On the stability of limit cycles for planar differential systems (original) (raw)
Some theorems on the existence, uniqueness, and nonexistence of limit cycles for quadratic systems
Journal of Differential Equations, 1987
Given a quadratic system (QS) with a focus or a center at the origin we write it in the form i= y+P,(x, y), j=-x+dy+ QI(x, y) where P, and Qz are homogeneous polynomials of degree 2. If we detine F(x, y) = (X-dy) P,(x, y) + yQz(x, y) and g(x, y)=xQ,(x, y)-yPz(.x, y) we give results of existence, nonexistence, and uniqueness of limit cycles if F(x, y) g(x, y) does not change of sign. Then, by using these results plus the properties on the evolution of the limit cycles of the semicomplete families of rotated vector fields we can study some particular families of QS, i.e., the QS with a unique finite singularity and the bounded QS with either one or two linite singularities.
On the number of limit cycles in piecewise planar quadratic differential systems
arXiv (Cornell University), 2023
We consider piecewise quadratic perturbations of centers of piecewise quadratic systems in two zones determined by a straight line through the origin. By means of expansions of the displacement map, we are able to find isolated zeros of it, without dealing with the unsurprising difficult integrals inherent in the usual averaging approach. We apply our technique to non-smooth perturbations of the four families of isochronous centers of the Loud family, S 1 , S 2 , S 3 , and S 4 , as well as to non-smooth perturbations of non-smooth centers given by putting different S i 's in each zone. To show the coverage of our approach, we apply its first order, which recovers the averaging method of the first order, in perturbations of the already mentioned centers considering all the straight lines through the origin. Then we apply its second order to perturbations of the above centers for a specific oblique straight line. Here in order to argue we introduce certain blow-ups in the perturbative parameters. As a consequence of our study, we obtain examples of piecewise quadratic systems with at least 12 limit cycles. By analyzing two previous works of the literature claiming much more limit cycles we found some mistakes in the calculations. Therefore, the best lower bound for the number of limit cycles of a piecewise quadratic system is up to now the 12 limit cycles found in the present paper.
Limit cycles for some families of smooth and non-smooth planar systems
Nonlinear Analysis
In this paper, we apply the averaging method via Brouwer degree in a class of planar systems given by a linear center perturbed by a sum of continuous homogeneous vector fields, to study lower bounds for their number of limit cycles. Our results can be applied to models where the smoothness is lost on the set Σ = {xy = 0}. We also apply them to present a variant of Hilbert 16th problem, where the goal is to bound the number of limit cycles in terms of the number of monomials of a family of polynomial vector fields, instead of doing this in terms of their degrees.
On the limit cycles of polynomial differential systems with homogeneous nonlinearities
Proceedings of the Edinburgh Mathematical Society, 2000
We consider three classes of polynomial differential equations of the form ẋ = y + establish Pn (x, y), ẏ = x + Qn (x, y), where establish Pn and Qn are homogeneous polynomials of degree n, having a non-Hamiltonian centre at the origin. By using a method different from the classical ones, we study the limit cycles that bifurcate from the periodic orbits of such centres when we perturb them inside the class of all polynomial differential systems of the above form. A more detailed study is made for the particular cases of degree n = 2 and n = 3.
Existence of limit cycles for real quadratic differential systems with an invariant cubic
Pacific Journal of Mathematics, 2006
This work is part of a wider study of the significance of the existence of invariant algebraic curves for planar polynomial differential systems. The class of real quadratic systems having a cubic invariant algebraic curve is examined. Using affine canonical forms for the members of this class we show that no system of this type has limit cycles except for two cases. For these cases, concrete examples are given with a limit cycle. We also include a simple and short proof on the nonexistence of quadratic systems with an algebraic limit cycle of third degree.
