arXiv subjects

Olga Podvigina

Publications and source records attributed to Olga Podvigina.

16 recordsLinked to original sources

arXiv2026

Asymptotic stability of heteroclinic cycles of type Y

We investigate stability of a new class of heteroclinic cycles that we call heteroclinic cycles of type Y. The cycles can be regarded as a generalisation of heteroclinic cycles of type Z introduced in [Podvigina, Nonlinearity 25, 2012]. The type Y cycles differ from the cycles of type Z in the following: The trajectories comprising a cycle of type Y belong to flow-invariant subspaces that can be of different dimensions. Unlike in the most studies of the stability of heteroclinic cycles, we do not require that the eigenvalues of the linearisations of the dynamical system near the equilibria are distinct. Instead of the common assumption that the cycles are robust, we prescribe flow-invariance of certain subspaces. Similarly to type Z cycles, asymptotic stability and fragmentary asymptotic stability of type Y cycles is determined by the eigenvalues and eigenvectors of transition matrices. The matrices are products of basic transition matrices that depend on the eigenvalues of linearisations and the dimensions of the contracting subspaces.

math.DS↗

arXiv2023

Stability of convective rolls in a horizontal layer rotating about an inclined axis

We present three results on stability of rolls in Boussinesq convection in a plane horizontal layer with rigid boundaries that is rotating about an inclined axis with the angular velocity $Ω=(Ω_1,Ω_2,Ω_3)$. i. We call the full problem the set of equations governing the temporal behaviour of the flow and temperature for an arbitrary $Ω$, and by the reduced problem the set of equations for the angular velocity $(Ω_1,0,Ω_3)$. Here $x,y$ are horizontal Cartesian coordinates in the layer and $z$ is the vertical one. We prove that a $y$-independent solution to one of the two problems is also a solution to the second one. ii. We calculate the critical Rayleigh number for the monotonic onset of convection. The instability mode in the form of rolls (a flow independent of a horizontal direction) is assumed. Let $β$ be the angle between the horizontal projection of $Ω$ and the rolls axes. We show that $β=0$ for the least stable mode. Taking i into account, we conclude that the critical Rayleigh number for the onset of convection is independent of $Ω_1$ and $Ω_2$ and the emerging flow are rolls with axis aligned with the horizontal component of the rotation vector. iii. We study the behaviour of convective flows by integrating numerically the three-dimensional equations of convection for $Ω=(0,Ω_2,Ω_3)$ and a range of the Rayleigh numbers, other parameters of the problem being fixed. We assume square horizontal periodicity cells, whose sides are equal to the period of the most unstable mode. The computations indicate that, in general, in the nonlinear regime convective rolls become more stable as $Ω_2$ increases. Namely, on increasing $Ω_2$, the interval of the Rayleigh numbers for which convective rolls are stable increases.

physics.flu-dyn↗

arXiv2023

An efficient Galerkin method for problems with physically realistic boundary conditions

The Galerkin method is often employed for numerical integration of evolutionary equations, such as the Navier-Stokes equation or the magnetic induction equation. Application of the method requires solving an equation of the form $P(Av-f)=0$ at each time step, where $v$ is an element of a finite-dimensional space $V$ with a basis satisfying boundary conditions, $P$ is the orthogonal projection on this space and $A$ is a linear operator. Usually the coefficients of $v$ expanded in the basis are found by calculating the matrix of $PA$ acting on $V$ and solving the respective system of linear equations. For physically realistic boundary conditions (such as the no-slip boundary conditions for the velocity, or for a dielectric outside the fluid volume for the magnetic field) the basis is often not orthogonal and solving the problem can be computationally demanding. We propose an algorithm giving an opportunity to reduce the computational cost for such a problem. Suppose there exists a space $W$ that contains $V$, the difference between the dimensions of $W$ and $V$ is small relative to the dimension of $V$, and solving the problem $P(Aw-f)=0$, where $w$ is an element of $W$, requires less operations than solving the original problem. The equation $P(Av-f)=0$ is then solved in two steps: we solve the problem $P(Aw-f)=0$ in $W$, find a correction $h=v-w$ that belongs to a complement to $V$ in $W$, and obtain the solution $w+h$. When the dimension of the complement is small the proposed algorithm is more efficient than the traditional one.

