Detection of Special Curves Via the Double Resultant

Ferragut, Antoni; García-Saldaña, Johanna D.; Gasull, Armengol

doi:10.1007/s12346-015-0180-x

Cited by 2 publications

(2 citation statements)

References 26 publications

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“…In our case, there are no multiple factors in the polynomials R x (y) and R y (x), so we have R x (y) = R x (y) and R y (x) = R y (x). Next, following the approach proposed in [11,12], we consider double resultants (see [? ] for a general introduction to this setting).…”

Section: Preliminary Results: Systems Of Polynomial Equationsmentioning

confidence: 99%

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Rational Parameterizations Approach for Solving Equations in Some Dynamical Systems Problems

Gasull

Lázaro

Torregrosa

2018

Qual. Theory Dyn. Syst.

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We show how the use of rational parameterizations facilitates the study of the number of solutions of many systems of equations involving polynomials and square roots of polynomials. We illustrate the effectiveness of this approach, applying it to several problems appearing in the study of some dynamical systems. Our examples include Abelian integrals, Melnikov functions and a couple of questions in Celestial Mechanics: the computation of some relative equilibria and the study of some central configurations.

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Section: Preliminary Results: Systems Of Polynomial Equationsmentioning

confidence: 99%

“…Proof. To eliminate the square roots in (11) we impose that k 2 + 1 = u 2 and 2 + 1 = v 2 , for some new variables u and v. Again, both planar curves have genus zero and can be parameterized through the rational parameterizations…”

Section: Theorem 41 For Non-vanishing Values Ofmentioning

confidence: 99%

Rational Parameterizations Approach for Solving Equations in Some Dynamical Systems Problems

Gasull

Lázaro

Torregrosa

2018

Qual. Theory Dyn. Syst.

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Exploiting delayed transitions to sustain semiarid ecosystems after catastrophic shifts

Vidiella

Sardanyés

Solé

2017

Preprint

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Semiarid ecosystems (including arid, semiarid and dry-subhumid ecosystems) span more than 40% of extant habitats and a similar percentage of human population. As a consequence of global warming, these habitats face future potential shifts towards the desert state characterized by an accelerated loss of diversity and stability leading to collapse. Such possibility has been raised by several mathematical and computational models, along with several early warning signal methods applied to spatial vegetation patterns. Here we show that just after a catastrophic shift has taken place an expected feature is the presence of a ghost, i.e., a delayed extinction associated to the underlying dynamical flows. As a consequence, a system might exhibit for very long times an apparent stationarity hiding in fact an inevitable collapse. Here we explore this problem showing that the ecological ghost is a generic feature of standard models of green-desert transitions including facilitation. If present, the ghost could hide warning signals, since statistical patterns are not be expected to display growing fluctuations over time. We propose and computationally test a novel intervention method based on the restoration of small fractions of desert areas with vegetation as a way to maintain the fragile ecosystem beyond the catastrophic shift caused by a saddle-node bifurcation, taking advantage of the delaying capacity of the ghost just after the bifurcation.

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Detection of Special Curves Via the Double Resultant

Cited by 2 publications

References 26 publications

Rational Parameterizations Approach for Solving Equations in Some Dynamical Systems Problems

Rational Parameterizations Approach for Solving Equations in Some Dynamical Systems Problems

Exploiting delayed transitions to sustain semiarid ecosystems after catastrophic shifts

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