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### Open Access Journals

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

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-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.

DCDS-B

Isagi et al introduced a model for masting, that is, the intermittent production of flowers and fruit by trees.
A tree produces flowers and fruit only when the stored energy exceeds a certain
threshold value. If flowers and fruit are not produced, the stored
energy increases by a certain fixed amount; if flowers and fruit are produced, the energy is depleted by an
amount proportional to the excess stored energy. Thus a one-dimensional model is derived for the amount
of stored energy. When the ratio of the amount of energy used for flowering and fruit production in a
reproductive year to the excess amount of stored energy before that year is small, the stored energy
approaches a constant value as time passes. However when this ratio is large, the amount of stored energy varies
unpredictably and as the ratio increases the range of possible values for the stored energy increases also.
In this article we describe this chaotic behavior precisely with complete proofs.

DCDS-B

A continuous map $f:[0,1]\rightarrow[0,1]$ is called an $n$-modal
map if there is a partition $0=z_0 < z_1 < ... < z_n=1$ such that
$f(z_{2i})=0$, $f(z_{2i+1})=1$ and, $f$ is (not necessarily strictly)
monotone on each $[z_{i},z_{i+1}]$. It is well-known that such a
map is topologically semi-conjugate to a piecewise linear map; however
here we prove that the topological semi-conjugacy is unique for this
class of maps; also our proof is constructive and yields a sequence
of easily computable piecewise linear maps which converges uniformly
to the semi-conjugacy. We also give equivalent conditions for the
semi-conjugacy to be a conjugacy as in Parry's theorem. Related work
was done by Fotiades and Boudourides and Banks, Dragan and Jones,
who however only considered cases where a conjugacy exists. Banks,
Dragan and Jones gave an algorithm to construct the conjugacy map
but only for one-hump maps.

DCDS

In this paper we study unimodal maps on the closed unit interval,
which have a stable period 3 orbit and an unstable period 3 orbit,
and give conditions under which all points in the open unit interval are either asymptotic to the stable period 3 orbit or land after a finite time on an invariant Cantor set $\Lambda$ on which the dynamics is conjugate to a subshift of finite type and is, in fact, chaotic.
For the particular value
of $\mu=3.839$, Devaney [3], following ideas of Smale and Williams, shows
that the logistic map $f(x)=\mu x(1-x)$ has this property.
In this case the stable and unstable period 3 orbits appear when $\mu=\mu_0=1+\sqrt{8}$. We use our theorem to show that the property holds for all values of $\mu>\mu_0$ for which the stable period 3 orbit persists.

keywords:
chaos
,
stability
,
subshift of finite type.
,
Schwarzian derivative
,
unimodal map
,
Period 3
,
hyperbolic

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