Alexey Ivanov
199034, St.Petersburg, Universitetskaya nab. 7/9
SaintPetersburg State University
Publications:
Gelfreikh N. G., Ivanov A. V.
SlowFast Systems with an Equilibrium Near the Folded Slow Manifold
2024, vol. 29, no. 2, pp. 376403
Abstract
We study a slowfast 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 “equilibriumfold” 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 perioddoubling bifurcation. The theory is
applied to a generalization of the FitzHugh – Nagumo system.

Ivanov A. V.
On $SL(2,\mathbb{R})$Cocycles over Irrational Rotations with Secondary Collisions
2023, vol. 28, no. 2, pp. 207226
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.

Ivanov A. V.
On Singularly Perturbed Linear Cocycles over Irrational Rotations
2021, vol. 26, no. 3, pp. 205221
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 quasiperiodic 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.

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. 392417
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.

Ivanov A. V.
Connecting Orbits near the Adiabatic Limit of Lagrangian Systems with Turning Points
2017, vol. 22, no. 5, pp. 479501
Abstract
We consider a natural Lagrangian system defined on a complete Riemannian manifold being subjected to action of a timeperiodic 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 $\varepsilon \in \mathcal{E}_{h}$.

Ivanov A. V.
Connecting Orbits of Lagrangian Systems in a Nonstationary Force Field
2016, vol. 21, no. 5, pp. 510521
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 $X_{+}$. 
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. 5394
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.

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. 329343
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.

Ivanov A. V.
Study of the double mathematical pendulum — I. Numerical investigation of homoclinic transversal intersections
1999, vol. 4, no. 1, pp. 104116
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.
