scholarly journals Synchronization in complete networks of reaction-diffusion equations of FitzHugh-Nagumo type with nonlinear coupling

2021 ◽  
Vol 13 (2) ◽  
pp. 43-51
Author(s):  
Van Long Em Phan
Keyword(s):  
Theoretical Result ◽  
Numerical Simulations ◽  
Coupling Strength ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Sufficient Condition ◽  
Nonlinear Coupling ◽  
Complete Network ◽  
Complete Networks

The synchronization in complete network consisting of  nodes is studied in this paper. Each node is connected to all other ones by nonlinear coupling and is represented by a reaction-diffusion system of FitzHugh-Nagumo type which can be obtained by simplifying the famous Hodgkin-Huxley model. From this complete network, the sufficient condition on the coupling strength to achieve the synchronization is found. The result shows that the networks with bigger in-degrees of nodes synchronize more easily. The paper also presents the numerical simulations for theoretical result and shows a compromise between the theoretical and numerical results.

scholarly journals Indentical synchronization in complete networks of reaction-diffusion equations of FitzHugh-Nagumo

2020 ◽  
Vol 8 (2) ◽  
pp. 45-53
Author(s):  
Phan Van Long Em
Keyword(s):  
Theoretical Result ◽  
Coupling Strength ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Sufficient Condition ◽  
Linear Coupling ◽  
Natural Systems ◽  
Complete Network ◽  
Complete Networks

Synchronization is a ubiquitous feature in many natural systems and nonlinear science. This paper studies the synchronization in complete network consisting of n nodes. Each node is connected to all other nodes by linear coupling and represented by a reaction-diffusion system of FitzHugh-Nagumo type which can be obtained by simplifying the famous Hodgkin-Huxley model. From this complete network, the author seeks a sufficient condition on the coupling strength to achieve synchronization. The result shows that the more easily the nodes synchronize, the bigger the degrees of the networks. Based on this consequence, the author will test the theoretical result numerically to see if there is a compromise.

scholarly journals Nonlinear Dynamics of a Toxin-Phytoplankton-Zooplankton System with Self- and Cross-Diffusion

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Pengfei Wang ◽  
Min Zhao ◽  
Hengguo Yu ◽  
Chuanjun Dai ◽  
Nan Wang ◽  
...  
Keyword(s):  
Nonlinear Dynamics ◽  
Spatial Distribution ◽  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Cross Diffusion ◽  
The Rich

A nonlinear system describing the interaction between toxin-producing phytoplankton and zooplankton was investigated analytically and numerically, where the system was represented by a couple of reaction-diffusion equations. We analyzed the effect of self- and cross-diffusion on the system. Some conditions for the local and global stability of the equilibrium were obtained based on the theoretical analysis. Furthermore, we found that the equilibrium lost its stability via Turing instability and patterns formation then occurred. In particular, the analysis indicated that cross-diffusion can play an important role in pattern formation. Subsequently, we performed a series of numerical simulations to further study the dynamics of the system, which demonstrated the rich dynamics induced by diffusion in the system. In addition, the numerical simulations indicated that the direction of cross-diffusion can influence the spatial distribution of the population and the population density. The numerical results agreed with the theoretical analysis. We hope that these results will prove useful in the study of toxic plankton systems.

Steady State Bifurcation and Patterns of Reaction–Diffusion Equations

2020 ◽  
Vol 30 (11) ◽  
pp. 2050215
Author(s):  
Chunrui Zhang ◽  
Baodong Zheng
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Spatial Region ◽  
Bifurcation Points ◽  
Steady State Bifurcation ◽  
Theoretical Results

In this paper, steady state bifurcations arising from the reaction–diffusion equations are investigated. Using the Lyapunov–Schmidt reduction on a square domain, a simple, and a double steady state bifurcation caused by the symmetry of spatial region is obtained. By examining the reduced bifurcation equations, complete bifurcation scenario and patterns at simple and double steady state bifurcation points are obtained. Numerical simulations support the theoretical results.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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