scholarly journals The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations

2016 ◽  
Vol 21 (10) ◽  
pp. 3793-3808
Author(s):  
Bin Long ◽  
Changrong Zhu
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Homoclinic Solution ◽  
Parabolic Differential Equations

scholarly journals Periodic solutions of certain nonlinear parabolic differential equations in domains with periodically moving boundaries

1978 ◽  
Vol 70 ◽  
pp. 111-123 ◽  
Author(s):  
Yoshio Yamada
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Moving Boundaries ◽  
Typical Problem ◽  
Parabolic Differential Equations ◽  
Periodic Problems ◽  
Nonlinear Parabolic

In this paper we consider the periodic problems for certain nonlinear parabolic differential equations in domains with periodically moving boundaries. The typical problem, which is going to be discussed in the present paper, is to solve the following:

From homoclinics to quasi-periodic solutions for ordinary differential equations

2015 ◽  
Vol 145 (5) ◽  
pp. 1091-1114
Author(s):  
Changrong Zhu
Keyword(s):  
Functional Analysis ◽  
Periodic Solution ◽  
Fixed Point ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Homoclinic Solution ◽  
Unperturbed System ◽  
Perturbed System ◽  
Hyperbolic Fixed Point ◽  
Parameter Values

We consider the quasi-periodic solutions bifurcated from a degenerate homoclinic solution. Assume that the unperturbed system has a homoclinic solution and a hyperbolic fixed point. The bifurcation function for the existence of a quasi-periodic solution of the perturbed system is obtained by functional analysis methods. The zeros of the bifurcation function correspond to the existence of the quasi-periodic solution at the non-zero parameter values. Some solvable conditions of the bifurcation equations are investigated. Two examples are given to illustrate the results.

scholarly journals Existence and Uniqueness of Homoclinic Solution for a Class of Nonlinear Second-Order Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Lijuan Chen ◽  
Shiping Lu
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Homoclinic Orbit ◽  
Limit Point ◽  
Existence And Uniqueness ◽  
Homoclinic Solution ◽  
Second Order ◽  
Continuation Theorem ◽  
Mawhin’S Continuation Theorem ◽  
Second Order Differential Equations

The authors study the existence and uniqueness of a set with2kT-periodic solutions for a class of second-order differential equations by using Mawhin's continuation theorem and some analysis methods, and then a unique homoclinic orbit is obtained as a limit point of the above set of2kT-periodic solutions.

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