arXiv subjects

Oleg Makarenkov

Publications and source records attributed to Oleg Makarenkov.

At least 19 recordsLinked to original sources

arXiv2025

Bifurcation of Limit Cycles from a Fold-Fold Singularity in a Glacial Cycles Model

We study the occurrence of limit cycles from a point on the discontinuity hyperplane $L$ between two smooth vector fields where the two vector fields both point towards one another. Generically, such a point (called switched equilibrium in control) is asymptotically stable, but we consider the situation where the two vector fields become tangent to L at the switched equilibrium under varying parameter making a degenerate fold-fold singularity. We prove that moving the parameter past such a singular value leads to the occurrence of an attracting limit cycle, which is exactly the dynamical mechanism we then discover in a conceptual model of glacial cycles.

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arXiv2025

Bifurcation of spiking oscillations from a center in resonate-and-fire neurons

The theta rhythm is important for many cognitive functions including spatial processing, memory encoding, and memory recall. The information processing underlying these functions is thought to rely on consistent, phase-specific spiking throughout a theta oscillation that may fluctuate significantly in baseline (center of oscillations), frequency, or amplitude. Experimental evidence shows that spikes can occur at specific phases even when the baseline membrane potential varies significantly, such that the integrity of phase-locking persists across a large variability in spike threshold. The mechanism of this precise spike timing during the theta rhythm is not yet known and previous mathematical models have not reflected the large variability in threshold potential seen experimentally. Here we introduce a straightforward mathematical neural model capable of demonstrating a phase-locked spiking in the face of significant baseline membrane potential fluctuation during theta rhythm. This novel approach incorporates a degenerate grazing bifurcation of an asymptotically stable oscillation. This model suggests a potential mechanism for how biological neurons can consistently produce spikes near the peak of a variable membrane potential oscillation.

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arXiv2024

Optimization of a lattice spring model with elastoplastic conducting springs: A case study

We consider a simple lattice spring model in which every spring is elastoplastic and is capable to conduct current. The elasticity bounds of spring $i$ are taken as $[-c_i,c_i]$ and the resistance of spring $i$ is taken as $1/c_i$, which allows us to compute the resistance of the system. The model is further subjected to a gradual stretching and, due to plasticity, the response force increases until a certain terminal value. We demonstrate that the recently developed sweeping process theory can be used to optimize the interplay between the terminal response force and the resistance on a physical domain of parameters $c_i.$ The proposed methodology can be used by practitioners for the design of multi-functional materials as an alternative to topological optimization.

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arXiv2024

Crossing limit cycles in piecewise smooth Kolmogorov systems: an application to Palomba's model

In this paper, we study the number of isolated crossing periodic orbits, so-called crossing limit cycles, for a class of piecewise smooth Kolmogorov systems defined in two zones separated by a straight line. In particular, we study the number of crossing limit cycles of small amplitude. They are all nested and surround one equilibrium point or a sliding segment. We denote by $\mathcal M_{K}^{p}(n)$ the maximum number of crossing limit cycles bifurcating from the equilibrium point via a degenerate Hopf bifurcation for a piecewise smooth Kolmogorov systems of degree $n=m+1$. We make a progress towards the determination of the lower bounds $M_K^p(n)$ of crossing limit cycles bifurcating from the equilibrium point via a degenerate Hopf bifurcation for a piecewise smooth Kolmogorov system of degree $n$. Specifically, we shot that $M_{K}^{p}(2)\geq 1$, $M_{K}^{p}(3)\geq 12$, and $M_{K}^{p}(4)\geq 18$. In particular, we show at least one crossing limit cycle in Palomba's economics model, considering it from a piecewise smooth point of view. To our knowledge, these are the best quotes of limit cycles for piecewise smooth polynomial Kolmogorov systems in the literature.

