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## FIRST-ORDER PERTURBATION FOR MULTI-PARAMETER CENTER FAMILIES

### Regilene Oliveira

#### ICMC-USP, Campus São Carlos, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak22d7e6460d67666e181e625de1ca7bc7').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy22d7e6460d67666e181e625de1ca7bc7 = 'r&#101;g&#105;l&#101;n&#101;' + '&#64;'; addy22d7e6460d67666e181e625de1ca7bc7 = addy22d7e6460d67666e181e625de1ca7bc7 + '&#105;cmc' + '&#46;' + '&#117;sp' + '&#46;' + 'br'; var addy_text22d7e6460d67666e181e625de1ca7bc7 = 'r&#101;g&#105;l&#101;n&#101;' + '&#64;' + '&#105;cmc' + '&#46;' + '&#117;sp' + '&#46;' + 'br';document.getElementById('cloak22d7e6460d67666e181e625de1ca7bc7').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy22d7e6460d67666e181e625de1ca7bc7 + '\'>'+addy_text22d7e6460d67666e181e625de1ca7bc7+'<\/a>';

In the weak 16th Hilbert problem, the Poincaré-Pontryagin-Melnikov function, $M_1(h)$, is used for obtaining isolated periodic orbits bifurcating from centers up to a first-order analysis. This problem becomes more difficult when a family of centers is considered. In this work we provide a compact expression for the first-order Taylor series of the function $M_1(h,a)$ with respect to $a$, being a the multi-parameter in the unperturbed center family. More concretely, when the center family has an explicit first integral or inverse integrating factor depending on $a$. We use this new bifurcation mechanism to increase the number of limit cycles appearing up to a first-order analysis without the difficulties that higher-order studies present. We show its effectiveness by applying it to some classical examples.

Joint work with Jackson Itikawa (UNIR, Brazil) and Joan Torregrosa (UAB, Spain).