Khái niệm cốt lõi
Lotka-Volterra tree-systems preserve measures and have rational integrals under Kahan discretisation.
Tóm tắt
Introduction to Lotka-Volterra Systems:
LV systems are autonomous n-dimensional systems.
Darboux polynomials are key for ODEs.
Tree-Systems and Measures:
Tree-systems associated with n-vertex trees preserve measures.
Density and reciprocal density are key for measure preservation.
Kahan Discretisation:
Kahan discretisation of tree-systems factorises and preserves measures.
Linear Darboux polynomials are preserved.
Jacobian Determinant and Integrals:
Explicit expressions for Jacobian determinant and integrals are provided.
Linear functions in Kahan maps correspond to preserved DPs.
Integrals for LV Systems on Graphs:
G-systems associated with graphs have multiple measures and integrals.
Propositions on integrals for G-systems with complete subgraphs are discussed.
Super-Integrability:
ODEs of G-systems are super-integrable.
Kahan discretisation is measure-preserving with at least one integral.
Graph Classification:
Different classes of G-systems and their associated graphs are classified.
Number of functionally independent integrals for each class is determined.
Thống kê
LV systems have n DPs.
LV systems with additional DPs have several integrals.
Kahan discretisation is explicitly given.
Trích dẫn
"We show that Lotka-Volterra T-systems are measure-preserving."