The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems
Abstract
:1. Introduction and Statement of the Main Results
2. Preliminaries
3. Proofs of Theorems 1 and 2
Algorithm 1 The rank of the Jacobian matrices (18) |
restart: with (Linear Algebra): |
num: = proc (a,m) |
local S, F, A, F1, B, BMy, M, c, i; |
S: = proc (n); local i, sum; sum = 0; for i from trunc (1/2 ∗ n + 1/2) to n do sum: = sum + (i)! ∗ ((n−i)! ∗ (2i−n)!)−1 ∗ (−a)n−1 ∗ 2 2i−n od sum end proc F: = proc (i,j)→(j+1) ∗ S(i+2−j); A: = proc (j) local AM; if j = 0 then AM: Matrix (1, 1, 1) else if j = 1 then AM: = Matrix ([4, 3]) else AM: = Matrix (2 ∗ j−2, 2 ∗ j, F) end if; end if; AM end proc; F1: = proc (i, j) → S (i + 2 − j); B: = proc (i) local BM; if i = 1 then BM: = Matrix ([2, 1]) else if i ≤ trunc (1/2 ∗ m) then BM: = Matrix (2 ∗ i − 2, 2 ∗ i, F1) else BM: = Matrix (2 ∗ i − 2, m, F1) end if; end if; BM end proc; BMy: = proc (i) local AM, j; AM: = Matrix (1, 1, 1); for j from 0 to i − 1 do AM: = MatrixMatrixMultiply(AM, A (j)) od; MatrixMatrixMultiply (AM, B (i)) end proc; M:= proc (i) if i ≤ trunc (1/2 ∗ m) then << 0 >|< BMy(i) >|< Matrix(1, m−2 ∗ i, 0)> > else << 0 >|< BMy(i) >> end if; end proc; c: = <<1>| Matrix (1, m, 0) > ; for i to trunc (3/4 ∗ m + 1/2) do c: = <c,M (i)> od; Rank (c) end proc; |
Algorithm 2 The rank of the Jacobian matrices (23) |
restart: with (Linear Algebra): |
rank: = proc (a,m) |
local k, PQ, l, i, divPQ, C, diPQ, cof, G, G1, E, e, ser; |
k: = trunc (1/2 * m + 1/2) * trunc (1/2 * m + 1); PQ: = 0; for l from 0 to m -1 do for i from 0 to l do PQ: = PQ + c[i, l-i] * x* y end do; end do; divPQ[0]: = PQ; C[0]: = subs ([x = 0, y = 0], divPQ[0]); ser: = proc (l) convert (series (1/(1-2 * x + a * x), x = 0, l), polynom) end proc; diPQ: = proc (l, F) local P, Q; P: = sort (collect ((subs (y = 0, F)-subs ([x = 0, y = 0], F)) * x* ser (2 * k-2 * l + 2), x)); Q: = sort (expand (simplify ((F-subs (y = 0, F)) * y)), [x, y]); sort (diff (P, x) + diff (Q, y), [x, y]) end proc; for i to k do divPQ[i]: = diPQ (i, divPQ[i -1]); C[i]: = (factorial (i + 1))* subs ([x = 0, y = 0], divPQ[i]) end do; cof: = proc (i) local f, l, j; f: = 0; for l from 0 to i -1 do for j from 0 to trunc (1/2 * l) do f: = f + c[l-2 * j, 2 * j] * x* y end do; end do; coeffs (f, [x, y]) end proc; G: = 0; for i from 0 to m + trunc (1/2 * (m -1)) -1 do G: = G + C[i] * x end do; G1: = coeffs (G, x); E, e: = GenerateMatrix (G1, [cof (m)]); Rank (E) end proc; |
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Wei, L.; Tian, Y.; Xu, Y. The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems. Mathematics 2022, 10, 1483. https://doi.org/10.3390/math10091483
Wei L, Tian Y, Xu Y. The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems. Mathematics. 2022; 10(9):1483. https://doi.org/10.3390/math10091483
Chicago/Turabian StyleWei, Lijun, Yun Tian, and Yancong Xu. 2022. "The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems" Mathematics 10, no. 9: 1483. https://doi.org/10.3390/math10091483
APA StyleWei, L., Tian, Y., & Xu, Y. (2022). The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems. Mathematics, 10(9), 1483. https://doi.org/10.3390/math10091483