math.NA↗

arXiv2021

Behaviour of trajectories near a two-cycle heteroclinic network

We study behaviour of trajectories near a type Z heteroclinic network which is a union of two cycles. Analytical and numerical studies indicate that attractiveness of this network can be associated with various kinds of dynamics in its vicinity, one or both of these cycle being fragmentarily asymptotically stable, or both being completely unstable. In the latter case trajectories can switch irregularly between the cycles, or they can make a certain number of turns around one of them before switching to the other one. Regular behaviour of trajectories near a heteroclinic network can be described using the notion of an omnicycle, which we introduce in this paper. In particular, we use it to prove that the network can be fragmentarily asymptotically stable even if both cycles are completely unstable.

nlin.CD↗

arXiv2019

Asymptotic stability of robust heteroclinic networks

We provide conditions guaranteeing that certain classes of robust heteroclinic networks are asymptotically stable. We study the asymptotic stability of ac-networks --- robust heteroclinic networks that exist in smooth ${\mathbb Z}^n_2$-equivariant dynamical systems defined in the positive orthant of ${\mathbb R}^n$. Generators of the group ${\mathbb Z}^n_2$ are the transformations that change the sign of one of the spatial coordinates. The ac-network is a union of hyperbolic equilibria and connecting trajectories, where all equilibria belong to the coordinate axes (not more than one equilibrium per axis) with unstable manifolds of dimension one or two. The classification of ac-networks is carried out by describing all possible types of associated graphs. We prove sufficient conditions for asymptotic stability of ac-networks. The proof is given as a series of theorems and lemmas that are applicable to the ac-networks and to more general types of networks. Finally, we apply these results to discuss the asymptotic stability of several examples of heteroclinic networks.

math.DS↗

arXiv2018

Stability of a heteroclinic network and its cycles: a case study from Boussinesq convection

This article is concerned with three heteroclinic cycles forming a heteroclinic network in ${\mathbb R}^6$. The stability of the cycles and of the network are studied. The cycles are of a type that has not been studied before, and provide an illustration for the difficulties arising in dealing with cycles and networks in high dimension. In order to obtain information on the stability for the present network and cycles, in addition to the information on eigenvalues and transition matrices, it is necessary to perform a detailed geometric analysis of return maps. Some general results and tools for this type of analysis are also developed here.

math.DS↗

arXiv2018

Simple heteroclinic networks in ${\mathbb R}^4$

We classify simple heteroclinic networks for a $Γ$-equivariant system in ${\mathbb R}^4$ with finite $Γ\subset {\rm O}(4)$, proceeding as follows: we define a graph associated with a given $Γ\subset {\rm O}(n)$ and identify all so-called simple graphs associated with subgroups of ${\rm O}(4)$. Then, knowing the graph associated with a given $Γ$, we determine the types of heteroclinic networks that the group admits. Our study is restricted to networks that are maximal in the sense that they have the highest possible number of connections -- any non-maximal network can then be derived by deleting one or more connections. Finally, for networks of type A, i.e., admitted by $Γ\subset {\rm SO}(4)$, we give necessary and sufficient conditions for fragmentary and essential asymptotic stability. (For other simple heteroclinic networks the conditions for stability are known.) The results are illustrated by a numerical example of a simple heteroclinic network that involves two subcycles that can be essentially asymptotically stable simultaneously.

math.DS↗

arXiv2018

Pseudo-simple heteroclinic cycles in $R^4$

We study pseudo-simple heteroclinic cycles for a $Γ$-equivariant system in $R^4$ with finite $Γ\subset O(4)$, and their nearby dynamics. In particular, in a first step towards a full classification - analogous to that which exists already for the class of simple cycles - we identify all finite subgroups of $O(4)$ admitting pseudo-simple cycles. To this end we introduce a constructive method to build equivariant dynamical systems possessing a robust heteroclinic cycle. Extending a previous study we also investigate the existence of periodic orbits close to a pseudo-simple cycle, which depends on the symmetry groups of equilibria in the cycle. Moreover, we identify subgroups $Γ\subset O(4)$, $Γ\not\subset SO(4)$, admitting fragmentarily asymptotically stable pseudo-simple heteroclinic cycles. (It has been previously shown that for $Γ\subset SO(4)$ pseudo-simple cycles generically are completely unstable.) Finally, we study a generalized heteroclinic cycle, which involves a pseudo-simple cycle as a subset.

