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2013
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# Alexei Ivanov

Depatment of Applied Mathematics, St.-Petersburg State University

## Publications:

 Ivanov A. V. Connecting Orbits near the Adiabatic Limit of Lagrangian Systems with Turning Points 2017, vol. 22, no. 5, pp.  479-501 Abstract We consider a natural Lagrangian system defined on a complete Riemannian manifold being subjected to action of a time-periodic force field with potential $U(q,t, \varepsilon) = f(\varepsilon t)V(q)$ depending slowly on time. It is assumed that the factor $f(\tau)$ is periodic and vanishes at least at one point on the period. Let $X_{c}$ denote a set of isolated critical points of $V(x)$ at which $V(x)$ distinguishes its maximum or minimum. In the adiabatic limit $\varepsilon \to 0$ we prove the existence of a set $\mathcal{E}_{h}$ such that the system possesses a rich class of doubly asymptotic trajectories connecting points of $X_{c}$  for $\varepsilon \in \mathcal{E}_{h}$. Keywords: connecting orbits, homoclinic and heteroclinic orbits, nonautonomous Lagrangian system, singular perturbation, exponential dichotomy Citation: Ivanov A. V.,  Connecting Orbits near the Adiabatic Limit of Lagrangian Systems with Turning Points, Regular and Chaotic Dynamics, 2017, vol. 22, no. 5, pp. 479-501 DOI:10.1134/S1560354717050021
 Ivanov A. V. Connecting Orbits of Lagrangian Systems in a Nonstationary Force Field 2016, vol. 21, no. 5, pp.  510-521 Abstract We study connecting orbits of 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 $\infty$ as $t\to\pm\infty$ and vanishes at a unique point $t_{0} \in \mathbb{R}$. Let $X_{+}$, $X_{-}$ denote the sets of isolated critical points of $V(x)$ at which $U(x, t)$ as a function of $x$ distinguishes its maximum for any fixed $t > t_{0}$ and $t < t_{0}$, respectively. Under nondegeneracy conditions on points of $X_\pm$ we prove the existence of infinitely many doubly asymptotic trajectories connecting $X_{-}$ and $X_{+}$. Keywords: connecting orbits, homoclinic and heteroclinic orbits, nonautonomous Lagrangian system, variational method Citation: Ivanov A. V.,  Connecting Orbits of Lagrangian Systems in a Nonstationary Force Field, Regular and Chaotic Dynamics, 2016, vol. 21, no. 5, pp. 510-521 DOI:10.1134/S1560354716050026
 Ivanov A. V. Study of the Double Mathematical Pendulum — IV. Quantitative Bounds on Values of the System Parameters when the Homoclinic Transversal Intersections Exist 2001, vol. 6, no. 1, pp.  53-94 Abstract We consider the double mathematical pendulum in the limit of small ratio of pendulum masses. Besides we assume that values of other two system parameters are close to the degenerate ones (i.e. zero or infinity). In these limit cases we prove asymptotic formulae for the homoclinic invariant of some special chosen homoclinic trajectories and obtain quantitative bounds on values of the system parameters when these formulae are valid. Citation: Ivanov A. V.,  Study of the Double Mathematical Pendulum — IV. Quantitative Bounds on Values of the System Parameters when the Homoclinic Transversal Intersections Exist, Regular and Chaotic Dynamics, 2001, vol. 6, no. 1, pp. 53-94 DOI:10.1070/RD2001v006n01ABEH000166
 Ivanov A. V. Study of the Double Mathematical Pendulum — III. Melnikov's Method Applied to the System In the Limit of Small Ratio of Pendulums Masses 2000, vol. 5, no. 3, pp.  329-343 Abstract We consider the double mathematical pendulum in the limit when the ratio of pendulums masses is close to zero and if the value of one of other system parameters is close to degenerate value (i.e. zero or infinity). We investigate homoclinic intersections, using Melnikov's method, and obtain an asymptotic formula for the homoclinic invariant in this case. Citation: Ivanov A. V.,  Study of the Double Mathematical Pendulum — III. Melnikov's Method Applied to the System In the Limit of Small Ratio of Pendulums Masses, Regular and Chaotic Dynamics, 2000, vol. 5, no. 3, pp. 329-343 DOI:10.1070/RD2000v005n03ABEH000152
 Ivanov A. V. Study of the double mathematical pendulum — I. Numerical investigation of homoclinic transversal intersections 1999, vol. 4, no. 1, pp.  104-116 Abstract We investigate the separatrices splitting of the double mathematical pendulum. The numerical method to find periodic hyperbolic trajectories, homoclinic transversal intersections of its separatreces is discussed. This method is realized for some values of the system parameters and it is found out that homoclinic invariants corresponding to these parameters are not equal to zero. Citation: Ivanov A. V.,  Study of the double mathematical pendulum — I. Numerical investigation of homoclinic transversal intersections, Regular and Chaotic Dynamics, 1999, vol. 4, no. 1, pp. 104-116 DOI:10.1070/RD1999v004n01ABEH000102