Birth of limit cycles for a class of continuous and discontinuous differential systems in (d+ 2)–dimension (original) (raw)
Related papers
2018
We study the periodic solutions bifurcating from periodic orbits of linear differential systems x′ = Mx, when they are perturbed inside a class of discontinuous piecewise polynomial differential systems with two zones. More precisely, we study the periodic solutions of the differential system x′ = Mx+ εF 1 (x) + ε F 2 (x), in R where ε is a small parameter, M is a (d+2)×(d+2) matrix having one pair of pure imaginary conjugate eigenvalues, m zeros eigenvalues, and d−m non–zero real eigenvalues. For solving this problem we need to extend the averaging theory for studying periodic solutions to a new class of non–autonomous d + 1-dimensional discontinuous piecewise smooth differential system.
Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems
Revista Matemática Iberoamericana, 2019
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 non-autonomous n-dimensional discontinuous piecewise smooth differential system. As a fundamental hypothesis, it is assumed that the unperturbed system has a manifold Z ⊂ R n of periodic solutions satisfying dim(Z) < n. Then, we apply this result to study limit cycles bifurcating from periodic solutions of linear differential systems, x = M x, when they are perturbed inside a class of discontinuous piecewise polynomial differential systems with two zones. More precisely, we study the periodic solutions of the following differential system x = M x + εF n 1 (x) + ε 2 F n 2 (x), in R d+2 where ε is a small parameter, M is a (d+2)×(d+2) matrix having one pair of pure imaginary conjugate eigenvalues, m zeros eigenvalues, and d − m non-zero real eigenvalues.
2018
We study limit cycles bifurcating from periodic solutions of linear differential systems, x'=Mx, when they are perturbed inside a class of discontinuous piecewise polynomial differential systems with two zones. More precisely, we study the periodic solutions of the differential system x'=Mx+ ε F_1^n(x)+ε^2F_2^n(x), in R^d+2 where ε is a small parameter, M is a (d+2)×(d+2) matrix having one pair of pure imaginary conjugate eigenvalues, m zeros eigenvalues, and d-m non-zero real eigenvalues. For solving this problem we need to extend the averaging theory for studying periodic solutions of a class of non-autonomous d+1-dimensional discontinuous piecewise smooth differential system.
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 .
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,
On the limit cycles of a class of piecewise linear differential systems in with two zones
Mathematics and Computers in Simulation, 2011
We study the bifurcation of limit cycles from the periodic orbits of a four-dimensional center in a class of piecewise linear differential systems with two zones. Our main result shows that three is an upper bound for the number of limit cycles that bifurcate from a center, up to first order expansion of the displacement function. Moreover, this upper bound is reached. The main technique used is the averaging method.
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 in uniform isochronous centers of discontinuous differential systems with four zones
Discrete & Continuous Dynamical Systems - B, 2017
We apply the averaging theory of first order for discontinuous differential systems to study the bifurcation of limit cycles from the periodic orbits of the uniform isochronous center of the differential systemsẋ = −y + x 2 ,ẏ = x + xy, andẋ = −y + x 2 y,ẏ = x + xy 2 , when they are perturbed inside the class of all discontinuous quadratic and cubic polynomials differential systems with four zones separately by the axes of coordinates, respectively. Using averaging theory of first order the maximum number of limit cycles that we can obtain is twice the maximum number of limit cycles obtained in a previous work for discontinuous quadratic differential systems perturbing the same uniform isochronous quadratic center at origin perturbed with two zones separately by a straight line, and 5 more limit cycles than those achieved in a prior result for discontinuous cubic differential systems with the same uniform isochronous cubic center at the origin perturbed with two zones separately by a straight line. Comparing our results with those obtained perturbing the mentioned centers by the continuous quadratic and cubic differential systems we obtain 8 and 9 more limit cycles respectively.