# Alexey Ivanov

199034, St.-Petersburg, Universitetskaya nab. 7/9
Saint-Petersburg State University

## Publications:

 Gelfreikh N. G., Ivanov A. V. Slow-Fast Systems with an Equilibrium Near the Folded Slow Manifold 2024, vol. 29, no. 2, pp.  376-403 Abstract We study a slow-fast system with two slow and one fast variables. We assume that the slow manifold of the system possesses a fold and there is an equilibrium of the system in a small neighborhood of the fold. We derive a normal form for the system in a neighborhood of the pair “equilibrium-fold” and study the dynamics of the normal form. In particular, as the ratio of two time scales tends to zero we obtain an asymptotic formula for the Poincaré map and calculate the parameter values for the first period-doubling bifurcation. The theory is applied to a generalization of the FitzHugh – Nagumo system. Keywords: slow-fast systems, period-doubling bifurcation Citation: Gelfreikh N. G., Ivanov A. V.,  Slow-Fast Systems with an Equilibrium Near the Folded Slow Manifold, Regular and Chaotic Dynamics, 2024, vol. 29, no. 2, pp. 376-403 DOI:10.1134/S156035472354002X
 Ivanov A. V. On $SL(2,\mathbb{R})$-Cocycles over Irrational Rotations with Secondary Collisions 2023, vol. 28, no. 2, pp.  207-226 Abstract We consider a skew product $F_{A} = (\sigma_{\omega}, A)$ over irrational rotation $\sigma_{\omega}(x) = x + \omega$ of a circle $\mathbb{T}^{1}$. It is supposed that the transformation $A: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ which is a $C^{1}$-map has the form $A(x) = R\big(\varphi(x)\big) Z\big(\lambda(x)\big)$, where $R(\varphi)$ is a rotation in $\mathbb{R}^{2}$ through the angle $\varphi$ and $Z(\lambda)= \text{diag}\{\lambda, \lambda^{-1}\}$ is a diagonal matrix. Assuming that $\lambda(x) \geqslant \lambda_{0} > 1$ with a sufficiently large constant $\lambda_{0}$ and the function $\varphi$ is such that $\cos \varphi(x)$ possesses only simple zeroes, we study hyperbolic properties of the cocycle generated by $F_{A}$. We apply the critical set method to show that, under some additional requirements on the derivative of the function $\varphi$, the secondary collisions compensate weakening of the hyperbolicity due to primary collisions and the cocycle generated by $F_{A}$ becomes uniformly hyperbolic in contrast to the case where secondary collisions can be partially eliminated. Keywords: linear cocycle, hyperbolicity, Lyapunov exponent, critical set Citation: Ivanov A. V.,  On $SL(2,\mathbb{R})$-Cocycles over Irrational Rotations with Secondary Collisions, Regular and Chaotic Dynamics, 2023, vol. 28, no. 2, pp. 207-226 DOI:10.1134/S1560354723020053
 Ivanov A. V. On Singularly Perturbed Linear Cocycles over Irrational Rotations 2021, vol. 26, no. 3, pp.  205-221 Abstract We study a linear cocycle over the irrational rotation $\sigma_{\omega}(x) = x + \omega$ of the circle~$\mathbb{T}^{1}$. It is supposed that the cocycle is generated by a $C^{2}$-map $A_{\varepsilon}: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ which depends on a small parameter $\varepsilon\ll 1$ and has the form of the Poincar\'e map corresponding to a singularly perturbed Hill equation with quasi-periodic potential. Under the assumption that the norm of the matrix $A_{\varepsilon}(x)$ is of order $\exp(\pm \lambda(x)/\varepsilon)$, where $\lambda(x)$ is a positive function, we examine the property of the cocycle to possess an exponential dichotomy (ED) with respect to the parameter~$\varepsilon$. We show that in the limit $\varepsilon\to 0$ the cocycle typically'' exhibits ED only if it is exponentially close to a constant cocycle. Conversely, if the cocycle is not close to a constant one, it does not possess ED, whereas the Lyapunov exponent is typically'' large. Keywords: exponential dichotomy, Lyapunov exponent, reducibility, linear cocycle Citation: Ivanov A. V.,  On Singularly Perturbed Linear Cocycles over Irrational Rotations, Regular and Chaotic Dynamics, 2021, vol. 26, no. 3, pp. 205-221 DOI:10.1134/S1560354721030011
 Ivanov A. V. On Transversal Connecting Orbits of Lagrangian Systems in a Nonstationary Force Field: the Newton – Kantorovich Approach 2019, vol. 24, no. 4, pp.  392-417 Abstract 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 $\infty$ 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 $x$ attains its maximum for any fixed $t> t_{0}$ and $t< t_{0}$, respectively. Under nondegeneracy conditions on points of $X_{\pm}$ 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 geodesics defined in a vicinity of the point $t_{0}$ to the whole real line. Keywords: connecting orbits, homoclinics, heteroclinics, nonautonomous Lagrangian system, Newton – Kantorovich method Citation: Ivanov A. V.,  On Transversal Connecting Orbits of Lagrangian Systems in a Nonstationary Force Field: the Newton – Kantorovich Approach, Regular and Chaotic Dynamics, 2019, vol. 24, no. 4, pp. 392-417 DOI:10.1134/S1560354719040038
 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. 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
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