nlin.CD↗

arXiv2015

Asymptotic stability of pseudo-simple heteroclinic cycles in R^4

Robust heteroclinic cycles in equivariant dynamical systems in R^4 have been a subject of intense scientific investigation because, unlike heteroclinic cycles in R^3, they can have an intricate geometric structure and complex asymptotic stability properties that are not yet completely understood. In a recent work, we have compiled an exhaustive list of finite subgroups of O(4) admitting the so-called simple heteroclinic cycles, and have identified a new class which we have called pseudo-simple heteroclinic cycles. By contrast with simple heteroclinic cycles, a pseudo-simple one has at least one equilibrium with an unstable manifold which has dimension 2 due to a symmetry. Here, we analyse the dynamics of nearby trajectories and asymptotic stability of pseudo-simple heteroclinic cycles in R^4.

math.DS↗

arXiv2015

Simple heteroclinic cycles in R^4

In generic dynamical systems heteroclinic cycles are invariant sets of codimension at least one, but they can be structurally stable in systems which are equivariant under the action of a symmetry group, due to the existence of flow-invariant subspaces. For dynamical systems in R^n the minimal dimension for which such robust heteroclinic cycles can exist is n=3. In this case the list of admissible symmetry groups is short and well-known. The situation is different and more interesting when n=4. In this paper we list all finite groups Gamma such that an open set of smooth Gamma-equivariant dynamical systems in R^4 possess a very simple heteroclinic cycle (a structurally stable heteroclinic cycle satisfying certain additional constraints). This work extends the results which were obtained by Sottocornola in the case when all equilibria in the heteroclinic cycle belong to the same Gamma-orbit (in this case one speaks of homoclinic cycles).

nlin.CD↗

arXiv2012

Stability and bifurcations of heteroclinic cycles of type Z

Dynamical systems that are invariant under the action of a non-trivial symmetry group can possess structurally stable heteroclinic cycles. In this paper we study stability properties of a class of structurally stable heteroclinic cycles in R^n which we call heteroclinic cycles of type Z. It is well-known that a heteroclinic cycle that is not asymptotically stable can attract nevertheless a positive measure set from its neighbourhood. We say that an invariant set X is fragmentarily asymptotically stable, if for any delta>0 the measure of its local basin of attraction B_delta(X) is positive. A local basin of attraction B_delta(X) is the set of such points that trajectories starting there remain in the delta-neighbourhood of X for all t>0, and are attracted by X as t\to\infty. Necessary and sufficient conditions for fragmentary asymptotic stability are expressed in terms of eigenvalues and eigenvectors of transition matrices. If all transverse eigenvalues of linearisations near steady states involved in the cycle are negative, then fragmentary asymptotic stability implies asymptotic stability. In the latter case the condition for asymptotic stability is that the transition matrices have an eigenvalue larger than one in absolute value. Finally, we discuss bifurcations occurring when the conditions for asymptotic stability or for fragmentary asymptotic stability are broken.

nlin.CD↗

arXiv2012

Classification and stability of simple homoclinic cycles in R^5

The paper presents a complete study of simple homoclinic cycles in R^5. We find all symmetry groups Gamma such that a Gamma-equivariant dynamical system in R^5 can possess a simple homoclinic cycle. We introduce a classification of simple homoclinic cycles in R^n based on the action of the system symmetry group. For systems in R^5, we list all classes of simple homoclinic cycles. For each class, we derive necessary and sufficient conditions for asymptotic stability and fragmentary asymptotic stability in terms of eigenvalues of linearisation near the steady state involved in the cycle. For any action of the groups Gamma which can give rise to a simple homoclinic cycle, we list classes to which the respective homoclinic cycles belong, thus determining conditions for asymptotic stability of these cycles.