Bulletin des Sciences Mathématiques, 2009
Let P k (x 1 ,. .. , x d) and Q k (x 1 ,. .. , x d) be polynomials of degree n k for k = 1, 2,. .. , d. Consider the polynomial differential system in R d defined bẏ x 1 = −x 2 + εP 1 (x 1 ,. .. , x d) + ε 2 Q 1 (x 1 ,. .. , x d), x 2 = x 1 + εP 2 (x 1 ,. .. , x d) + ε 2 Q 2 (x 1 ,. .. , x d), x k = εP k (x 1 ,. .. , x d) + ε 2 Q k (x 1 ,. .. , x d), for k = 3,. .. , d. Suppose that n k = n 2 for k = 1, 2,. .. , d. Then, by applying the first order averaging method this system has at most (n − 1)n d−2 /2 limit cycles bifurcating from the periodic orbits of the same system with ε = 0; and by applying the second order averaging method it has at most (n − 1)(2n − 1) d−2 limit cycles bifurcating from the periodic orbits of the same system with ε = 0. We provide polynomial differential systems reaching these upper bounds. In fact our results are more general, they provide the number of limit cycles for arbitrary n k .
Transversal conics and the existence of limit cycles
Journal of Mathematical Analysis and Applications, 2015
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 Poincaré-Bendixson regions by using transversal conics. We present several examples of known systems in the literature showing different features about limit cycles: hyperbolicity, Hopf bifurcation, sky-blue bifurcation, rotated vector fields, . . . for which the obtained Poincaré-Bendixson region allows to locate the limit cycles. Our method gives bounds for the bifurcation values of parametrical families of planar vector fields and intervals of existence of limit cycles.
International Journal of Bifurcation and Chaos, 2022
First, we study the planar continuous piecewise differential systems separated by the straight line [Formula: see text] formed by a linear isochronous center in [Formula: see text] and an isochronous quadratic center in [Formula: see text]. We prove that these piecewise differential systems cannot have crossing periodic orbits, and consequently they do not have crossing limit cycles. Second, we study the crossing periodic orbits and limit cycles of the planar continuous piecewise differential systems separated by the straight line [Formula: see text] having in [Formula: see text] the general quadratic isochronous center [Formula: see text], [Formula: see text] after an affine transformation, and in [Formula: see text] an arbitrary quadratic isochronous center. For these kind of continuous piecewise differential systems the maximum number of crossing limit cycles is one, and there are examples having one crossing limit cycles. In short for these families of continuous piecewise diffe...
On a Liouville Integrable Planar Differential System with Non-Algebraic Limit Cycle
2021 International Conference on Recent Advances in Mathematics and Informatics (ICRAMI)
In this paper, we prove that a class of differential system of degree nine is Liouville integrable by transforming it into a Bernoulli differential equation and we determine exactly its first integral. This allows us to show that this class admits an explicit non-algebraic limit cycle enclosing the origin, here a non-elementary singular point. For singularities, at infinity, this class does not possess singular points.
Robustness and Stability of Limit Cycles in a Class of Planar Dynamical Systems
2013
Using a macroeconomic example, the paper proposes an algorithm to symbolically construct the topological normal form of Andronov-Hopf bifurcation. It also offers a program, using the Computer Algebra System `Maxima', to apply this algorithm. In case the limit cycle turns out to be unstable, the possibilities of the dynamics converging to another limit cycle is explored.
A Note on the Lyapunov and Period Constants
Qualitative Theory of Dynamical Systems, 2020
It is well known that the number of small amplitude limit cycles that can bifurcate from the origin of a weak focus or a non degenerated center for a family of planar polynomial vector fields is governed by the structure of the so called Lyapunov constants, that are polynomials in the parameters of the system. These constants are essentially the coefficients of the odd terms of the Taylor development at zero of the displacement map. Although many authors use that the coefficients of the even terms of this map belong to the ideal generated by the previous odd terms, we have not found a proof in the literature. In this paper we present a simple proof of this fact based on a general property of the composition of one-dimensional analytic reversing orientation diffeomorphisms with themselves. We also prove similar results for the period constants. These facts, together with some classical tools like the Weirstrass preparation Theorem, or the theory of extended Chebyshev systems, are used to revisit some classical results on cyclicity and criticality for polynomial families of planar differential equations.
