Related papers: Limit cycle enumeration in random vector fields
We prove that the number of limit cycles generated by a small non-conservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing…
A central limit theorem is established for a sum of random variables belonging to a sequence of random fields. The fields are assumed to have zero mean conditional on the past history and to satisfy certain conditional $\alpha$-mixing…
Motivated by open problems in applied and computational algebraic topology, we establish multivariate normal approximation theorems for three random vectors which arise organically in the study of random clique complexes. These are: (1) the…
We study the maximum number of crossing limit cycles in piecewise-smooth vector fields on the two-dimensional torus, where the discontinuity set is the boundary of the fundamental square. Under the assumption of a polynomial first integral…
Avila recently introduced a new method for the study of the discrete Schr\"odinger Operator with limit periodic potential. I adapt this method to the context of orthogonal polynomials in the unit circle with limit periodic Verblunsky…
In this paper, we study the bifurcation of limit cycles near a homoclinic cuspidal loop in a planar cubic near-Hamiltonian system by high-order Melnikov functions. We present a method combining the algebraic structure of Abelian integrals…
In 2021, Gupta and Suzumura proposed a novel algorithm for enumerating all bounded-length simple cycles in directed graphs. In this work, we present concrete examples demonstrating that the proposed algorithm fails to enumerate certain…
We propose a program for finding the cyclicity of period annuli of quadratic systems with centers of genus one. As a first step, we classify all such systems and determine the essential one-parameter quadratic perturbations which produce…
We study the asymptotic distributions of the number of crossings and the number of simple chords in a random chord diagram. Using size-bias coupling and Stein's method, we obtain bounds on the Kolmogorov distance between the distribution of…
The goal of this paper is to study the number of sliding limit cycles of a regularized piecewise linear $VI_3$ two-fold using the notion of slow divergence integral. We focus on limit cycles produced by canard cycles located in the…
The averaging theory has been extensively employed for studying periodic solutions of smooth and nonsmooth differential systems. Here, we extend the averaging theory for studying periodic solutions a class of regularly perturbed…
We study scaling limits of a family of planar random growth processes in which clusters grow by the successive aggregation of small particles. In these models, clusters are encoded as a composition of conformal maps and the location of each…
The restricted version of the Hilbert 16th problem for quadratic vector fields requires an upper estimate of the number of limit cycles through a vector parameter that characterizes the vector fields considered and the limit cycles to be…
Benjamini and Schramm introduced the notion of distributional limit of a sequence of graphs with uniformly bounded valence and studied such limits in the case that the involved graphs are planar. We investigate distributional limits of…
Recent work by Arpin, Chen, Lauter, Scheidler, Stange, and Tran counted the number of cycles of length $r$ in supersingular $\ell$-isogeny graphs. In this paper, we extend this work to count the number of cycles that occur along the spine.…
Consider the class of Ensemble Square Root filtering algorithms for the numerical approximation of the posterior distribution of nonlinear Markovian signals partially observed with linear observations corrupted with independent measurement…
For real planar polynomial differential systems there appeared a simple version of the $16$th Hilbert problem on algebraic limit cycles: {\it Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of…
We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime, the limiting free energy exists together…
An upper bound for the number of Hamiltonian cycles of symmetric diagraphs is established first in this paper, which is tighter than the famous Minc's bound and the Br$\acute{e}$gman's bound. A transformation on graphs is proposed, so that…
We study the number of limit cycles and the bifurcation diagram in the Poincar\'{e} sphere of a one-parameter family of planar differential equations of degree five $\dot {\bf x}=X_b({\bf x})$ which has been already considered in previous…