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arXiv2023

Stable manifolds for periodically perturbed maps

We prove that if a certain entry in the map of the Hadamard-Perron theorem is $T$-periodic in one of the variables, then the stable manifold guaranteed by the Hadamard-Perron theorem is a graph of a $T$-periodic function. As an application, we extend the classical Levinson's result about the occurrence of an attracting closed invariant curve near a stable cycle of a system of autonomous equations under periodic perturbations to hybrid differential equations.

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arXiv2023

Sweeping process approach to stress analysis in elastoplastic Lattice Springs Models with applications to Hyperuniform Network Materials

Disordered network materials abound in both nature and synthetic situations while rigorous analysis of their nonlinear mechanical behaviors still is very challenging. The purpose of this paper is to connect the mathematical framework of sweeping process originally proposed by Moreau to a generic class of Lattice Spring Models with plasticity phenomenon. We explicitly construct a sweeping process and provide numerical schemes to find the evolution of stresses in a Lattice Spring Model with infinitesimal strains and perfect plasticity. In particular, we develop a highly efficient "leapfrog" computational framework that allow ones to rigorously track the progression of plastic events in the system based on the sweeping process theory. The utility of our framework is demonstrated by analyzing the elastoplastic stresses in a novel class of disordered network materials exhibiting the property of hyperuniformity, in which the infinite wave-length density fluctuations associated with the distribution of network nodes are completely suppressed. We find enhanced mechanical properties such as increasing stiffness, yield strength and tensile strength as the degree of hyperuniformity of the material system increases. Our results have implications for optimal network material design and our event-based framework can be readily generalized for nonlinear stress analysis of other heterogeneous material systems. We also include some insights to the model from the viewpoint of the rigidity theory.

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arXiv2023

Topological properties of elastoplastic lattice spring models that determine terminal distributions of plastic deformations

A recent result by Gudoshnikov et al [SIAM J. Control Optim. 2022] ensures finite-time convergence of the stress-vector of an elastoplastic lattice spring model under assumption that the vector $g'(t)$ of the applied displacement controlled-loading lies strictly inside the normal cone to the associated polyhedral set (that depends on mechanical parameters of the springs). Determination of the terminal distribution of stresses has been thereby linked to a problem of spotting a face on the boundary of the polyhedral set where the normal cone contains vector $g'(t)$. In this paper the above-mentioned problem of spotting an eligible face is converted into a search for an eligible set of springs that have a certain topological property with respect to the entire graph (of springs). Specifically, we prove that eligible springs are those that keep non-zero lengths after their nodes are collapsed with the nodes of the displacement-controlled loading after a finite number of eligible displacements of the nodes of the graph. The proposed result allows to judge about possible distribution of plastic deformations in elastoplastic lattice spring models directly from the topology of the associated graph of springs. A benchmark example is provided.

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arXiv2020

Finite-time stability of polyhedral sweeping processes with application to elastoplastic systems

We use the ideas of Adly-Attoych-Cabot [Adv. Mech. Math., 12, Springer, 2006] on finite-time stabilization of dry friction oscillators to establish a theorem on finite-time stabilization of differential inclusions with a moving polyhedral constraint (known as polyhedral sweeping processes) of the form $C+c(t).$ We then employ the ideas of Moreau [New variational techniques in mathematical physics, CIME, 1973] to apply our theorem to a system of elastoplastic springs with a displacement-controlled loading. We show that verifying the condition of the theorem ultimately leads to the following two problems: (i) identifying the active vertex ``A'' or the active face ``A'' of the polyhedron that the vector $c'(t)$ points at; (ii) computing the distance from $c'(t)$ to the normal cone to the polyhedron at ``A''. We provide a computational guide to implement steps (i)-(ii) in the case of an arbitrary elastoplastic system and apply the guide to a particular example. Due to the simplicity of the particular example, we can solve (i)-(ii) by the methods of linear algebra and minor combinatorics.

