scholarly journals Periodic Solutions of a Class of Fourth-Order Superlinear Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Yanyan Li ◽  
Yuhua Long
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Nonlinear Term ◽  
Single Equation ◽  
Mountain Pass ◽  
Mountain Pass Lemma ◽  
Variational Techniques ◽  
Superlinear Growth ◽  
Symmetric Mountain Pass Lemma

This paper deals with the periodic solutions of a class of fourth-order superlinear differential equations. By using the classical variational techniques and symmetric mountain pass lemma, the periodic solutions of a single equation in literature are extended to that of equations, and also, the cubic growth of nonlinear term is extended to a general form of superlinear growth.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

23.—Oscillation Phenomena in the Hodgkin-Huxley Equations

1976 ◽  
Vol 74 ◽  
pp. 299-310 ◽  
Author(s):  
William C. Troy ◽  
J. B. McLeod
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Nerve Conduction ◽  
Critical Value ◽  
Squid Axon ◽  
Current Clamp ◽  
Linear Partial Differential Equations ◽  
Current Parameter

SynopsisA widely accepted model of nerve conduction in the squid axon is the systemof four non-linear partial differential equations developed by Hodgkin and Huxley. Under space clamp and current clamp conditions these equations are reduced to a system of ordinary differential equations.We find that under appropriate assumptions on the functions and parameters in the resulting fourth order Hodgkin-Huxley equations there occurs a bifurcation of periodic solutions from the steady state. This bifurcation takes place as the current parameter, I, passes through a critical value.

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