Abstract
In this paper, we study the periodic orbits of a type of odd order differential delay system with 2k−1 lags via the S1 index theory and the variational method. This type of system has not been studied by others. Our results provide a new and more accurate method for counting the number of periodic orbits.
Highlights
IntroductionThe delay differential equations have useful applications in various fields such as age-structured population growth, life sciences, control theory, and any model involving responses with non-zero delays [1,2]
The delay differential equations have useful applications in various fields such as age-structured population growth, life sciences, control theory, and any model involving responses with non-zero delays [1,2].The problem of periodic solutions for multi-delay differential equations started from the research work of Kaplan and Yorke [3] in 1974
I =1 in which f (− x ) = − f ( x ), x f ( x ) > 0, x 6= 0. They proved the existence of the periodic solutions of (1) when n = 1 and n = 2, respectively
Summary
The delay differential equations have useful applications in various fields such as age-structured population growth, life sciences, control theory, and any model involving responses with non-zero delays [1,2]. In 2006, Fei used the variational method to study the number of periodic solutions of Equation (1). He studied the cases of n = 2k − 1 and n = 2k respectively in literature [4,5] since the specific research methods and details vary greatly. All of these researches are about first order differential equations and did not give the precise counting method for the number of periodic solutions. The counting method for the number of periodic orbits in our results only depends on the eigenvalues of A∞ and A0.
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