Smoothness of unordered curves in two-dimensional strongly competitive systems

Janusz Mierczyński

Applicationes Mathematicae (1999)

  • Volume: 25, Issue: 4, page 449-455
  • ISSN: 1233-7234

Abstract

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It is known that in two-dimensional systems of ODEs of the form i = x i f i ( x ) with ∂ f i / ∂ x j < 0 (strongly competitive systems), boundaries of the basins of repulsion of equilibria consist of invariant Lipschitz curves, unordered with respect to the coordinatewise (partial) order. We prove that such curves are in fact of class C 1 .

How to cite

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Mierczyński, Janusz. "Smoothness of unordered curves in two-dimensional strongly competitive systems." Applicationes Mathematicae 25.4 (1999): 449-455. <http://eudml.org/doc/219218>.

@article{Mierczyński1999,
abstract = {It is known that in two-dimensional systems of ODEs of the form $^i=x^if^i(x)$ with $\{\partial f^i\}/\{\partial x^j\} < 0$ (strongly competitive systems), boundaries of the basins of repulsion of equilibria consist of invariant Lipschitz curves, unordered with respect to the coordinatewise (partial) order. We prove that such curves are in fact of class $C^1$.},
author = {Mierczyński, Janusz},
journal = {Applicationes Mathematicae},
keywords = {strongly competitive system of ordinary differential equations; invariant manifold; d-curve; Lotka-Volterra system; smoothness; unordered curves; two-dimensional strongly competitive systems; equilibria; invariant Lipschitz curves},
language = {eng},
number = {4},
pages = {449-455},
title = {Smoothness of unordered curves in two-dimensional strongly competitive systems},
url = {http://eudml.org/doc/219218},
volume = {25},
year = {1999},
}

TY - JOUR
AU - Mierczyński, Janusz
TI - Smoothness of unordered curves in two-dimensional strongly competitive systems
JO - Applicationes Mathematicae
PY - 1999
VL - 25
IS - 4
SP - 449
EP - 455
AB - It is known that in two-dimensional systems of ODEs of the form $^i=x^if^i(x)$ with ${\partial f^i}/{\partial x^j} < 0$ (strongly competitive systems), boundaries of the basins of repulsion of equilibria consist of invariant Lipschitz curves, unordered with respect to the coordinatewise (partial) order. We prove that such curves are in fact of class $C^1$.
LA - eng
KW - strongly competitive system of ordinary differential equations; invariant manifold; d-curve; Lotka-Volterra system; smoothness; unordered curves; two-dimensional strongly competitive systems; equilibria; invariant Lipschitz curves
UR - http://eudml.org/doc/219218
ER -

References

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