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Search: MSC category 34C ( Qualitative theory [See also 37-XX] )

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1. CMB Online first

Llibre, Jaume; Zhang, Xiang
On the limit cycles of linear differential systems with homogeneous nonlinearities
We consider the class of polynomial differential systems of the form $\dot x= \lambda x-y+P_n(x,y)$, $\dot y=x+\lambda y+ Q_n(x,y),$ where $P_n$ and $Q_n$ are homogeneous polynomials of degree $n$. For this class of differential systems we summarize the known results for the existence of limit cycles, and we provide new results for their nonexistence and existence.

Keywords:polynomial differential system, limit cycles, differential equations on the cylinder
Categories:34C35, 34D30

2. CMB 2011 (vol 56 pp. 39)

Ben Amara, Jamel
Comparison Theorem for Conjugate Points of a Fourth-order Linear Differential Equation
In 1961, J. Barrett showed that if the first conjugate point $\eta_1(a)$ exists for the differential equation $(r(x)y'')''= p(x)y,$ where $r(x)\gt 0$ and $p(x)\gt 0$, then so does the first systems-conjugate point $\widehat\eta_1(a)$. The aim of this note is to extend this result to the general equation with middle term $(q(x)y')'$ without further restriction on $q(x)$, other than continuity.

Keywords:fourth-order linear differential equation, conjugate points, system-conjugate points, subwronskians
Categories:47E05, 34B05, 34C10

3. CMB 2011 (vol 56 pp. 366)

Kyritsi, Sophia Th.; Papageorgiou, Nikolaos S.
Multiple Solutions for Nonlinear Periodic Problems
We consider a nonlinear periodic problem driven by a nonlinear nonhomogeneous differential operator and a Carathéodory reaction term $f(t,x)$ that exhibits a $(p-1)$-superlinear growth in $x \in \mathbb{R}$ near $\pm\infty$ and near zero. A special case of the differential operator is the scalar $p$-Laplacian. Using a combination of variational methods based on the critical point theory with Morse theory (critical groups), we show that the problem has three nontrivial solutions, two of which have constant sign (one positive, the other negative).

Keywords:$C$-condition, mountain pass theorem, critical groups, strong deformation retract, contractible space, homotopy invariance
Categories:34B15, 34B18, 34C25, 58E05

4. CMB 2011 (vol 55 pp. 3)

Agarwal, Ravi P.; Mustafa, Octavian G.
On a Local Theory of Asymptotic Integration for Nonlinear Differential Equations
We improve several recent results in the asymptotic integration theory of nonlinear ordinary differential equations via a variant of the method devised by J. K. Hale and N. Onuchic The results are used for investigating the existence of positive solutions to certain reaction-diffusion equations.

Keywords:asymptotic integration, Emden-Fowler differential equation, reaction-diffusion equation
Categories:34E10, 34C10, 35Q35

5. CMB 2009 (vol 53 pp. 193)

Agarwal, Ravi P.; Avramescu, Cezar; Mustafa, Octavian G.
On the Oscillation of a Second Order Strictly Sublinear Differential Equation
We establish a flexible oscillation criterion based on an averaging technique that improves upon a result due to C.~G. Philos.

Keywords:oscillation theory, averaging method
Categories:34C10, 34C15, 34C29

6. CMB 2009 (vol 52 pp. 315)

Yi, Taishan; Zou, Xingfu
Generic Quasi-Convergence for Essentially Strongly Order-Preserving Semiflows
By employing the limit set dichotomy for essentially strongly order-preserving semiflows and the assumption that limit sets have infima and suprema in the state space, we prove a generic quasi-convergence principle implying the existence of an open and dense set of stable quasi-convergent points. We also apply this generic quasi-convergence principle to a model for biochemical feedback in protein synthesis and obtain some results about the model which are of theoretical and realistic significance.

Keywords:Essentially strongly order-preserving semiflow, compactness, quasi-convergence
Categories:34C12, 34K25

7. CMB 2007 (vol 50 pp. 377)

Gutierrez, C.; Jarque, X.; Llibre, J.; Teixeira, M. A.
Global Injectivity of $C^1$ Maps of the Real Plane, Inseparable Leaves and the Palais--Smale Condition
We study two sufficient conditions that imply global injectivity for a $C^1$ map $X\colon \R^2\to \R^2$ such that its Jacobian at any point of $\R^2$ is not zero. One is based on the notion of half-Reeb component and the other on the Palais--Smale condition. We improve the first condition using the notion of inseparable leaves. We provide a new proof of the sufficiency of the second condition. We prove that both conditions are not equivalent, more precisely we show that the Palais--Smale condition implies the nonexistence of inseparable leaves, but the converse is not true. Finally, we show that the Palais--Smale condition it is not a necessary condition for the global injectivity of the map $X$.

Categories:34C35, 34H05

8. CMB 2001 (vol 44 pp. 323)

Schuman, Bertrand
Une classe d'hamiltoniens polynomiaux isochrones
Soit $H_0 = \frac{x^2+y^2}{2}$ un hamiltonien isochrone du plan $\Rset^2$. On met en \'evidence une classe d'hamiltoniens isochrones qui sont des perturbations polynomiales de $H_0$. On obtient alors une condition n\'ecessaire d'isochronisme, et un crit\`ere de choix pour les hamiltoniens isochrones. On voit ce r\'esultat comme \'etant une g\'en\'eralisation du caract\`ere isochrone des perturbations hamiltoniennes homog\`enes consid\'er\'ees dans [L], [P], [S]. Let $H_0 = \frac{x^2+y^2}{2}$ be an isochronous Hamiltonian of the plane $\Rset^2$. We obtain a necessary condition for a system to be isochronous. We can think of this result as a generalization of the isochronous behaviour of the homogeneous polynomial perturbation of the Hamiltonian $H_0$ considered in [L], [P], [S].

Keywords:Hamiltonian system, normal forms, resonance, linearization
Categories:34C20, 58F05, 58F22, 58F30

9. CMB 1997 (vol 40 pp. 448)

Kaczynski, Tomasz; Mrozek, Marian
Stable index pairs for discrete dynamical systems
A new shorter proof of the existence of index pairs for discrete dynamical systems is given. Moreover, the index pairs defined in that proof are stable with respect to small perturbations of the generating map. The existence of stable index pairs was previously known in the case of diffeomorphisms and flows generated by smooth vector fields but it was an open question in the general discrete case.

Categories:54H20, 54C60, 34C35

10. CMB 1997 (vol 40 pp. 276)

Chouikha, Raouf
Fonctions elliptiques et équations différentielles ordinaires
In this paper, we detail some results of a previous note concerning a trigonometric expansion of the Weierstrass elliptic function $\{\wp(z);\, 2\omega, 2\omega'\}$. In particular, this implies its classical Fourier expansion. We use a direct integration method of the ODE $$(E)\left\{\matrix{{d^2u \over dt^2} = P(u, \lambda)\hfill \cr u(0) = \sigma\hfill \cr {du \over dt}(0) = \tau\hfill \cr}\right.$$ where $P(u)$ is a polynomial of degree $n = 2$ or $3$. In this case, the bifurcations of $(E)$ depend on one parameter only. Moreover, this global method seems not to apply to the cases $n > 3$.

Categories:33E05, 34A05, 33E20, 33E30, 34A20, 34C23

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