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### Open Access Journals

DCDS-S

We follow a functional analytic approach to study the problem of chaotic behaviour in time-perturbed impact systems whose unperturbed part has a piecewise continuous impact homoclinic solution that transversally enters the discontinuity manifold. We show that if a certain Melnikov function has a simple zero at some point, then the system has impact solutions that behave chaotically. Applications of this result to quasi periodic systems are also given.

DCDS

We consider the singularly perturbed system
$\dot\xi = f_0(\xi) + \varepsilon f_1(\xi,\eta,\varepsilon)$,
$\dot\eta = \varepsilon
g(\xi,\eta,\varepsilon )$ where $\xi\in\Omega\subset\mathbb R^n$, $\eta\in\mathbb R$ and
$\varepsilon\in\mathbb R$ is a small real parameter. We assume that $\dot\xi =
f_{0}(\xi)$ has a non degenerate heteroclinic solution $\g(t)$ and
that the Melnikov function $\int_{-\infty}^{+\infty} \psi^{*}(t)
f_{1}(\g(t),\alpha,0)\dt$ has a double zero at some point $\alpha_{0}$.
Using a functional analytic approach we show that if a suitable
second order Melnikov function is not zero, the above system has, in
a neighborhood of $\{\gamma(t)\}\times\mathbb R$, two heteroclinic orbits for
$\varepsilon$ on one side of $\varepsilon=0$ and none for $\varepsilon$ on the other side.
We also study the transversality of the intersection of the
center-stable and the center-unstable manifolds along these orbits.

CPAA

In this note we study Sil'nikov saddle-focus homoclinic orbits paying particular attention to four and higher dimensions where two additional conditions are needed. We give equivalent conditions in terms of subspaces associated with the variational equation along the orbit. Then we review Rodriguez's
construction of a three-dimensional system with Sil'nikov saddle-focus
homoclinic orbits and finally show how to construct higher-dimensional systems
with such orbits.

keywords:
exponential dichotomy.
,
homoclinic orbits
,
invariant manifolds
,
Sil'nikov chaos
,
transversality

DCDS-B

Higher-dimensional nonlinear and perturbed systems of implicit ordinary differential equations are studied by means of methods of dynamical systems. Namely, the persistence of solutions are studied under nonautonomous perturbations connecting either impasse points with IK-singularities or two impasse points. Important parts of the paper are applications of the theory to concrete perturbed fully nonlinear RLC circuits.

DCDS-B

We consider a singularly perturbed system with two normally hyperbolic centre manifolds. We derive one bifurcation function, the zeros of which correspond to heteroclinic connections near such a connection for the unperturbed system, and a second bifurcation function the zeros of which correspond to the vectors in the intersection of the tangent spaces to the centre-unstable and centre-stable manifolds along the heteroclinic connections.

DCDS

We consider the problem of existence of homoclinic orbits
in systems like $x_{n+1}=f(x_n)+\mu g(x_n,\mu )$, $x\in \mathbb{R} ^N$,
$\mu \in \mathbb{R}$, when the unperturbed system $x_{n+1}=f(x_n)$ has an
orbit ${ \gamma _n } _{n\in \mathbb{Z}}$ homoclinic to an expanding fixed
point (snap-back repeller) and such that $f'(\gamma _n)$ is
invertible for any $n \ne 0$ but $f'(\gamma _0)$ is not. We show that,
if a certain analytical condition is satisfied, homoclinic orbits of
the perturbed equation occur in pair on one side of $\mu =0$ while
are not present on the other side.

DCDS-B

We consider a singularly perturbed system with a normally hyperbolic centre manifold. Assuming the existence of a fast homoclinic orbit to a point of the centre manifold belonging to a hyperbolic periodic solution for the slow system, we prove an old and a new result concerning the existence of solutions of the singularly perturbed system that are homoclinic to a periodic solution of the system on the centre manifold. We also give examples in dimensions greater than three of Sil'nikov [16] periodic-to-periodic homoclinic orbits.

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