arXiv subjects

Yuri A. Kuznetsov

Publications and source records attributed to Yuri A. Kuznetsov.

14 recordsLinked to original sources

arXiv2026

Center Manifolds and Normal Forms for Nonlinearly Periodically Forced DDEs

The aim of this paper is to provide an effective framework for analysing bifurcations of equilibria in nonlinearly periodically forced delay differential equations. First, we establish the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic equilibrium using the rigorous functional analytic framework of dual semigroups (sun-star calculus). Second, we construct a center manifold parametrization that allows us to describe the local dynamics on the center manifold near the equilibrium in terms of periodically forced normal forms. Third, we present a normalization method to derive explicit computational formulas for the critical normal form coefficients at a bifurcation of interest. In particular, we obtain such formulas for the periodically forced fold and nonresonant Hopf bifurcation. Several examples and indications from the literature confirm the validity and effectiveness of our approach.

math.DS↗

arXiv2023

Periodic Center Manifolds for Nonhyperbolic Limit Cycles in ODEs

In this paper, we deal with a classical object, namely, a nonhyperbolic limit cycle in a system of smooth autonomous ordinary differential equations. While the existence of a center manifold near such a cycle was assumed in several studies on cycle bifurcations based on periodic normal forms, no proofs were available in the literature until recently. The main goal of this paper is to give an elementary proof of the existence of a periodic smooth locally invariant center manifold near a nonhyperbolic cycle in finite-dimensional ordinary differential equations by using the Lyapunov-Perron method. In addition, we provide several explicit examples of analytic vector fields admitting (non)-unique, (non)-$C^{\infty}$-smooth and (non)-analytic periodic center manifolds.

math.DS↗

arXiv2023

Bifurcations of heteroclinic contours in two-parameter planar systems: Overview and explicit examples

Smooth planar vector fields containing two hyperbolic saddles may possess contours formed by heteroclinic connections between these saddles. We present an overview of the bifurcations of these contours based on papers by J.W. Reyn and A.V. Dukov. Additionally, two new explicit polynomial systems containing such contours are derived, which are studied using the bifurcation software MatCont and are shown to exhibit the theoretically predicted phenomena, including series of heteroclinic connections.

math.DS↗

arXiv2023

Bifurcations of Neural Fields on the Sphere

A common model to study pattern formation in large groups of neurons is the neural field. We investigate a neural field with excitatory and inhibitory neurons, like Wilson and Cowan (1972), with transmission delays and gap junctions. We build on the work of Visser et al. (2017) by investigating pattern formation in these models on the sphere. Specifically, we investigate how different amounts of gap-junctions, modelled by a diffusion term, influence the behaviour of the neural field. We look in detail at the periodic orbits which are generated by Hopf bifurcation in the presence of spherical symmetry. For this end, we derive general formulas to compute the normal form coefficients of these bifurcations up to third order and predict the stability of the resulting branches, and formulate a novel numerical method to solve delay equations on the sphere.

math.DS↗

arXiv2021

Interplay between normal forms and center manifold reduction for homoclinic predictors near Bogdanov-Takens bifurcation

This paper provides for the first time correct third-order homoclinic predictors in $n$-dimensional ODEs near a generic Bogdanov-Takens bifurcation point. To achieve this, higher-order time approximations to the nonlinear time transformation in the Lindstedt-Poincaré method are essential. Moreover, a correct transform between approximations to solutions in the normal form and approximations to solutions on the parameter-dependent center manifold is needed. A detailed comparison is done between applying different normal forms (smooth and orbital), different phase conditions, and different perturbation methods (regular and Lindstedt-Poincaré) to approximate the homoclinic solution near Bogdanov-Takens points. Examples demonstrating the correctness of the predictors are given. The new homoclinic predictors are implemented in the open-source MATLAB/GNU Octave continuation package MatCont.

math.NA↗

arXiv2021

Neural Field Models with Transmission Delays and Diffusion

A neural field models the large scale behaviour of large groups of neurons. We extend results of van Gils et al. [2013] and Dijkstra et al. [2015] by including a diffusion term into the neural field, which models direct, electrical connections. We extend known and prove new sun-star calculus results for delay equations to be able to include diffusion and explicitly characterise the essential spectrum. For a certain class of connectivity functions in the neural field model, we are able to compute its spectral properties and the first Lyapunov coefficient of a Hopf bifurcation. By examining a numerical example, we find that the addition of diffusion suppresses non-synchronised steady-states, while favouring synchronised oscillatory modes.

math.DS↗

arXiv2019

Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations

In this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful tool to study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models.

math.DS↗

arXiv2017

Homoclinic saddle to saddle-focus transitions in 4D systems

A saddle to saddle-focus homoclinic transition when the stable leading eigenspace is 3-dimensional (called the 3DL bifurcation) is analyzed. Here a pair of complex eigenvalues and a real eigenvalue exchange their position relative to the imaginary axis, giving rise to a 3-dimensional stable leading eigenspace. This transition is different from the standard Belyakov bifurcation, where a double real eigenvalue splits either into a pair of complex-conjugate eigenvalues or two distinct real eigenvalues. In the wild case, we obtain sets of codimension 1 and 2 bifurcation curves and points that asymptotically approach the 3DL bifurcation point and have a structure that differs from that the standard Belyakov case. We also give an example of this bifurcation in the wild case occuring in a perturbed Lorenz-Stenflo 4D ODE model.

