In this paper, we study the propagation and its failure to propagate (pinning) of a travelling wave in a Nagumo type equation, an equation that describes impulse propagation in nerve axons that also models population ...In this paper, we study the propagation and its failure to propagate (pinning) of a travelling wave in a Nagumo type equation, an equation that describes impulse propagation in nerve axons that also models population growth with Allee effect. An analytical solution is derived for the traveling wave and the work is extended to a discrete formulation with a piecewise linear reaction function. We propose an operator splitting numerical scheme to solve the equation and demonstrate that the wave either propagates or gets pinned based on how the spatial mesh is chosen.展开更多
For a coupled slow-fast FitzHugh-Nagumo(FHN)equation derived from a reaction-diffusionmechanics(RDM)model,Holzer et al.(2013)studied the existence and stability of the travelling pulse,which consists of two fast orbit...For a coupled slow-fast FitzHugh-Nagumo(FHN)equation derived from a reaction-diffusionmechanics(RDM)model,Holzer et al.(2013)studied the existence and stability of the travelling pulse,which consists of two fast orbit arcs and two slow ones,where one fast segment passes the unique fold point with algebraic decreasing and two slow ones follow normally hyperbolic critical curve segments.Shen and Zhang(2021)obtained the existence of the travelling pulse,whose two fast orbit arcs both exponentially decrease,and one of the slow orbit arcs could be normally hyperbolic or not at the origin.Here,we characterize both the nonlinear and spectral stability of this travelling pulse.展开更多
In this paper, the dynamical behavior of coupled map lattices (CMLs) discretised from Nagumo equation was considered, and it was proved that there exist various periodic solutions in the CMLs.
The main objective of this article is to obtain the new analytical and numerical solutions of fractional Fitzhugh-Nagumo equation which arises as a model of reaction-ifusion systems,transmission of nerve impulses,circ...The main objective of this article is to obtain the new analytical and numerical solutions of fractional Fitzhugh-Nagumo equation which arises as a model of reaction-ifusion systems,transmission of nerve impulses,circuit theory,biology and the area of population genetics.For this aim conformable derivative with fractional order which is a well behaved,understandable and applicable definition is used as a tool.The analytical solutions were got by utilizing the fact that the conformable fractional derivative provided the chain rule.By the help of this feature which is not provided by other popular fractional derivatives,nonlinear fractional partial differential equation is turned into an integer order differential equation.The numerical solutions which is obtained with the aid of residual power series method are compared with the analytical results that obtained by performing sub equation method.This comparison is made both with the help of three-dimensional graphical representations and tables for different values of theγ.展开更多
In this paper,We study the global attractor and its properties on in nite lattice dynamical system FitzHugh-Nagumo in a weighted space lσ^2×lσ^2.We prove the existence and uniqueness of the solution to the latt...In this paper,We study the global attractor and its properties on in nite lattice dynamical system FitzHugh-Nagumo in a weighted space lσ^2×lσ^2.We prove the existence and uniqueness of the solution to the lattice dynamical system FitzHugh-Nagumo in lσ^2×lσ^2.Then we get a bounded absorbing set,which suggests the existence of global attractors.Finally,we study the uniform boundedness and the upper semicontinuity of the global attractor.展开更多
In this work, the authors considered the periodic optimal control problem of Fitzhugh-Nagumo equation. They firstly prove the existence of time-periodic solution to Fitzhugh-Nagumo equation. Then they show the existen...In this work, the authors considered the periodic optimal control problem of Fitzhugh-Nagumo equation. They firstly prove the existence of time-periodic solution to Fitzhugh-Nagumo equation. Then they show the existence of optimal solution to the optimal control problem, and finally the first order necessary condition is obtained by constructing an appropriate penalty function.展开更多
In this work, a new modified fractional form of the Nagumo equation has been presented and deeply analyzed. Using the Caputo–Fabrizio and Atangana–Baleanu time-fractional derivatives, classical Nagumo model is trans...In this work, a new modified fractional form of the Nagumo equation has been presented and deeply analyzed. Using the Caputo–Fabrizio and Atangana–Baleanu time-fractional derivatives, classical Nagumo model is transformed to a new fractional version. The modified equation has been solved by using the homotopy analysis transform method. The convergence analysis has been also examined with the help of the so-called hh-curves and average residual error. Comparing the obtained approximate solution with the exact solution leaves no doubt believing that the proposed technique is very efficient and converges toward the exact solution very rapidly.展开更多
In this paper, a close-loop feedback control is imposed locally on the Fitzhugh-Nagumo （FHN） system to suppress the stable spirals and spatiotemporal chaos according to the principle of self-adaptive coupling intera...In this paper, a close-loop feedback control is imposed locally on the Fitzhugh-Nagumo （FHN） system to suppress the stable spirals and spatiotemporal chaos according to the principle of self-adaptive coupling interaction. The simulation results show that an expanding target wave is stimulated by the spiral waves under dynamic control period when a local area. of 5 × 5 grids is controlled, or the spiral tip is driven to the board of the system, It is also found that the spatiotemporal chaos can be suppressed to get a stable homogeneous state within 50 time units as two local grids are controlled mutually. The mechanism of the scheme is briefly discussed.展开更多
文摘In this paper, we study the propagation and its failure to propagate (pinning) of a travelling wave in a Nagumo type equation, an equation that describes impulse propagation in nerve axons that also models population growth with Allee effect. An analytical solution is derived for the traveling wave and the work is extended to a discrete formulation with a piecewise linear reaction function. We propose an operator splitting numerical scheme to solve the equation and demonstrate that the wave either propagates or gets pinned based on how the spatial mesh is chosen.
