Polynomial Functional Dynamical Systems

The book is about the global stability and bifurcation of equilibriums in polynomial functional systems. Appearing and switching bifurcations of simple and higher-order equilibriums in the polynomial functional systems are discussed, and such bifurcations of equilibriums are not only for simple equilibriums but for higher-order equilibriums. The third-order sink and source bifurcations for simple equilibriums are presented in the polynomial functional systems. The third-order sink and source switching bifurcations for saddle and nodes are also presented, and the fourth-order upper-saddle and lower-saddle switching and appearing bifurcations are presented for two second-order upper-saddles and two second-order lower-saddles, respectively. In general, the (2�� + 1)th-order sink and source switching bifurcations for (2����)th-order saddles and (2���� +1)-order nodes are also presented, and the (2��)th-order upper-saddle and lower-saddle switching and appearing bifurcations are presented for (2����)th-order upper-saddles and (2����)th-order lower-saddles (��, �� = 1,2, ...). The vector fields in nonlinear dynamical systems are polynomial functional. Complex dynamical systems can be constructed with polynomial algebraic structures, and the corresponding singularity and motion complexity can be easily determined.

這本書探討了多項式函數系統中全球穩定性和平衡分歧的問題。討論了多項式函數系統中簡單和高階平衡分歧的出現和切換，這些平衡分歧不僅適用於簡單的平衡，也適用於高階平衡。書中介紹了多項式函數系統中簡單平衡的三階吸引子和源分歧，以及鞍點和節點的三階吸引子和源切換分歧。同樣地，書中還介紹了二階鞍點的四階上鞍點和下鞍點切換和出現分歧，以及二階下鞍點的四階上鞍點和下鞍點切換和出現分歧。一般而言，書中還介紹了(2n+1)階鞍點和(n+1)階節點的(2n+1)階吸引子和源切換分歧，以及(2n)階上鞍點和下鞍點的(2n)階切換和出現分歧，其中(n, n+1 = 1,2, ...)。非線性動力系統中的向量場是多項式函數的。可以使用多項式代數結構構建複雜的動力系統，並且可以輕鬆確定相應的奇異點和運動複雜性。