Related papers: Limit cycles of linear vector fields on $(\mathbb{…
We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in ${\mathbb R}^{n}$ permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincare--Bendixson…
In this paper we complete the global qualitative analysis of a quartic ecological model. In particular, studying global bifurcations of singular points and limit cycles, we prove that the corresponding dynamical system has at most two limit…
By using the Picard-Fuchs equation and the property of Chebyshev space to the discontinuous differential system, we obtain an upper bound of the number of limit cycles for the nongeneric quadratic reversible system when it is perturbed…
In this paper, we complete the global qualitative analysis of a quartic family of planar vector fields corresponding to a rational Holling-type dynamical system which models the dynamics of the populations of predators and their prey in a…
In this paper, we consider the bifurcation of limit cycles for generic L-V system ($\dot{x}=y+x^2-y^2\pm\frac{4}{\sqrt{3}}xy,~\dot{y}=-x+2xy$) and B-T system ($\dot{x}=y,~\dot{y}=-x+x^2$) under perturbations of piecewise smooth polynomials…
This paper studies the number of limit cycles that may bifurcate from an equilibrium of an autonomous system of differential equations. The system in question is assumed to be of dimension $n$, have a zero-Hopf equilibrium at the origin,…
A sequence of bifurcations is studied in a one-dimensional pattern forming system subject to the variation of two experimental control parameters: a dimensionless electrical forcing number ${\cal R}$ and a shear Reynolds number ${\rm Re}$.…
This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincar\'e--Bendixson regions by using transversal conics. We present several…
New criteria are established for upper bounds on the number of limit cycles of periodic Abel differential equations having two periodic invariant curves, one of them bounded. The criteria are applied to obtain upper bounds of either zero or…
In this paper we produce a lower bound for the number of periodic orbits of certain Hamiltonian vector fields near Bott-nondegenerate symplectic critical submanifolds. This result is then related to the problem of finding closed orbits of…
Our main goal in this paper is to study the number of small-amplitude isolated periodic orbits, so-called limit cycles, surrounding only one equilibrium point a class of polynomial Kolmogorov systems. We denote by $\mathcal M_{K}(n)$ the…
We study bifurcations of vector fields on 2-manifolds with handles in generic one-parameter families unfolding vector fields with a separatrix loop of a hyperbolic saddle. These bifurcations can differ drastically from the analogous…
This paper studies the number of centers and limit cycles of the family of planar quartic polynomial vector fields that has the invariant algebraic curve $(4x^2-1)(4y^2-1)=0.$ The main interest for this type of vector fields comes from…
A linear recurrence sequence in a cyclotomic field produces a sequence of the generating fields of each term. We show that the later sequence is periodic after removing the first finite terms, and give a bound of its period. This can be…
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…
In this article, we investigate the existence of induced cycles in Levi graphs associated to line arrangements in $\mathbb{P}_{\mathbb{C}}^2$. We also look at the problem of finding the length of a longest induced cycle in Levi graphs…
A universal cycle is a compact listing of a class of combinatorial objects. In this paper, we prove the existence of universal cycles of classes of labeled graphs, including simple graphs, trees, graphs with m edges, graphs with loops,…
A square lattice distribution of coupled oscillators that have heteroclinic cycle attractors is studied. In this system, we find a novel type of patterns that is spatially disordered and periodic in time. These patterns are limit cycle…
In this paper, using our bifurcational geometric approach, we solve the problem on the maximum number and distribution of limit cycles in the Kukles system representing a planar polynomial dynamical system with arbitrary linear and cubic…
Our object of study is non smooth vector fields on $\R^2$. We apply the techniques of geometric singular perturbations in non smooth vector fields after regularization and a blow$-$up. In this way we are able to bring out some results that…