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Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov-Schmidt reduction. (English) Zbl 1381.34057

In this work, the authors provide sufficient conditions to assure the persistence of some zeros of functions having the form \[ g(z,\varepsilon)=g_0(z)+\sum_{i=1}^k \varepsilon^i g_i(z)+\mathcal{O}(\varepsilon^{k+1}) \] for \(|\varepsilon|\neq0\) sufficiently small using the Lyapunov-Schmidt resuction method. Here, \(g_i:\mathcal{D}\rightarrow\mathbb{R}^n\), for \(i=0,1,\ldots,k\), are smooth functions, \(\mathcal{D}\subset \mathbb{R}^n\) is an open bounded set. This result is given in Theorem A.
Using the above result the authors provide sufficient conditions to assure the existence of periodic solutions of the following \(T\)-periodic smooth differential system \[ x'=F_0(t,x)+\sum_{i=1}^k \varepsilon^i F_i(t,x)+\mathcal{O}(\varepsilon^{k+1}), \quad (t,z)\in\mathbb{S}^1\times\mathcal{D}, \] using the Browder degree theory. It is assumed that the unperturbed differential system has a sub-manifold of periodic solutions \(\mathcal{Z}\), with \(\dim(\mathcal{Z})\leq n\). See Theorem B.
Finally, the authors also study the case when the bifurcation functions have a continuum of zeros. In this situation they also provide some results about the stability of the limit cycles.

MSC:

34C25 Periodic solutions to ordinary differential equations
34C29 Averaging method for ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34C23 Bifurcation theory for ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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References:

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