Existence of periodic solutions for Liénard equations with finite delay

Wei, Junjie; Huang, Qichang

doi:10.1007/bf02882832

Cited by 6 publications

(6 citation statements)

References 5 publications

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“…All the chemicals and solvents were of analytical grade Pyridoxamine (PM),7 dianhydride of diethylenetriaminepentaacetic acid (DTPAA),8 3‐ O ‐benzylglycerol (3‐ O ‐Bz‐GLYC),9, 10 pentaerythritol monobenzaldehyde ( S ‐BzA‐PETO),11 N ‐benzyldiethanolamine ( N ‐Bz‐DEA)12 and the solution of phosgene in toluene (2.0 M)13 were prepared according to the cited literatures.…”

Section: Methodsmentioning

confidence: 99%

“…In order to increase the hydrophilicity of polyester ligands, polymerizations were carried out adding ethylene glycol (EG) into the polymerization systems to yield polyesters with two different diol units in the backbone chain P(DTPA–GLYC–EG), P(DTPA–PETO–EG) and P(DTPA–DEA–EG). Pyridoxamine (PM) was chosen as a liver‐targeting group 5–7. It was first chlorocarbonylated and then incorporated into polyesters to obtain the polyester ligands containing pyridoxamine group with chemical structures shown in Fig 1.…”

Section: Introductionmentioning

confidence: 99%

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Hepatic targeting of macromolecular MRI contrast agents

Yan

Zhuo

et al. 2002

Polymer International

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Water‐soluble polyester ligands were synthesized by the polycondensation of diethylenetriaminepentaacetic dianhydride (DTPAA) with protected polyalcohol 3‐O‐benzyl‐glycerol (3‐O‐Bz‐GLYC), pentaerythritol monobenzaldehyde (S‐Bz‐PETO) and N‐benzyldiethanolamine (N‐Bz‐DEA), respectively, and the protecting groups were then removed by hydrogenation to give polyesters P(DTPA–GLYC), P(DTPA–PETO) and P(DTPA–DEA). In the same manner, by adding ethylene glycol (EG) monomer into the polymerization system, polyesters P(DTPA–GLYC–EG), P(DTPA–PETO–EG) and P(DTPA–DEA–EG) were also synthesized. Pyridoxamine was chosen as a liver‐targeting group and it was first chlorocarbonylated and then incorporated into polyesters. The polyester ligands containing pyridoxamine group thus prepared were further reacted with GdCl3 in water at room temperature to give the corresponding hepatic‐targeting polyester gadolinium complexes. These macromolecular ligands and their gadolinium complexes were characterized by 1H NMR, IR, UV and elementary analysis. Relaxivity studies showed that these polyester gadolinium complexes possess higher relaxation effectiveness than that of the clinically used small molecular gadolinium complex Gd‐DTPA. In vitro cytotoxicity assay demonstrated that these macromolecular ligands and their gadolinium complexes have low cytotoxicity to the human normal liver cells (L‐02). Magnetic resonance imaging of the liver in rats indicated that these polyester MRI contrast agents containing pyridoxamine group exhibit liver‐targeting property. © 2002 Society of Chemical Industry

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Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Hepatic targeting of macromolecular MRI contrast agents

Yan

Zhuo

et al. 2002

Polymer International

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“…Take T = 2π. Clearly, the average γ / = 2 and, for any q ∈ R, w(q) = 1 2π .g., [1,2,25,30,[32][33][34]. Clearly, Proposition 5.3 applies (for the non-delayed case, see [28]).…”

Section: Sunflower-like Equationmentioning

confidence: 96%

Sunflower model: time-dependent coefficients and topology of the periodic solutions set

Bisconti

Spadini

2015

Nonlinear Differ. Equ. Appl.

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We investigate the structure of the set of periodic solutions of a time-dependent generalized version of the sunflower equation (in fact of the delayed Liénard equation), where the coefficients can vary periodically, thus allowing for environmental oscillations. Our result stems from a more general analysis, based on fixed point index and degreetheoretic methods, of the set of T-periodic solutions of T-periodically perturbed coupled delay differential equations on differentiable manifolds.

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“…Recently, a great deal of research has been devoted to the topics. One of the methods used in them is the ejective fixed point argument developed by [16], which has been successfully used to obtain the global existence of periodic solutions bifurcating from the Hopf bifurcation by many researchers (see, for example, [1,2,7,10,14,17,22,24,30]). The other is the global Hopf bifurcation theorem due to Erbe et al [4], which was established using a purely topological argument.…”

Section: Introductionmentioning

confidence: 99%

Local Hopf bifurcation and global periodic solutions in a delayed predator–prey system

Song

Wei

2005

Journal of Mathematical Analysis and Applications

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181

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We consider a delayed predator-prey system. We first consider the existence of local Hopf bifurcations, and then derive explicit formulas which enable us to determine the stability and the direction of periodic solutions bifurcating from Hopf bifurcations, using the normal form theory and center manifold argument. Special attention is paid to the global existence of periodic solutions bifurcating from Hopf bifurcations. By using a global Hopf bifurcation result due to Wu [Trans. Amer. Math. Soc. 350 (1998) 4799], we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. Finally, several numerical simulations supporting the theoretical analysis are also given.  2004 Elsevier Inc. All rights reserved.

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Existence of periodic solutions for Liénard equations with finite delay

Cited by 6 publications

References 5 publications

Hepatic targeting of macromolecular MRI contrast agents

Hepatic targeting of macromolecular MRI contrast agents

Sunflower model: time-dependent coefficients and topology of the periodic solutions set

Local Hopf bifurcation and global periodic solutions in a delayed predator–prey system

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