2,039 research outputs found

    Shilnikov problem in Filippov dynamical systems

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    In this paper we introduce the concept of sliding Shilnikov orbits for 33D Filippov systems. In short, such an orbit is a piecewise smooth closed curve, composed by Filippov trajectories, which slides on the switching surface and connects a Filippov equilibrium to itself, namely a pseudo saddle-focus. A version of the Shilnikov's Theorem is provided for such systems. Particularly, we show that sliding Shilnikov orbits occur in generic one-parameter families of Filippov systems, and that arbitrarily close to a sliding Shilnikov orbit there exist countably infinitely many sliding periodic orbits. Here, no additional Shilnikov-like assumption is needed in order to get this last result. In addition, we show the existence of sliding Shilnikov orbits in discontinuous piecewise linear differential systems. As far as we know, the examples of Fillippov systems provided in this paper are the first exhibiting such a sliding phenomenon

    On nonsmooth perturbations of nondegenerate planar centers

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    We provide sufficient conditions for the existence of limit cycles of non-smooth perturbed planar centers when the discontinuity set is a union of regular curves. We introduce a mechanism which allows us to deal with such systems. The main tool used in this paper is the averaging method. Some applications are explained with careful details

    Higher order stroboscopic averaged functions: a general relationship with Melnikov functions

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    In the research literature, one can find distinct notions for higher order averaged functions of regularly perturbed non-autonomous TT- periodic differential equations of the kind x=εF(t,x,ε)x'=\varepsilon F(t,x,\varepsilon). By one hand, the classical (stroboscopic) averaging method provides asymptotic estimates for its solutions in terms of some uniquely defined functions gi\mathbf{g}_i's, called averaged functions, which are obtained through near-identity stroboscopic transformations and by solving homological equations. On the other hand, a Melnikov procedure is employed to obtain bifurcation functions fi\mathbf{f}_i's which controls in some sense the existence of isolated TT-periodic solutions of the differential equation above. In the research literature, the bifurcation functions fi\mathbf{f}_i's are sometimes likewise called averaged functions, nevertheless, they also receive the name of Poincar\'{e}-Pontryagin-Melnikov functions or just Melnikov functions. While it is known that f1=Tg1,\mathbf{f}_1=T \mathbf{g}_1, a general relationship between gi\mathbf{g}_i and fi\mathbf{f}_i is not known so far for i2.i\geq 2. Here, such a general relationship between these two distinct notions of averaged functions is provided, which allows the computation of the stroboscopic averaged functions of any order avoiding the necessity of dealing with near-identity transformations and homological equations. In addition, an Appendix is provided with implemented Mathematica algorithms for computing both higher order averaging functions.Comment: To appear in Electronic Journal of Qualitative Theory of Differential Equations, 202

    Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov-Schmidt reduction

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    In this work we first provide sufficient conditions to assure the persistence of some zeros of functions having the form g(z,ε)=g0(z)+i=1kεigi(z)+O(εk+1),g(z,\varepsilon)=g_0(z)+\sum_{i=1}^k \varepsilon^i g_i(z)+\mathcal{O}(\varepsilon^{k+1}), for ε0|\varepsilon|\neq0 sufficiently small. Here gi:DRng_i:\mathcal{D}\rightarrow\mathbb{R}^n, for i=0,1,,ki=0,1,\ldots,k, are smooth functions being DRn\mathcal{D}\subset \mathbb{R}^n an open bounded set. Then we use this result to compute the bifurcation functions which controls the periodic solutions of the following TT-periodic smooth differential system x=F0(t,x)+i=1kεiFi(t,x)+O(εk+1),(t,z)S1×D. 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}. It is assumed that the unperturbed differential system has a sub-manifold of periodic solutions Z\mathcal{Z}, dim(Z)n\textrm{dim}(\mathcal{Z})\leq n. We also study the case when the bifurcation functions have a continuum of zeros. Finally we provide the explicit expressions of the bifurcation functions up to order 5

    Bifurcations from families of periodic solutions in piecewise differential systems

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    Consider a differential system of the form x=F0(t,x)+i=1kεiFi(t,x)+εk+1R(t,x,ε), x'=F_0(t,x)+\sum_{i=1}^k \varepsilon^i F_i(t,x)+\varepsilon^{k+1} R(t,x,\varepsilon), where Fi:S1×DRmF_i:\mathbb{S}^1 \times D \to \mathbb{R}^m and R:S1×D×(ε0,ε0)RmR:\mathbb{S}^1 \times D \times (-\varepsilon_0,\varepsilon_0) \to \mathbb{R}^m are piecewise Ck+1C^{k+1} functions and TT-periodic in the variable tt. Assuming that the unperturbed system x=F0(t,x)x'=F_0(t,x) has a dd-dimensional submanifold of periodic solutions with d<md<m, we use the Lyapunov-Schmidt reduction and the averaging theory to study the existence of isolated TT-periodic solutions of the above differential system

    On extended chebyshev systems with positive accuracy

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    Agraïments: The first author is supported by a FAPESP-BRAZIL grant 2013/16492-0. The second author is supported by UNAB13-4E-1604 grant.A classical necessary condition for an ordered set of n+1 functions F to be an ECT-system in a closed interval is that all the Wronskians do not vanish. With this condition all the elements of Span(F) have at most n zeros taking into account the multiplicity. Here the problem of bounding the number of zeros of Span(F) is considered as well as the effectiveness of the upper bound when some Wronskians vanish. For this case we also study the possible configurations of zeros that can be realized by elements of Span(F). An application to count the number of isolated periodic orbits for a family of nonsmooth systems is performed

    Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields

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    In the present study we consider planar piecewise linear vector fields with two zones separated by the straight line x=0x=0. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has center, a real one for y<0y<0 and a virtual one for y>0y>0, and such that the real center is a global center. Then, working with a first order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one adittional crossing limit cycle can appear. Furthermore, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov method, the Extended Chebyshev systems with positive accuracy, and the Bendixson transformation.Comment: 24 pages, 7 figure

    Smoothing of homoclinic-like connections to regular tangential singularities in Filippov systems

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    In this paper, we are concerned about smoothing of a class of Σ\Sigma-polycycles of Filippov systems, namely homoclinic-like connections to regular-tangential singularities. Conditions are stablished in order to guarantee the existence of limit cycles bifurcating from such connections.Comment: arXiv admin note: text overlap with arXiv:2003.0954
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