基金supported by National Key R&D Program of China(Grant No.2022YFA1005900)National Natural Science Foundation of China(Grant Nos.12071284 and 12161131001)+1 种基金supported by National Natural Science Foundation of China(Grant No.11871334)Innovation Program of Shanghai Municipal Education Commission(Grant No.2021-01-07-00-02-E00087)。
文摘For a coupled slow-fast FitzHugh-Nagumo(FHN)equation derived from a reaction-diffusionmechanics(RDM)model,Holzer et al.(2013)studied the existence and stability of the travelling pulse,which consists of two fast orbit arcs and two slow ones,where one fast segment passes the unique fold point with algebraic decreasing and two slow ones follow normally hyperbolic critical curve segments.Shen and Zhang(2021)obtained the existence of the travelling pulse,whose two fast orbit arcs both exponentially decrease,and one of the slow orbit arcs could be normally hyperbolic or not at the origin.Here,we characterize both the nonlinear and spectral stability of this travelling pulse.
文摘In this paper, the dynamical behavior of coupled map lattices (CMLs) discretised from Nagumo equation was considered, and it was proved that there exist various periodic solutions in the CMLs.
文摘The main objective of this article is to obtain the new analytical and numerical solutions of fractional Fitzhugh-Nagumo equation which arises as a model of reaction-ifusion systems,transmission of nerve impulses,circuit theory,biology and the area of population genetics.For this aim conformable derivative with fractional order which is a well behaved,understandable and applicable definition is used as a tool.The analytical solutions were got by utilizing the fact that the conformable fractional derivative provided the chain rule.By the help of this feature which is not provided by other popular fractional derivatives,nonlinear fractional partial differential equation is turned into an integer order differential equation.The numerical solutions which is obtained with the aid of residual power series method are compared with the analytical results that obtained by performing sub equation method.This comparison is made both with the help of three-dimensional graphical representations and tables for different values of theγ.
基金Supported by The Scientic Research Foundation Funded by Hunan Provincial Education Department under grant 19A503Partially supported by Hunan Provincial Exploration of Undergraduate Research Learning and Innovative Experiment Project:2018XTUSJ008Hunan Provincial Natural Science Foundation of China under grant 2015JJ2144.
文摘In this paper,We study the global attractor and its properties on in nite lattice dynamical system FitzHugh-Nagumo in a weighted space lσ^2×lσ^2.We prove the existence and uniqueness of the solution to the lattice dynamical system FitzHugh-Nagumo in lσ^2×lσ^2.Then we get a bounded absorbing set,which suggests the existence of global attractors.Finally,we study the uniform boundedness and the upper semicontinuity of the global attractor.
基金supported by the Fundamental Research Funds for the Central Universities,ChinaUniversity of Geosciences(Wuhan)(CUGSX01).
文摘In this work, the authors considered the periodic optimal control problem of Fitzhugh-Nagumo equation. They firstly prove the existence of time-periodic solution to Fitzhugh-Nagumo equation. Then they show the existence of optimal solution to the optimal control problem, and finally the first order necessary condition is obtained by constructing an appropriate penalty function.
文摘In this work, a new modified fractional form of the Nagumo equation has been presented and deeply analyzed. Using the Caputo–Fabrizio and Atangana–Baleanu time-fractional derivatives, classical Nagumo model is transformed to a new fractional version. The modified equation has been solved by using the homotopy analysis transform method. The convergence analysis has been also examined with the help of the so-called hh-curves and average residual error. Comparing the obtained approximate solution with the exact solution leaves no doubt believing that the proposed technique is very efficient and converges toward the exact solution very rapidly.
基金The project supported partially by National Natural Science Foundation of China under Grant No. 90303010 We would like to thank H.Zhang for valuable discussions.
文摘In this paper, a close-loop feedback control is imposed locally on the Fitzhugh-Nagumo （FHN） system to suppress the stable spirals and spatiotemporal chaos according to the principle of self-adaptive coupling interaction. The simulation results show that an expanding target wave is stimulated by the spiral waves under dynamic control period when a local area. of 5 × 5 grids is controlled, or the spiral tip is driven to the board of the system, It is also found that the spatiotemporal chaos can be suppressed to get a stable homogeneous state within 50 time units as two local grids are controlled mutually. The mechanism of the scheme is briefly discussed.
