Approximation to invariant manifolds under pseudo-hyperbolicity

作者：Du, Zhengdong; Zhang, Weinian^*; Deng, Shengfu

来源：IMA Journal of Applied Mathematics, 2016, 81(1): 132-151.

DOI：10.1093/imamat/hxv030

• invariant manifold
• pseudo-hyperbolicity
• approximation
• locality radius

As known from the theory of centre manifolds, it is important to approximate invariant manifolds when discussing high order terms of the reduced systems in the investigation of bifurcations. In this paper, we give a method to approximate invariant manifolds which are obtained under pseudo-hyperbolicity. This method enables us to avoid the trouble of increasing dimension of the so-called approximate inertial manifold for smaller thickness of attractive neighbourhood. Unlike centre manifolds, exponential expansion and contraction may exist on such manifolds. Consequently, the technique of the contraction principle cannot be used directly. We not only overcome the difficulty so as to approximate such C-1 manifolds at a possibly higher order but also estimate, for non-local bifurcations, the lower bound of their locality radius such that the approximation is valid. Our approximation is demonstrated by examples of a wave equation and an ordinary differential equation.

• 单位
  岭南师范学院; 四川大学