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arXiv2019

Existence and stability of a limit cycle in the model of a planar passive biped walking down a slope

We consider the simplest model of a passive biped walking down a slope given by the equations of switched coupled pendula (McGeer, 1990). Following the fundamental work by Garcia et al (1998), we view the slope of the ground as a small parameter $γ\ge 0$. When $γ=0$ the system can be solved in closed form and the existence of a family of limit cycles (i.e. potential walking cycles) can be established explicitly. As observed in Garcia et al (1998), the family of limit cycles disappears when $γ$ increases and only isolated asymptotically stable cycles (walking cycles) persist. However, no rigorous proofs of such a bifurcation (often referred to as Melnikov bifurcation) have ever been reported. The present paper fills in this gap in the field and offers the required proof.

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arXiv2019

Stabilization of the response of cyclically loaded lattice spring models with plasticity

This paper develops an analytic framework to design both stress-controlled and displacement-controlled T-periodic loadings which make the quasistatic evolution of a one-dimensional network of elastoplastic springs converging to a unique periodic regime. The solution of such an evolution problem is a function t-> (e(t),p(t)), where e_i(t) and p_i(t) are the elastic and plastic deformations of spring i, defined on [t0,\infty) by the initial condition (e(t0),p(t0)). After we rigorously convert the problem into a Moreau sweeping process with a moving polyhedron C(t) in a vector space E of dimension d, it becomes natural to expect (based on a result by Krejci) that the solution t->(e(t),p(t)) always converges to a T-periodic function. The achievement of this paper is in spotting a class of loadings where the Krejci's limit doesn't depend on the initial condition (e(t0),p(t0)) and so all the trajectories approach the same T-periodic regime. The proposed class of sweeping processes is the one for which the normal vectors of any d different facets of the moving polyhedron C(t) are linearly independent. We further link this geometric condition to mechanical properties of the given network of springs. We discover that the normal vectors of any d different facets of the moving polyhedron C(t) are linearly independent, if the number of displacement-controlled loadings is two less the number of nodes of the given network of springs and when the magnitude of the stress-controlled loading is sufficiently large (but admissible). The result can be viewed as an analogue of the high-gain control method for elastoplastic systems. In continuum theory of plasticity, the respective result is known as Frederick-Armstrong theorem.

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arXiv2019

Structurally stable families of periodic solutions in sweeping processes of networks of elastoplastic springs

Networks of elastoplastic springs (elastoplastic systems) have been linked to differential equations with polyhedral constraints in the pioneering paper by Moreau (1974). Periodic loading of an elastoplastic system, therefore, corresponds to a periodic motion of the polyhedral constraint. According to Krejci (1996), every solution of a sweeping process with a periodically moving constraint asymptotically converges to a periodic orbit. Understanding whether such an asymptotic periodic orbit is unique or there can be an entire family of asymptotic periodic orbits (that form a periodic attractor) has been an open problem since then. Since suitable small perturbation of a polyhedral constraint seems to be always capable to destroy a potential family of periodic orbits, it is expected that none of potential periodic attractor is structurally stable. In the present paper we give a simple example to prove that even though the periodic attractor (of non-stationary periodic solutions) can be destroyed by little perturbation of the moving constraint, the periodic attractor resists perturbations of the physical parameters of the mechanical model (i.e. the parameters of the network of elastoplastic springs).

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arXiv2018

Global asymptotic stability of nonconvex sweeping processes

Building upon the technique that we developed earlier for perturbed sweeping processes with convex moving constraints and monotone vector fields (Kamenskii et al, Nonlinear Anal. Hybrid Syst. 30, 2018), the present paper establishes global asymptotic stability of global and periodic solutions to perturbed sweeping processes with prox-regular moving constraint. Our conclusion can be formulated as follows: closer the constraint to a convex one, weaker monotonicity is required to keep the sweeping process globally asymptotically stable. We explain why the proposed technique is not capable to prove global asymptotic stability of a periodic regime in a crowd motion model (Cao-Mordukhovich, DCDS-B 22, 2017). We introduce and analyze a toy model which clarifies the extent of applicability of our result.

