Injectivity of Local Diffeomorphisms from Nearly Spectral Conditions

Abstract. In this paper we examine matrices which arise naturally as Jacobians in chemical dynamics. We are particularly interested in when these Jacobians are P matrices (up to a sign change), ensuring certain bounds on their eigenvalues, precluding certain behaviour such as multiple equilibria, and sometimes implying stability. We first explore reaction systems and derive results which provide a deep connection between system structure and the P matrix property. We then examine a class of systems consisting of reactions coupled to an external rate-dependent negative feedback process, and characterise conditions which ensure the P matrix property survives the negative feedback. The techniques presented are applied to examples published in the mathematical and biological literature.

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“…From Theorem 2 in [20], this condition on the Jacobian guarantees global injectivity of the function.…”

Section: Implications Of a Pmentioning

confidence: 95%

“…the preimage of any compact set is compact. Recent elegant work such as that in [9] and [20] provides conditions (not all spectral) which ensure that a function is globally injective.…”

Section: Implications Of a Pmentioning

confidence: 99%

PMatrix Properties, Injectivity, and Stability in Chemical Reaction Systems

Banaji¹,

Donnell²,

Baigent³

2007

SIAM J. Appl. Math.

111

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“…If the assumptions are relaxed to 0eSpecðX Þ; then the conclusion, even for polynomial maps X ; need no longer be true, as shown by Pinchuck's counterexample [22]. Also Smyth and Xavier [27,Theorem 4] proved that there exist integers n42 and non-injective polynomial maps P : R n -R n with SpecðPÞ-½0; NÞ ¼ |:…”

mentioning

confidence: 96%

Global asymptotic stability for differentiable vector fields of R2

Fernandes

Gutiérrez

Rabanal

2004

Journal of Differential Equations

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a) Let X : R 2 -R 2 be a differentiable map (not necessarily C 1 ) and let SpecðX Þ be the set of (complex) eigenvalues of the derivative DX p when p varies in R 2 : If, for some e40; SpecðX Þ-½0; eÞ ¼ | then X is injective.(b) Let X : R 2 -R 2 be a differentiable vector field such that X ð0Þ ¼ 0 and SpecðX ÞCfzAC : RðzÞo0g: Then, for all pAR 2 ; there is a unique positive trajectory starting at p; moreover the o-limit set of p is equal to f0g: r 2004 Elsevier Inc. All rights reserved.

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“…A variety of such conditions, spectral and otherwise, have been found ( [1,2] for example). The explicit aim of such work is often to find restrictions on the Jacobian which guarantee injectivity of certain classes of functions (e.g.…”

Section: Introductionmentioning

confidence: 96%