math.DS↗

arXiv2012

On Local Bifurcations in Neural Field Models with Transmission Delays

Neural field models with transmission delay may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results.

math.DS↗

arXiv2012

Numerical periodic normalization for codim 2 bifurcations of limit cycles with center manifold of dimension higher than 3

Explicit computational formulas for coefficients of the periodic normal forms of the three most complex codim 2 bifurcations of limit cycles with dimension of the center manifold equal to 4 or to 5 in generic autonomous ODEs are derived. The resulting formulas are independent of the dimension of the phase space and involve solutions of certain boundary-value problems as well as multilinear functions from the Taylor expansion of the ODE right-hand side near the cycle. The formulas allow one to distinguish between the complicated bifurcation scenarios which can happen near these codim 2 bifurcations of limit cycles, where 3-tori and 4-tori can be present. We apply our techniques to the study of a known laser model, a novel model from population biology, and one for mechanical vibrations; these models exhibit Limit Point--Neimark-Sacker, Period-Doubling--Neimark-Sacker and double Neimark-Sacker bifurcations. Lyapunov exponents are computed to numerically confirm the results of the normal form analysis, in particular with respect to the existence of stable invariant tori of various dimensions and chaos.

math.DS↗

arXiv2011

Numerical Periodic Normalization for Codim 2 Bifurcations of Limit Cycles

Periodic normal forms for the codim 2 bifurcations of limit cycles up to a 3-dimensional center manifold in generic autonomous ODEs and computational formulas for their coefficients are derived. The formulas are independent of the dimension of the phase space and involve solutions of certain boundary-value problems on the interval [0, T ], where T is the period of the critical cycle, as well as multilinear functions from the Taylor expansion of the right-hand sides near the cycle. The formulas allow us to distinguish between various bifurcation scenarios near codim 2 bifurcations. Our formulation makes it possible to use robust numerical boundary-value algorithms based on orthogonal collocation, rather than shooting techniques, which greatly expands its applicability. The actual implementation is described in detail with numerical examples.

math.DS↗

arXiv2003

Monte Carlo simulations of infinitely dilute solutions of amphiphilic diblock star copolymers

Single-chain Monte Carlo simulations of amphiphilic diblock star copolymers were carried out in continuous space using implicit solvents. Two distinct architectures were studied: stars with the hydrophobic blocks attached to the core, and stars with the polar blocks attached to the core, with all arms being of equal length. The ratio of the lengths of the hydrophobic block to the length of the polar block was varied from 0 to 1. Stars with 3, 6, 9 or 12 arms, each of length 10, 15, 25, 50, 75 and 100 Kuhn segments were analysed. Four distinct types of conformations were observed for these systems. These, apart from studying the snapshots from the simulations, have been quantitatively characterised in terms of the mean-squared radii of gyration, mean-squared distances of monomers from the centre-of-mass, asphericity indices, static scattering form factors in the Kratky representation as well as the intra-chain monomer-monomer radial distribution functions.

cond-mat.soft↗

arXiv2001

Intra-chain correlation functions and shapes of homopolymers with different architectures in dilute solution

We present results of Monte Carlo study of the monomer-monomer correlation functions, static structure factor and asphericity characteristics of a single homopolymer in the coil and globular states for three distinct architectures of the chain: ring, open and star. To rationalise the results we introduce the dimensionless correlation functions rescaled via the corresponding mean-squared distances between monomers. For flexible chains with some architectures these functions exhibit a large degree of universality by falling onto a single or several distinct master curves. In the repulsive regime, where a stretched exponential times a power law form (de Cloizeaux scaling) can be applied, the corresponding exponents $δ$ and $θ$ have been obtained. The exponent $δ=1/ν$ is found to be universal for flexible strongly repulsive coils and in agreement with the theoretical prediction from improved higher-order Borel-resummed renormalisation group calculations. The short-distance exponents $θ_{\upsilon}$ of an open flexible chain are in a good agreement with the theoretical predictions in the strongly repulsive regime also. However, increasing the Kuhn length in relation to the monomer size leads to their fast cross-over towards the Gaussian behaviour. Likewise, a strong sensitivity of various exponents $θ_{ij}$ on the stiffness of the chain, or on the number of arms in star polymers, is observed. The correlation functions in the globular state are found to have a more complicated oscillating behaviour and their degree of universality has been reviewed. Average shapes of the polymers in terms of the asphericity characteristics, as well as the universal behaviour in the static structure factors, have been also investigated.

cond-mat.soft↗

arXiv2001

Comparison of the energy histograms and conformations between different models of copolymers on lattice

Based on large-scale Monte Carlo simulations on lattice the energy probability distribution functions are investigated for a large set of primary sequences in distinct models of copolymers at low temperatures below transitions to compacted states. Amphiphilic copolymers with hydrophobic and hydrophilic units are found to produce a single or double peak energy distributions corresponding to mono- or multi-meric micellar conformations. However, copolymers with short ranged random `charge' interactions in some cases are found to produce energy distribution functions with a well pronounced lowest energy state and a gap separating it from the rest of the spectrum. These, however have rather peculiar conformations corresponding to effectively immiscible domains comprised from monomers of likewise species. Relevance of these observations for coarse-grained models for protein folding is discussed.

cond-mat.soft↗