The Geometric Properties of Homogeneous Polynomial Vector Fields

Paper [1] studied a generalized characteristic value problem of homogeneous polynomial vector field f ( x ) = f 1 ( x ) , ⋯ , f n ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq6431.tif"/> , where x = x 1 , ⋯ , x n ∈ R n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq6432.tif"/> , and each f_i(x) being a real homogeneous polynomial of degree m(m > 1). The necessary condition for asymptotic stability of the flow about f(x) is that each eigenvalue of f(x) is negative. The key method is the coordinate transformation: x = x i y , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq6433.tif"/>