materials, O＇ -SiAION-BN （ Si2-z AlzO1 ＋z N2-z, z= 0. 3） composite was synthesized by reaction sintering. According to theoretical proportion ratio： n（ SiO2）/n（ α-Si3N4） = 1, the effects of two sintering aid composites, Y2O3 ＋ B2O3 and Y2O3 ＋ TiO2 at 1700℃ for 2h, were studied. The results indicate that Y2 O3 ＋ TiO2 as sintering aid can accelerate reaction sintering of O＇ -SiAION-BN more effectively than Y2O3 ＋ B2O3, and the relative density of the composites declined with the increase of BN addition （10%, 20% and 30% respectively）; XRD analysis found that excessive β-Si3N4 existed in the O＇ -SiAION-BN composite. Therefore, in order to get more pure O＇ -SiAION and BN phases in the composites ore SiO2 is needed. When Y2O3 ＋ TiO2 was used as sintering aid and addition of BN was 10%, the result of cross experiment on condition of A- n（SiO2）/n（α-Si3N4） was 1.05, 1.1 and 1.2; B-- addition of sintering aid was 2%, 4% and 6% ; C-- firing temperature was 1600℃, 1650℃ and 1700℃ ; D--soaking time was 1h, 2h and 3h, shows that the sintering properties were influenced by factors of firing temperature, soaking time, addition of sintering aid and n（ SiO2 ）/n（α-SisN4） in order of importance. In addition, the technical parameter A s B 3 C s D3 can achieve the highest relative density. Besides, using Pattern Recognition method, the optimized parameter range to form pure O＇ -SiAION and BN without β-Si3N4 remained was determined as Y 〉 1024X^2 - 230. 400X ＋ 11.088 （ X = 0. 9999A -0. 0006C - 0. 0163D, Y = 0. 0163A ＋ 0. 009B -0. 0014C ＋0. 9999D）.
基金supported by the National Natural Science Foundation of China(Grant Nos.11205044 and 11405042)the Research Foundation of Education Bureau of Hebei Province,China(Grant Nos.Y2012009 and ZD2015025)+1 种基金the Program for Young Principal Investigators of Hebei Province,Chinathe Midwest Universities Comprehensive Strength Promotion Project
文摘Pattern formations in an Oregonator model with superdiffusion are studied in two-dimensional(2D) numerical simulations. Stability analyses are performed by applying Fourier and Laplace transforms to the space fractional reaction–diffusion systems. Antispiral, stable turing patterns, and travelling patterns are observed by changing the diffusion index of the activator. Analyses of Floquet multipliers show that the limit cycle solution loses stability at the wave number of the primitive vector of the travelling hexagonal pattern. We also observed a transition between antispiral and spiral by changing the diffusion index of the inhibitor.
基金The National Natural Science Foundation of China,Grant No.19392300
文摘Based on the principle given in nonlinear diffusion-reaction dynamics, a new dynamic model for dislocation patterning is proposed by introducing a relaxation time to the relation between dislocation density and dislocation flux. The so-called chemical potential like quantities, which appear in the model can be derived from variation principle for free energy functional of dislocated media, where the free energy density function is expressed in terms of not only the dislocation density itself but also their spatial gradients. The Linear stability analysis on the governing equations of a simple dislocation density shows that there exists an intrinsic wave number leading to bifurcation of space structure of dislocation density. At the same time, the numerical results also demonstrate the coexistence and transition between different dislocation patterns.
文摘In this paper, a spatial tri-trophic food chain model with ratio-dependent Michaelis-Menten type functional response under homogeneous Neumann boundary conditions is studied. Conditions for Hopf and Turing bifurcation are derived. Sufficient conditions for the emergence of spatial patterns are obtained. The results of numerical simulations reveal the formation of labyrinth patterns and the coexistence of spotted and stripe-like patterns.
