And μ m is called the mth singular point quantity at the origin on center manifold of system (17) or (8) or (2).
μ k is called the k th singular point quantity of the origin of system (2.1) and τ k is called the k th period constant of the origin of system (2.1).
where μ m is the mth singular point quantity at infinity, '∼' is the symbol of algebraic equivalence, c 00 = 1, c ( 2 n + 1 ) k, ( 2 n + 1 ) k can be taken arbitrarily, k = 1, 2, … .
Based on the developed algorithm of a singular point quantity in [12, 13] and the generalized algorithm of period constants in [11], we find all integrable and linearizable necessary conditions.
Definition 1.1 For any positive integer k, μ k is called the k th singular point quantity of the origin of system (4), and the origin of system (4) is called the generalized center, i.e., system (4) is integrable at the origin if μ k = 0, i.e., p k = q k, k = 1, 2, 3, … .
where c 00 = 1, c k k = 0, k = 1, 2, … , f 5 k ( z, w ) = ∑ α + β = 5 k c α β z α w β, and for any positive integer m, the mth singular point quantity at the origin μ m ( 0 ) can be determined by the following recursion formulas: (3.3) (3.3).
For the mth singular point quantity and the mth focal value at the origin on center manifold of system (8), i.e. µ m and v2m+1, m = 1, 2,..., we have the following relation: v 2 m + 1 ( 2 π ) = i π μ m + i π ∑ k = 1 m - 1 ξ m ( k ) μ k (20).
For the mth Liapunov constant of the origin for system (2) and the mth singular point quantity of the origin for system (8) or (17), i.e. V2mand µ m, m = 1, 2,..., there exists the following relation: V 2 m = i π σ m μ m + i π σ m ∑ k = 1 m - 1 ξ m ( k ) μ k (29).
and for any positive integer m, λ m is determined uniquely by the following recursive formula: λ m = [ ( m q + q − 1 ) a 2 − 3 ( m + 1 ) b 1 ] c m q + q − 1, 3 m + 3 + [ ( m + 1 ) q a 1 − ( 3 m + 2 ) b 2 ] c m q + q, 3 m + 2 ; (17). in the above expressions, if α < 0 or β < 0, let c α β = 0, and λ m is the mth singular point quantity of the origin of system (3).
In Sections 3 and 4, the first nine singular point quantities at degenerate singular point and the first seven singular point quantities at infinity are deduced.
