Sentence examples for weak focus of from inspiring English sources

They proved that nine limit cycles could bifurcate from the origin when the origin is a weak focus of order eight.

For the succession function in (13), if v2(2π) = v3(2π) = · · · = v2k(2π) = 0 and v2k+1(2π) ≠ 0, then the origin is called the fine focus or weak focus of order k, and the quantity of v2k+1(2π) is called the k th focal value at the origin on center manifold of system (8) or (2), k = 1, 2,.... Remark 1.

Next we will prove that the third-order nilpotent singular O ( 0, 0 ) is at most a weak focus of order five, moreover, based on this conclusion, the perturbed system of (1.2) can produce five limit cycles enclosing an elementary node at the origin.

Further, Theorem 3.1.3 in [15] shows that if the origin of system (4.3)(|_{delta=varepsison=0}) is a weak focus of order m, then, when (0

Further, Theorem 3.1.3 in [15] yields that if the origin of system (3.4)(mid_{delta =varepsilon =0}) is a weak focus of order m, then, when (0

When the origin of system (1.1) is a 11th-order weak focus, the first Lyapunov constant of system (5.3) at the origin is V_{1}=2 (2 a_{22} + 3 b_{13}) varepsilon^{3} (2 + b40varepsilonlambda )^{2}neq0, when (varepsilonrightarrow0). Similarly, summarizing the above results yields the following theorem.

When the origin of system (1.7) is a 12th-order weak focus, the first Lyapunov constant of system (3.7) at origin is V_{1}=-frac{1}{4}b_{21}varepsilon +o varepsilon) neq 0 when (varepsilon rightarrow 0).

So λ13 ≠ 0, the origin of system (1.2) is a 13-order weak focus.

So, when condition (3.3) holds, the origin of system (1.7) is a 12th-order weak focus.

2. If ν2 -2π) ≠ 0, the origin of system (2.2) is called 1-order weak focus.

So when condition in Theorem 4.1 holds, the origin of system (1.1) is a 11th-order weak focus.