nlin.CD↗

arXiv2012

Optimal transport by omni-potential flow and cosmological reconstruction

One of the simplest models used in studying the dynamics of large-scale structure in cosmology, known as the Zeldovich approximation, is equivalent to the three-dimensional inviscid Burgers equation for potential flow. For smooth initial data and sufficiently short times it has the property that the mapping of the positions of fluid particles at any time $t_1$ to their positions at any time $t_2\ge t_1$ is the gradient of a convex potential, a property we call omni-potentiality. Are there other flows with this property, that are not straightforward generalizations of Zeldovich flows? This is answered in the affirmative in both two and three dimensions. How general are such flows? Using a WKB technique we show that in two dimensions, for sufficiently short times, there are omni-potential flows with arbitrary smooth initial velocity. Mappings with a convex potential are known to be associated with the quadratic-cost optimal transport problem. This has important implications for the problem of reconstructing the dynamical history of the Universe from the knowledge of the present mass distribution.

math.AP↗

arXiv2011

On local attraction properties and a stability index for heteroclinic connections

Some invariant sets may attract a nearby set of initial conditions but nonetheless repel a complementary nearby set of initial conditions. For a given invariant set $X\subset\R^n$ with a basin of attraction $N$, we define a stability index $σ(x)$ of a point $x\in X$ that characterizes the local extent of the basin. Let $B_ε$ denote a ball of radius $ε$ about $x$. If $σ(x)>0$, then the measure of $B_ε\setminus N$ relative the measure of the ball is $O(ε^{|σ(x)|})$, while if $σ(x)<0$, then the measure of $B_ε\cap N$ relative the measure of the ball is of the same order. We show that this index is constant along trajectories, and relate this orbit invariant to other notions of stability such as Milnor attraction, essential asymptotic stability and asymptotic stability relative to a positive measure set. We adapt the definition to local basins of attraction (i.e. where $N$ is defined as the set of initial conditions that are in the basin and whose trajectories remain local to $X$). This stability index is particularly useful for discussing the stability of robust heteroclinic cycles, where several authors have studied the appearance of cusps of instability near cycles that are Milnor attractors. We study simple (robust heteroclinic) cycles in $\R^4$ and show that the local stability indices (and hence local stability properties) can be calculated in terms of the eigenvalues of the linearization of the vector field at steady states on the cycle. In doing this, we extend previous results of Krupa and Melbourne (1995,2004) and give criteria for simple heteroclinic cycles in $\R^4$ to be Milnor attractors.

nlin.CD↗

arXiv2011

The onset of convection in a rotating layer of viscous fluid with an imposed magnetic field: dependence on the Prandtl numbers

We consider the onset of Boussinesq convection in a horizontal layer of electrically conducting incompressible fluid with rigid electrically insulating horizontal boundaries. The fluid is heated from below and rotates about a vertical axis; vertical magnetic field is imposed. The instability of the trivial steady state (fluid at rest) can be monotonic or oscillatory depending on parameters of the problem (the kinematic and magnetic Prandtl, Taylor and Chandrasekhar numbers). If the Taylor number is sufficiently large, convective rolls, emerging in the monotonic instability, experience the Küppers-Lortz instability. We study how the critical Rayleigh number, the type of instability of the trivial steady state and the critical Taylor number for the Küppers-Lortz instability depend on the Prandtl numbers. We consider Prandtl numbers not exceeding one, which is typical for the inner core of the Earth.

nlin.PS↗

arXiv2010

On stability of rolls near the onset of convection in a layer with stress-free boundaries

We consider a classical problem of linear stability of convective rolls in a plane layer with stress-free horizontal boundaries near the onset of convection. The problem has been studied by a number of authors, who have shown that rolls of wave number $k$ are unstable with respect to perturbations of different types, if some inequalities relating $k$ and the Rayleigh number $R$ are satisfied. The perturbations involve a large-scale mode. Certain asymptotic dependencies between wave numbers of the mode and overcriticality are always assumed in the available proofs of instability. We analyse the stability analytically following the approach of Podvigina (2008) without making a priori assumptions concerning asymptotic relations between small parameters characterising the problem. Instability of rolls to short-scale modes is also considered. Therefore, our analytical results on stability to space-periodic perturbations are exhaustive; they allow to identify the areas in the $(k,R)$ plane, where convective rolls are stable near the onset. The analytical results are compared with numerical solutions to the eigenvalue problem determining stability of rolls.

nlin.CD↗