115,579 research outputs found
Strange attractors in periodically-kicked degenerate Hopf bifurcations
We prove that spiral sinks (stable foci of vector fields) can be transformed
into strange attractors exhibiting sustained, observable chaos if subjected to
periodic pulsatile forcing. We show that this phenomenon occurs in the context
of periodically-kicked degenerate supercritical Hopf bifurcations. The results
and their proofs make use of a new multi-parameter version of the theory of
rank one maps developed by Wang and Young.Comment: 16 page
Complex patterns on the plane: different types of basin fractalization in a two-dimensional mapping
Basins generated by a noninvertible mapping formed by two symmetrically
coupled logistic maps are studied when the only parameter \lambda of the system
is modified. Complex patterns on the plane are visualised as a consequence of
basins' bifurcations. According to the already established nomenclature in the
literature, we present the relevant phenomenology organised in different
scenarios: fractal islands disaggregation, finite disaggregation, infinitely
disconnected basin, infinitely many converging sequences of lakes, countable
self-similar disaggregation and sharp fractal boundary. By use of critical
curves, we determine the influence of zones with different number of first rank
preimages in the mechanisms of basin fractalization.Comment: 19 pages, 11 figure
Convergence in law in the second Wiener/Wigner chaos
Let L be the class of limiting laws associated with sequences in the second
Wiener chaos. We exhibit a large subset L_0 of L satisfying that, for any
F_infinity in L_0, the convergence of only a finite number of cumulants
suffices to imply the convergence in law of any sequence in the second Wiener
chaos to F_infinity. This result is in the spirit of the seminal paper by
Nualart and Peccati, in which the authors discovered the surprising fact that
convergence in law for sequences of multiple Wiener-It\^o integrals to the
Gaussian is equivalent to convergence of just the fourth cumulant. Also, we
offer analogues of this result in the case of free Brownian motion and double
Wigner integrals, in the context of free probability.Comment: 14 pages. This version corrects an error which, unfortunately,
appears in the published version in EC
Hierarchical adaptive polynomial chaos expansions
Polynomial chaos expansions (PCE) are widely used in the framework of
uncertainty quantification. However, when dealing with high dimensional complex
problems, challenging issues need to be faced. For instance, high-order
polynomials may be required, which leads to a large polynomial basis whereas
usually only a few of the basis functions are in fact significant. Taking into
account the sparse structure of the model, advanced techniques such as sparse
PCE (SPCE), have been recently proposed to alleviate the computational issue.
In this paper, we propose a novel approach to SPCE, which allows one to exploit
the model's hierarchical structure. The proposed approach is based on the
adaptive enrichment of the polynomial basis using the so-called principle of
heredity. As a result, one can reduce the computational burden related to a
large pre-defined candidate set while obtaining higher accuracy with the same
computational budget
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