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1.
In this note we give a family of planar polynomial differential systems with a prescribed hyperbolic limit cycle. This family constitutes a corrected and wider version of an example given in the work [M.A. Abdelkader, Relaxation oscillators with exact limit cycles, J. Math. Anal. Appl. 218 (1998) 308-312]. The result given in this note may be used to construct models of Liénard differential equations exhibiting a desired limit cycle.  相似文献   

2.
In this paper, we study the number of zeros of Abelian integrals and the monotonicity of period functions for planar quasihomogeneous Hamiltonian vector fields. The result for Abelian integrals extends the recent work of Li et al. [C. Li, W. Li, J. Llibre, Z. Zhang, Polynomial systems: A lower bound for the weakened 16th Hilbert problem, Extracta Math. 16 (3) (2001) 441–447] and Llibre and Zhang [J. Llibre, X. Zhang, On the number of limit cycles for some perturbed Hamiltonian polynomial systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 8 (2) (2001) 161–181].  相似文献   

3.
Determining the number of limit cycles for a continuous culture vessel system is always useful in analyzing the system. We prove the conditions that guarantee there exist three limit cycles for the chemostat with variable yield that was first proposed by Huang [Limit cycles in a continuous fermentation model, J. Math. Chem. 5 (1990) 287–296] and by Pilyugin and Waltman [Multiple limit cycles in the chemostat with variable yield, Math. Boisci. 182 (2003) 151–166].  相似文献   

4.
In this note we introduce the study of the global behaviour of the network-based SIS epidemic model recently proposed by Pastor-Satorras and Vespignani [Epidemic spreading in scale-free networks, Phys. Rev. Lett. 86 (2001) 3200], characterized in case of homogeneous scale-free networks by a very small epidemic threshold, and extended by Olinky and Stone [Unexpected epidemic threshold in heterogeneous networks: the role of disease transmission, Phys. Rev. E 70 (2004) 03902(r)]. We show that the above model may be read as a particular case of the classical multi-group SIS model proposed by Lajmainovitch and Yorke [A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28 (1976) 221] and extended by Aronsson and Mellander [A deterministic model in biomathematics. Asymptotic behaviour and threshold conditions, Math. Biosci. 49 (1980) 207]. Thus, by applying the methods used for SIS multi-group models, we straightforwardly show, for the first time, that the local conditions identified in the physics literature also determine the global behaviour of a disease spreading on a network. Finally, we briefly study the case in which the force of infection is non-linear, by showing that multiple coexisting equilibria are possible, and by giving a global threshold condition for the extinction.  相似文献   

5.
For three-dimensional competitive Lotka–Volterra systems, Zeeman [M.L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka–Volterra systems, Dyn. Stab. Syst. 8 (1993) 189–217] identified 33 stable nullcline equivalence classes. Among these, only classes 26–31 may have limit cycles. Hofbauer and So [J. Hofbauer, J.W.-H. So, Multiple limit cycles for three dimensional Lotka–Volterra equations, Appl. Math. Lett. 7 (1994) 65–70] conjectured that the number of limit cycles is at most two for these systems. In this paper, we construct three limit cycles for class 29 without a heteroclinic polycycle in Zeeman’s classification.  相似文献   

6.
In this paper we describe how techniques of asymptotic analysis can be used in a systematic way to perform ‘aggregation’ of variables, based on a separation of different time scales, in a population model with age and space structure. The main result of the paper is proving the convergence of the formal asymptotic expansion to the solution of the original equation. This result improves and clarifies earlier results of Arino et al. (SIAM J Appl Math 60(2):408–436, 1999), Auger et al. (Structured population models in biology and epidemiology. Springer Verlag, Berlin, 2008), Lisi and Totaro (Math Biosci 196(2):153–186, 2005).  相似文献   

7.
Among the six classes of Zeeman's classification for three-dimensional Lotka-Volterra competitive systems with limit cycles, besides the classes 26, 27, 28 and 29, multiple limit cycles are found in classes 30 and 31 by an algorithmic method proposed by Hofbauer and So [J. Hofbauer, J.W. So, Multiple limit cycles for three-dimensional Lotka-Volterra equations, Appl. Math. Lett. 7 (1994) 65-70]. This also gives an answer to a problem proposed in [J. Hofbauer, J.W. So, Multiple limit cycles for three-dimensional Lotka-Volterra equations, Appl. Math. Lett. 7 (1994) 65-70].  相似文献   

8.
In [SIAM J. Appl. Math. 41 (1981) 70-93], Majda proposed a model for the interaction between chemical reactions and compressible fluid dynamics. This model is a low Mach number limit of the one-component reactive Navier-Stokes equations [SIAM J. Appl. Math. 43 (1983) 1086-1118] and was extended to the case where the diffusion coefficient is positive by Larrouturou [Nonlinear Anal. 77 (2001) 405-418]. In this paper, the existence of a one-dimensional Chapman-Jouguet detonation wave, or equivalently a heteroclinic orbit, for the extended model is proven. The proof is based on an application of topological arguments to a system of ordinary differential equations which is obtained from the partial differential equations describing the interaction.  相似文献   

9.
The Littelmann path model gives a realization of the crystals of integrable representations of symmetrizable Kac–Moody Lie algebras. Recent work of Gaussent and Littelmann [S. Gaussent, P. Littelmann, LS galleries, the path model, and MV cycles, Duke Math. J. 127 (1) (2005) 35–88] and others [A. Braverman, D. Gaitsgory, Crystals via the affine Grassmannian, Duke Math. J. 107 (3) (2001) 561–575; S. Gaussent, G. Rousseau, Kac–Moody groups, hovels and Littelmann's paths, preprint, arXiv: math.GR/0703639, 2007] has demonstrated a connection between this model and the geometry of the loop Grassmanian. The alcove walk model is a version of the path model which is intimately connected to the combinatorics of the affine Hecke algebra. In this paper we define a refined alcove walk model which encodes the points of the affine flag variety. We show that this combinatorial indexing naturally indexes the cells in generalized Mirkovi?–Vilonen intersections.  相似文献   

10.
In this paper we prove the conjecture of J.-C. Bermond (Ann. Discrete Math.36 (1978), 21–28): If two graphs are decomposable into Hamiltonian cycles, then their lexicographic product is decomposable, too.  相似文献   

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