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arXiv2018

A continuation principle for periodic BV-continuous state-dependent sweeping processes

We consider a Caratheodory differential equation with a state-dependent convex constraint that changes BV-continuously in time (a perturbed BV-continuous state-dependent sweeping processes). By setting up an appropriate catching-up algorithm we prove solvability of the initial value problem. Then, for sweeping processes with $T$-periodic right-hand-sides, we prove the existence of at least one $T$-periodic solution. Finally, we further consider a $T$-periodic sweeping process which is close to an autonomous sweeping process with a constant constraint and prove the existence of a $T$-periodic solution specifically located near the boundary switched equilibrium of the autonomous sweeping process.

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arXiv2018

Bifurcations of finite-time stable limit cycles from focus boundary equilibria in impacting systems, Filippov systems and sweeping processes

We establish a theorem on bifurcation of limit cycles from a focus boundary equilibrium of an impacting system, which is universally applicable to prove bifurcation of limit cycles from focus boundary equilibria in other types of piecewise-smooth systems, such as Filippov systems and sweeping processes. Specifically, we assume that one of the subsystems of the piecewise-smooth system under consideration admits a focus equilibrium that lie on the switching manifold at the bifurcation value of the parameter. In each of the three cases, we derive a linearized system which is capable to conclude about the occurrence of a finite-time stable limit cycle from the above-mentioned focus equilibrium when the parameter crosses the bifurcation value. Examples illustrate how conditions of our theorems lead to closed-form formulas for the coefficients of the linearized system.

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arXiv2018

A linear state feedback switching rule for global stabilization of switched nonlinear systems about a nonequilibrium point

A switched equilibrium of a switched system of two subsystems is a such a point where the vector fields of the two subsystems point strictly towards one another. Using the concept of stable convex combination that was developed by Wicks-Peleties-DeCarlo (1998) for linear systems, Bolzern-Spinelli (2004) offered a design of a state feedback switching rule that is capable to stabilize an affine switched system to any switched equilibrium. The state feedback switching rule of Bolzern-Spinelli gives a nonlinear (quadratic) switching threshold passing through the switched equilibrium. In this paper we prove that the switching threshold (i.e. the associated switching rule) can be chosen linear, if each of the subsystems of the switched system under consideration are stable.

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arXiv2018

Bifurcation of limit cycles from a switched equilibrium in planar switched systems and its application to power converters

We consider a switched system of two subsystems that are activated as the trajectory enters the regions $\{(x,y):x>\bar x\}$ and $\{(x,y):x<-\bar x\}$ respectively, where $\bar x$ is a positive parameter. We prove that a regular asymptotically stable equilibrium of the associated Filippov equation of sliding motion (corresponding to $\bar x=0$) yields an orbitally stable limit cycle for all $\bar x>0$ sufficiently small. The research is motivated by an application to a dc-dc power converter, where $\bar x>0$ is used in place of $\bar x=0$ to avoid sliding motions.

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arXiv2017

Dwell time for local stability of switched systems with application to non-spiking neuron models

For switched systems that switch between distinct globally stable equilibria, we offer closed-form formulas that lock oscillations in the required neighborhood of the equilibria. Motivated by non-spiking neuron models, the main focus of the paper is on the case of planar switched affine systems, where we use properties of nested cylinders coming from quadratic Lyapunov functions. In particular, for the first time ever, we use the dwell-time concept in order to give an explicit condition for non-spiking of linear neuron models with periodically switching current. An extension to the general nonlinear case is also given.

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arXiv2017

Global stability of almost periodic solutions of monotone sweeping processes and their response to non-monotone perturbations

We develop a theory which allows making qualitative conclusions about the dynamics of both monotone and non-monotone Moreau sweeping processes. Specifically, we first prove that any sweeping processes with almost periodic monotone right-hand-sides admits a globally exponentially stable almost periodic solution. And then we describe the extent to which such a globally stable solution persists under non-monotone perturbations.

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