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Crossing limit cycles of planar piecewise linear Hamiltonian systems without equilibrium points
https://ddd.uab.cat/record/228103
In this paper, we study the existence of limit cycles of planar piecewise linear Hamiltonian systems without equilibrium points. Firstly, we prove that if these systems are separated by a parabola, they can have at most two crossing limit cycles, and if they are separated by a hyperbola or an ellipse, they can have at most three crossing limit cycles. Additionally, we prove that these upper bounds are reached. Secondly, we show that there is an example of two crossing limit cycles when these systems have four zones separated by three straight lines. Benterki, RebihaWed, 15 Jul 2020 12:00:34 GMThttps://ddd.uab.cat/record/2281032020Periodic solutions of a class of Duffing differential equations
https://ddd.uab.cat/record/222635
In this work we study the existence of new periodic solutions for the well knwon class of Duffing differential equation of the form x" + cx' + a(t)x + b(t)x3 = h(t), where c is a real parameter, a(t), b(t) and h(t) are continuous T-periodic functions. Our results are proved using three different results on the averaging theory of first order. Benterki, RebihaFri, 29 May 2020 19:06:42 GMThttps://ddd.uab.cat/record/2226352019Periodic solutions of the Duffing differential equation revisited via the averaging theory
https://ddd.uab.cat/record/222630
We use three different results of the averaging theory of first order for studying the existence of new periodic solutions in the two Duffing differential equations ¨ y + asiny = bsint and ¨ y + ay−cy3 = bsint, where a, b and c are real parameters. Benterki, RebihaFri, 29 May 2020 16:07:38 GMThttps://ddd.uab.cat/record/2226302019Phase portraits of quadratic polynomial differential systems having as solution some classical planar algebraic curves of degree 4
https://ddd.uab.cat/record/222620
We classify the phase portraits of quadratic polynomial differential systems having some relevant classic quartic algebraic curves as invariant algebraic curves, i. e. these curves are formed by solution curves of a quadratic polynomial differential system. We show the existence of 25 different global phase portraits in the Poincaré disc for such quadratic polynomial differential systems realizing exactly 14 different invariant algebraic curves of degree 4. Benterki, RebihaThu, 28 May 2020 18:15:35 GMThttps://ddd.uab.cat/record/2226202019The centers and their cyclicity for a class of polynomial differential systems of degree 7
https://ddd.uab.cat/record/221329
We classify the global phase portraits in the Poincaré disc of the generalized Kukles systems ẋ=−y,ẏ=x+axy6+bx3y4+cx5y2+dx7,which are symmetric with respect to both axes of coordinates. Moreover using the averaging theory up to sixth order, we study the cyclicity of the center located at the origin of coordinates, i. e. how many limit cycles can bifurcate from the origin of coordinates of the previous differential system when we perturb it inside the class of all polynomial differential systems of degree 7. Benterki, RebihaWed, 15 Apr 2020 15:23:07 GMThttps://ddd.uab.cat/record/2213292020Centers and limit cycles of polynomial differential systems of degree 4 via averaging theory
https://ddd.uab.cat/record/169499
In this paper we classify the phase portraits in the Poincar\'e disc of the centers of the generalized class of Kukles systems \[ =-y,=x ax^3y bxy^3, \] symmetric with respect to the y-axis, and we study, using the averaging theory up to sixth order, the limit cycles which bifurcate from the periodic solutions of these centers when we perturb them inside the class of all polynomial differential systems of degree 4. Benterki, RebihaMon, 23 Jan 2017 15:21:48 GMThttps://ddd.uab.cat/record/1694992017Limit cycles of polynomial differential equations with quintic homogenous nonlinearities
https://ddd.uab.cat/record/150593
In this paper we mainly study the number of limit cycles which can bifurcate from the periodic orbits of the two centers x˙ = −y, y˙ = x; x˙ = −y(1 − (x2 + y2)2), y˙ = x(1 − (x2 + y2)2); when they are perturbed inside the class of all polynomial differential systems with quintic homogenous nonlinearities. We do this study using the averaging theory of first, second and third order. Benterki, RebihaFri, 06 May 2016 08:54:14 GMThttps://ddd.uab.cat/record/1505932013Polynomial differential systems with explicit non-algebraic limit cycles
https://ddd.uab.cat/record/150529
Up to know all the examples of polynomial differential systems for which non-algebraic limit cycles are known explicitly have degree ≥ 5. Here we show that already there are polynomial differential systems of degree ≥ 3 exhibiting explicit non-algebraic limit cycles. It is well known that polynomial differential systems of degree 1 (i. e. linear differential systems) has no limit cycles. It remains the open question to determine if the polynomial differential systems of degree 2 can exhibit explicit non-algebraic limit cycles. Benterki, RebihaFri, 06 May 2016 08:49:22 GMThttps://ddd.uab.cat/record/1505292012