Limit cycles for rigid cubic systems

Gasull, Armengol; Prohens, R.; Torregrosa, Joan

doi:10.1016/j.jmaa.2004.07.030

Cited by 46 publications

(29 citation statements)

References 12 publications

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“…The following result due to Collins [6] in 1997, also obtained by Devlin, Lloyd and Pearson [8] in 1998, and by Gasull, Prohens and Torregrosa [11] in 2005 characterizes the uniform isochronous centers of cubic polynomial systems.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 88%

“…In the last decades several works about the bifurcation of limit cycles in planar differential systems having a uniform isochronous center have been published see for instance [1,9,11]. Aside from its importance in physical applications, isochronicity is closely related to the uniqueness and existence of solutions for some boundary value, perturbation, or bifurcation problems.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers

Llibre

Itikawa

2015

Journal of Computational and Applied Mathematics

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Abstract. 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 Poincaré disc for the uniform isochronous cubic centers.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 88%

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers

Llibre

Itikawa

2015

Journal of Computational and Applied Mathematics

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“…These planar systems whose angular speed is constant are usually called rigid or uniformly isochronous (see e.g. [11]). Of course, for these differential systems all their centers are isochronous, see e.g.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Darboux integrability and algebraic limit cycles for a class of polynomial differential systems

2014

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Abstract. This paper deals with the existence of Darboux first integrals for the planar polynomial differential systemsẋ = λx − y + P n+1 (x, y) + xF 2n (x, y),ẏ = x + λy + Q n+1 (x, y)+yF 2n (x, y), where P i (x, y), Q i (x, y) and F i (x, y) are homogeneous polynomials of degree i. Inside this class we identify some new Darboux integrable systems having either a focus or a center at the origin. For such Darboux integrable systems having degrees 5 and 9 we give the explicit expressions of their algebraic limit cycles. For the systems having degrees 3, 5, 7 and 9 we present necessary and sufficient conditions for being Darboux integrable.

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“…When P, Q are quadratic, this problem is equivalent to the determination of limit cycles of x ′ = c 1 x + c 2 (t)x 2 + c 3 (t)x 3 , where c 1 ∈ R and c 2 (t), c 3 (t) are trigonometric polynomials [17]. Some higher degree planar systems, in particular rigid systems, can also be reduced to generalized Abel equations [9] [12,13].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Existence of non-trivial limit cycles in Abel equations with symmetries

Alvarez

Bravo

Fernández

2013

Nonlinear Analysis: Theory, Methods & Applications

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Abstract. We study the periodic solutions of the generalized Abel equation x ′ = a1A1(t)x n 1 +a2A2(t)x n 2 +a3A3(t)x n 3 , where n1, n2, n3 > 1 are distinct integers, a1, a2, a3 ∈ R, and A1, A2, A3 are 2π-periodic analytic functions such that A1(t) sin t, A2(t) cos t, A3(t) sin t cos t are π-periodic positive even functions.When (n3 − n1)(n3 − n2) < 0 we prove that the equation has no nontrivial (different from zero) limit cycle for any value of the parameters a1, a2, a3.When (n3 − n1)(n3 − n2) > 0 we obtain under additional conditions the existence of non-trivial limit cycles. In particular, we obtain limit cycles not detected by Abelian integrals.

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Limit cycles for rigid cubic systems

Cited by 46 publications

References 12 publications

Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers

Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers

Darboux integrability and algebraic limit cycles for a class of polynomial differential systems

Existence of non-trivial limit cycles in Abel equations with symmetries

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