Averaging principle and a variational approach to the problem of bifurcation of periodic solutions from nonisolated equilibria of the averaged equation
Consider the differential equation \[ {dx\over dt}= Ax+ f\Biggl({t\over\varepsilon}, x\Biggr),\tag{\(*\)} \] where \(A\) is a nonsingular \(n\times n\)-matrix, \(f\in C(\mathbb{R}\times\mathbb{R}^n,\mathbb{R}^n)\) is \(T\)-periodic in the first variable and twice differentiable in the second variable, \(0\leq \varepsilon\ll 1\). Under the additional assumptions (A1), the averaged equation \[ {dx\over dt}= Ax+{1\over T} \int^T_0 f(t,x)\,dx \] has a one-dimensional connected manifold of equilibria \(\{x_0(\theta): \theta\in [0,1]\}\), where \(x_0\) is twice differentiable, and it holds \(x_0'(\theta)\neq 0\,\forall\theta\), and (A2), the zero eigenvalue of the operator \[ A+{1\over T} \int^T_0 {\partial f\over\partial x} (t,x_0(\theta))\,dt \] is simple, and by using the abstract Malkin bifurcation function \(M\) defined in the paper, the authors prove the following averaging principle. Theorem. Let the assumptions (A1) and (A2) be satisfied. Additionally it is assumed that there is \(\theta_0\in [0,1]\) such that \(M(\theta_0)= 0\), \(M'(\theta_0)= 0\). Then, for sufficiently small \(\varepsilon\), there exists a family of \(\varepsilon T\)-periodic solutions \(\{x(\varepsilon)\}\) of \((*)\) such that \(x(\varepsilon)\to x_0(\theta_0)\) as \(\varepsilon\to 0\). Moreover, a variational approach is presented to solve the same problem.
scientific article; zbMATH DE number 6065258