Some computable results on the existence of periodic solutions for singular non-autonomous third order systems
Introduction
There have been extensive researches on the existence of periodic solution for the nonlinear non-autonomous system of ODEs. One of the standard methods employed for this, is to transform the system into a Fredholm's type integral operator by making use of a Green or generalized Green function with a suitable periodic or semi-periodic boundary conditions, and then to apply the topological fixed point theorems like one of Banach or Leray–Schauder to prove the existence of a solution for the integral operator, see for example [1], [2], [3]. When the system has a trivial solution, applying fixed point theorems to establish the existence of a periodic solution for the system is more complicated than the case in which the system doesn't have any trivial solution, and for this reason many authors have employed the topological degree theory to treat these cases; see for example [4], [5], [6]. However, the analysis of differential equations with a singular term is more complicated than the analysis of regular differential equations where defining the topological degree is simply possible. Singular terms are usually encountered in the studies of celestial mechanics specially the studies of periodic behaviour of heavens and galaxies.
Here we are concerned with the following non-autonomous system:where , f is periodic with respect to t with period ω, Λ is a constant positive diagonal n×n matrix as diag[λ12,…,λn2] and γ is also a diagonal matrix. As usual it is assumed that is smooth enough to guarantee the existence and uniqueness of the solution. We first prove the existence of a periodic solution for the nonsingular subsystem of Eq. (1.1) where γ=0 just for the sufficiently small norm of f. For this let us consider the following parametric system:where ϵ is considered as a parameter. Note that when ω is equal to one of the ratios for λi∈{λ1,…,λn}, then the solution of the Eq. (1.2) for linear systems is of the type of resonance which would be unbounded when t→∞. What is interesting here is that we can obtain some conditions under which the nonlinear system has a periodic solution if {λ1,…,λn} has a greatest common divisor λ, i.e., λ=gcd{λ1,…,λn} and . Considering the nonlinearity as a term dependent on a parameter has this advantage that makes it possible to study the deformation, which is caused by the change of the value of the parameter in the system, of a periodic orbit which is related to the first variation of the system.
Section snippets
Main theorem
Let us consider Eq. (1.2) with Λ=I the identity matrix, just for notation simplicity. Applying the method for the case when Λ=diag[λ1,…,λn] has a greatest common divisor as λ is similar. Now note that ω=2π in this case and the necessary and sufficient condition for the existence of a periodic solution for Eq. (1.2) isThe solution of Eq. (1.2) can be written aswhere A, B and C are vectors of
Numerical treatment
Now let us consider an equation for the numerical analysis. Consider the following system:where and L(x,x′,x′′) is a linear term with Sx, Sx′, Sx′′ as its n×n coefficient matrices of x, x′, x′′ respectively, i.e.,and f is a 2π periodic vector function respect to t. The difficulty arising in the computation of Eqs. , for Eq. (3.1) is due to the singular term presented in the equation. Let us denote by the linear
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