Regular and Chaotic Dynamics

, Volume 24, Issue 4, pp 392–417 | Cite as

On Transversal Connecting Orbits of Lagrangian Systems in a Nonstationary Force Field: the Newton – Kantorovich Approach

  • Alexey V. IvanovEmail author


We consider a natural Lagrangian system defined on a complete Riemannian manifold subjected to the action of a nonstationary force field with potential U(q,t) = f(t)V(q). It is assumed that the factor f(t) tends to ∞ as t → ±∞ and vanishes at a unique point t0 ∈ ℝ. Let X+, X denote the sets of isolated critical points of V(x) at which U(x,t) as a function of x attains its maximum for any fixed t > t0 and t < t0, respectively. Under nondegeneracy conditions on points of X± we apply the Newton – Kantorovich type method to study the existence of transversal doubly asymptotic trajectories connecting X and X+. Conditions on the Riemannian manifold and the potential which guarantee the existence of such orbits are presented. Such connecting trajectories are obtained by continuation of geodesies defined in a vicinity of the point t0 to the whole real line.


connecting orbits homoclinics heteroclinics nonautonomous Lagrangian system Newton – Kantorovich method 

MSC2010 numbers

37J45 37C29 58K45 65P10 


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This research was supported by RFBR grant (project No. 17-01-00668/19).


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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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