A Characterization of theω-Limit Sets of Planar Continuous Dynamical Systems
Journal of Differential Equations, 1998
If the family of curves G=[1 i ] i and the set of points S are given, we find necessary and sufficient conditions for the existence of a C k map f from the plane into itself, 0 k (if k=0 we also assume that f is locally Lipschitz), such that the curves from G and the points from S are respectively nondegenerate trajectories and singular points of the system x$= f (x), and additionally S _ ( i 1 i ) is the |-limit set of some trajectory of the system. In such cases, we also provide a detailed geometric description of S _ ( i 1 i ) (after contraction of the connected components of S).
Dynamical Systems, 2015
The orbits of the reversible differential systemẋ = −y,ẏ = x,ż = 0, with x, y ∈ R and z ∈ R d , are periodic with the exception of the equilibrium points (0, 0, z). We compute the maximum number of limit cycles which bifurcate from the periodic orbits of the systemẋ = −y,ẏ = x,ż = 0, using the averaging theory of first order, when this system is perturbed, first inside the class of all polynomial differential systems of degree n, and second inside the class of all discontinuous piecewise polynomial differential systems of degree n with two pieces, one in y > 0 and the other in y < 0. In the first case this maximum number is n d (n − 1)/2, and in the second is n d+1 .
Limit cycles of a class of polynomial vector fields in the plane
Journal of Differential Equations, 1986
This work was inspired by the recent paper of D. E. Koditschek and K. S. Narendra [6] in which they study the existence and uniqueness of limit cycles for a class of quadratic vector fields on the plane whose flows have a unique stationary point. After a translation a vector field X in this class takes the form X(u) = Av +f(o) Bu, where A and B are 2 x 2 matrices andfis a homogeneous linear function. There are two hypotheses. The first hypothesis HI states that the origin is a spiral hyperbolic source, i.e., the eigenvalues of A are complex conjugates a +_ bi with a > 0 and b #O. To specify the second hypothesis let J denote the symplectic 2 x 2 matrix (y-A) and for a 2 x 2 matrix C let C* denote the transpose of C. Recall that the symmetric part of C is given by [Cl, = $(C + C*). The second hypothesis Hz states that [JB], and [B*JA], are sign definite and agree in sign. Expressed geometrically H2 requires the eigenvalues of B to be complex and the vector fields Av and Bv to be nowhere parallel. The theorem of Koditschek and Narendra is as follows: THEOREM A. If X satisfies H, and H, then X has a unique stationary point at the origin and a unique hyperbolic stable limit cycle surrounding the origin.
Limit Cycles for a Class of Polynomial Differential Systems Via Averaging Theory
Journal of Siberian Federal University. Mathematics & Physics, 2019
One of the main problems in the qualitative theory of real planar differential equations is to determinate the number of limit cycles for a given planar differential system. As we all know, this is a very difficult problem for a general polynomial system. Therefore, many mathematicians study some systems with special conditions. To obtain the number of limit cycles as many as possible for a planar differential system, we usually take in consideration of the bifurcation theory. In recent decades, many new results have been obtained (see [9, 10]). The number of medium amplitude limit cycles bifurcating from the linear center ẋ= y, ẏ = −x for the following three kind of generalized polynomial Liénard differential systems { ẋ = y,
Upper bounds for the number of limit cycles of some planar polynomial differential systems
Discrete and Continuous Dynamical Systems, 2010
We give an effective method for controlling the maximum number of limit cycles of some planar polynomial systems. It is based on a suitable choice of a Dulac function and the application of the well-known Bendixson-Dulac Criterion for multiple connected regions. The key point is a new approach to control the sign of the functions involved in the criterion. The method is applied to several examples.