2,039 research outputs found

### Shilnikov problem in Filippov dynamical systems

In this paper we introduce the concept of sliding Shilnikov orbits for $3$D
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

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

In the research literature, one can find distinct notions for higher order
averaged functions of regularly perturbed non-autonomous $T$- periodic
differential equations of the kind $x'=\varepsilon F(t,x,\varepsilon)$. By one
hand, the classical (stroboscopic) averaging method provides asymptotic
estimates for its solutions in terms of some uniquely defined functions
$\mathbf{g}_i$'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 $\mathbf{f}_i$'s which controls in some sense the
existence of isolated $T$-periodic solutions of the differential equation
above. In the research literature, the bifurcation functions $\mathbf{f}_i$'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 $\mathbf{f}_1=T \mathbf{g}_1,$ a general
relationship between $\mathbf{g}_i$ and $\mathbf{f}_i$ is not known so far for
$i\geq 2.$ 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

In this work we first 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. Here $g_i:\mathcal{D}\rightarrow\mathbb{R}^n$, for $i=0,1,\ldots,k$, are
smooth functions being $\mathcal{D}\subset \mathbb{R}^n$ an open bounded set.
Then we use this result to compute the bifurcation functions which controls the
periodic solutions of the following $T$-periodic smooth differential system $x'=F_0(t,x)+\sum_{i=1}^k \varepsilon^i F_i(t,x)+\mathcal{O}(\varepsilon^{k+1}),
\quad (t,z)\in\mathbb{S}^1\times\mathcal{D}.$ It is assumed that the
unperturbed differential system has a sub-manifold of periodic solutions
$\mathcal{Z}$, $\textrm{dim}(\mathcal{Z})\leq n$. 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

Consider a differential system of the form $x'=F_0(t,x)+\sum_{i=1}^k
\varepsilon^i F_i(t,x)+\varepsilon^{k+1} R(t,x,\varepsilon),$ where
$F_i:\mathbb{S}^1 \times D \to \mathbb{R}^m$ and $R:\mathbb{S}^1 \times D
\times (-\varepsilon_0,\varepsilon_0) \to \mathbb{R}^m$ are piecewise $C^{k+1}$
functions and $T$-periodic in the variable $t$. Assuming that the unperturbed
system $x'=F_0(t,x)$ has a $d$-dimensional submanifold of periodic solutions
with $d<m$, we use the Lyapunov-Schmidt reduction and the averaging theory to
study the existence of isolated $T$-periodic solutions of the above
differential system

### On extended chebyshev systems with positive accuracy

AgraÃ¯ments: 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

In the present study we consider planar piecewise linear vector fields with
two zones separated by the straight line $x=0$. 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 $y<0$ and a virtual one for
$y>0$, 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

In this paper, we are concerned about smoothing of a class of
$\Sigma$-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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