6533b82bfe1ef96bd128e26a
RESEARCH PRODUCT
Infinite orbit depth and length of Melnikov functions
L. Ortiz-bobadillaJessie Pontigo-herreraDmitry NovikovPavao Mardešićsubject
PolynomialDynamical Systems (math.DS)Iterated integrals01 natural sciencesHamiltonian system03 medical and health sciences0302 clinical medicineFOS: MathematicsCenter problem030212 general & internal medicine0101 mathematicsMathematics - Dynamical Systems[MATH]Mathematics [math]Mathematical PhysicsMathematical physicsPoincaré mapPhysicsConjecturePlane (geometry)Applied Mathematics010102 general mathematicsMSC : primary 34C07 ; secondary 34C05 ; 34C08Loop (topology)Bounded functionMAPOrbit (control theory)Analysis34C07 34C05 34C08description
Abstract In this paper we study polynomial Hamiltonian systems d F = 0 in the plane and their small perturbations: d F + ϵ ω = 0 . The first nonzero Melnikov function M μ = M μ ( F , γ , ω ) of the Poincare map along a loop γ of d F = 0 is given by an iterated integral [3] . In [7] , we bounded the length of the iterated integral M μ by a geometric number k = k ( F , γ ) which we call orbit depth. We conjectured that the bound is optimal. Here, we give a simple example of a Hamiltonian system F and its orbit γ having infinite orbit depth. If our conjecture is true, for this example there should exist deformations d F + ϵ ω with arbitrary high length first nonzero Melnikov function M μ along γ. We construct deformations d F + ϵ ω = 0 whose first nonzero Melnikov function M μ is of length three and explain the difficulties in constructing deformations having high length first nonzero Melnikov functions M μ .
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2019-11-01 |