Many periodic solutions for a second order cubic periodic differential equation
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Many periodic solutions for a second order cubic periodic differential equation Adriana Buica˘ 1
· Armengol Gasull2
Received: 3 March 2020 / Accepted: 20 May 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract The aim of this work is to provide results that assure the existence of many isolated T -periodic solutions for T -periodic second-order differential equations of the form x = a(t)x +b(t)x 2 +c(t)x 3 . We use bifurcation methods, including Malkin functions and results of Fonda, Sabatini and Zanolin. In addition, we give a general result that assures the existence of a T -periodic perturbation of a non-isochronous center with an arbitrary number of T -periodic solutions. Keywords Second order differential equation · Cubic · Periodic · Bifurcation methods Mathematics Subject Classification 34C25 · 34A34 · 34C23 · 34C29
1 Introduction The aim of this work is to provide results that assure the existence of many T -periodic solutions for the class of T -periodic second-order differential equations
Communicated by Adrian Constantin. This work was supported by Ministerio de Ciencia, Innovación y Universidades of the Spanish Government by grants MTM2016-77278-P (MINECO/AEI/FEDER, UE) and 2017-SGR-1617 from AGAUR, Generalitat de Catalunya.
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Adriana Buic˘a [email protected] Armengol Gasull [email protected]
1
Departamentul de Matematic˘a, Universitatea Babe¸s–Bolyai, Str. Kog˘alniceanu 1, 400084 Cluj-Napoca, Romania
2
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
123
A. Buic˘a, A. Gasull
x = a(t)x + b(t)x 2 + c(t)x 3 .
(1)
We assume that a, b, c ∈ C(R) are T-periodic functions, and T > 0 is a fixed real number. Recall that, when in the above equation we replace x by x , we obtain the classical Abel equation. Hence Eq. (1) can be seen as a generalized Abel equation, and our goal is similar to the celebrated result of Lins–Neto [14] where the author proved that there is no upper bound for the number of isolated T -periodic solutions for the classical T -periodic Abel differential equations. Another motivation to look for such results came after reading the papers [11] and [7] where sufficient conditions are given in order to assure the existence of 1 and, respectively, 2 non-null T -periodic solutions. We mention that the authors of [7,11] were motivated to study equations of this form by Austin [1] who proposed a similar equation as a biomathematical model of an aneurysm, and by Cronin [5] who was the first to study the existence of periodic solutions of it. In the class of equations of the form (1) we distinguish the ones with constant coefficients a, b, c ∈ R. Here T > 0 is not related with the coefficients. Note that, whenever a = 0 and b2 − 4ac > 0, Eq. (1) has exactly 2 non-null constant (thus T -periodic for any T > 0) isolated solutions. The equation x = 0 has any constant function as T -periodic solution, while the solutions of the equation x = −x are all 2π -periodic. Another interesting example
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