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A new result on averaging theory for a class of discontinuous planar differential systems with applications
https://ddd.uab.cat/record/221382
We develop the averaging theory at any order for computing the periodic solutions of periodic discontinuous piecewise differential system of the form dr/dθ= r'={F+(θ, r, ϵ) if 0≤ θ ≤ α, F-(θ, r, ϵ) if α ≤ θ ≤ 2π, where F±(θ, r, ϵ) = Σk i=1 ϵiF± i (θ, r) + ϵk+1R ± (θ, r, ϵ) with θ ϵ S and r ϵ D, where D is an open interval of ℝ+, and ϵ is a small real parameter. Applying this theory, we provide lower bounds for the maximum number of limit cycles that bifurcate from the origin of quartic polynomial differential systems of the form x = -y+xp(x, y), y = x+yp(x, y), with p(x, y) a polynomial of degree 3 without constant term, when they are perturbed, either inside the class of all continuous quartic polynomial differential systems, or inside the class of all discontinuous piecewise quartic polynomial differential systems with two zones separated by the straight line y = 0. Itikawa, JacksonThu, 16 Apr 2020 05:59:19 GMThttps://ddd.uab.cat/record/2213822017Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones
https://ddd.uab.cat/record/221378
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 systems ẋ = -y+x, y = x + xy, and ẋ = -y + xy, y = x + xy, 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. Itikawa, JacksonThu, 16 Apr 2020 05:59:19 GMThttps://ddd.uab.cat/record/2213782017Centers and uniform isochronous centers of planar polynomial differential systems
https://ddd.uab.cat/record/199330
For planar polynomial vector fields of the form \[ (-y X(x,y)) x (x Y(x,y)) y, \] where X and Y start at least with terms of second order in the variables x and y, we determine necessary and sufficient conditions under which the origin is a center or a uniform isochronous centers. Llibre, JaumeMon, 12 Nov 2018 12:11:49 GMThttps://ddd.uab.cat/record/1993302018An inverse approach to the center-focus problem for polynomial differential system with homogenous nonlinearities
https://ddd.uab.cat/record/182507
We consider polynomial vector fields of the form \[ \X=(-y X_m) x (x Y_m) y, \] where X_m=X_m(x,y) and Y_m=Y_m(x,y) are homogenous polynomials of degree m. It is well--known that \X has a center at the origin if and only if \X has an analytic first integral of the form \[ H=12(x^2 y^2) _j=3^ H_j, \] where H_j=H_j(x,y) is a homogenous polynomial of degree j. The classical center-focus problem already studied by H. Poincar\'e consists in distinguishing when the origin of \X is either a center or a focus. In this paper we study the inverse center-focus problem. In particular for a given analytic function H defined in a neighborhood of the origin we want to determine the homogenous polynomials X_m and Y_m in such a way that H is a first integral of \X and consequently the origin of \X will be a center. Moreover, we study the case when \[H=12(x^2 y^2)(1 _j=1^ \Upsilon_j),\] where \Upsilon_j is a convenient homogenous polynomial of degree j for j 1. The solution of the inverse center problem for polynomial differential systems with homogenous nonlinearities, provides a new mechanism to study the center problem, which is equivalent to Liapunov's Theorem and Reeb's criterion. Llibre, JaumeTue, 28 Nov 2017 07:46:28 GMThttps://ddd.uab.cat/record/1825072017Uniform isochronous cubic and quartic centers: Revisited
https://ddd.uab.cat/record/169500
In this paper we completed the classification of the phase portraits in the Poincaré disc of uniform isochronous cubic and quartic centers previously studied by several authors. There are three and fourteen different topological phase portraits for the uniform isochronous cubic and quartic centers respectively. Artés, Joan CarlesMon, 23 Jan 2017 15:21:48 GMThttps://ddd.uab.cat/record/1695002017Global phase portraits of uniform isochronous centers with quartic homogeneous polynomial nonlinearities
https://ddd.uab.cat/record/169461
We classify the global phase portraits in the Poincar\'e disc of the differential systems =-y xf(x,y), =x yf(x,y), where f(x,y) is a homogeneous polynomial of degree 3. These systems have a uniform isochronous center at the origin. This paper together with the results presented in IL2 completes the classification of the global phase portraits in the Poincar\'e disc of all quartic polynomial differential systems with a uniform isochronous center at the origin. Itikawa, JacksonMon, 23 Jan 2017 15:21:46 GMThttps://ddd.uab.cat/record/1694612016Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers
https://ddd.uab.cat/record/145326
Let p be a uniform isochronous cubic polynomial center. We study the maximum number of small or medium limit cycles that bifurcate from p or from the periodic solutions surrounding p respectively, when they are perturbed, either inside the class of all continuous cubic polynomial differential systems, or inside the class of all discontinuous differential systems formed by two cubic differential systems separated by the straight line y = 0. In the case of continuous perturbations using the averaging theory of order 6 we show that the maximum number of small limit cycles that can appear in a Hopf bifurcation at p is 3, and this number can be reached. For a subfamily of these systems using the averaging theory of first order we prove that at most 3 medium limit cycles can bifurcate from the periodic solutions surrounding p, and this number can be reached. In the case of discontinuous perturbations using the averaging theory of order 6 we prove that the maximum number of small limit cycles that can appear in a Hopf bifurcation at p is 5, and this number can be reached. For a subfamily of these systems using the averaging method of first order we show that the maximum number of medium limit cycles that can bifurcate from the periodic solutions surrounding p is 7, and this number can be reached. We also provide all the first integrals and the phase portraits in the Poincar'e disc for the uniform isochronous cubic centers. Llibre, JaumeTue, 12 Jan 2016 16:38:09 GMThttps://ddd.uab.cat/record/1453262015Phase portraits of uniform isochronous quartic centers
https://ddd.uab.cat/record/145315
In this paper we classify the global phase portraits in the Poincaré disc of all quartic polynomial differential systems with a uniform isochronous center at the origin such that their nonlinear part is not homogeneous. Itikawa, JacksonTue, 12 Jan 2016 16:38:08 GMThttps://ddd.uab.cat/record/1453152015Limit cycles bifurcanting from the period annulus of a uniform isochronous center in a quartic polynomial differential system
https://ddd.uab.cat/record/145311
We study the number of limit cycles that bifurcate from the periodic solutions surrounding a uniform isochronous center located at the origin of the quartic polynomial differential system =-y xy(x^2 y^2), =x y^2(x^2 y^2), when it is perturbed inside the class of all quartic polynomial differential systems. Using the averaging theory of first order we show that at least 8 limit cycles can bifurcate from the period annulus of the considered center. Recently this problem was studied in Electron. J. Differ. Equ. 95 (2014), 1--14 where the authors only found 3 limit cycles. Itikawa, JacksonTue, 12 Jan 2016 16:38:08 GMThttps://ddd.uab.cat/record/1453112015