1. CJM 2015 (vol 67 pp. 1247)
 Barros, Carlos Braga; Rocha, Victor; Souza, Josiney

Lyapunov Stability and Attraction Under Equivariant Maps
Let $M$ and $N$ be admissible Hausdorff topological spaces endowed
with
admissible families of open coverings. Assume that $\mathcal{S}$ is a
semigroup acting on both $M$ and $N$. In this paper we study the behavior of
limit sets, prolongations, prolongational limit sets, attracting sets,
attractors and Lyapunov stable sets (all concepts defined for the action of
the semigroup $\mathcal{S}$) under equivariant maps and semiconjugations
from $M$ to $N$.
Keywords:Lyapunov stability, semigroup actions, generalized flows, equivariant maps, admissible topological spaces Categories:37B25, 37C75, 34C27, 34D05 

2. CJM 2013 (vol 67 pp. 450)
 Santoprete, Manuele; Scheurle, Jürgen; Walcher, Sebastian

Motion in a Symmetric Potential on the Hyperbolic Plane
We study the motion of a particle in the hyperbolic plane (embedded in Minkowski space), under the action of a potential that depends only on one variable. This problem is the analogous to the spherical pendulum in a unidirectional force field. However, for the discussion of the hyperbolic plane one has to distinguish three inequivalent cases, depending on the direction of the force field. Symmetry reduction, with respect to groups that are not necessarily compact or even reductive, is carried out by way of Poisson varieties and Hilbert maps. For each case the dynamics is discussed, with special attention to linear potentials.
Keywords:Hamiltonian systems with symmetry, symmetries, noncompact symmetry groups, singular reduction Categories:37J15, 70H33, 70F99, 37C80, 34C14, , 20G20 

3. CJM 2012 (vol 65 pp. 808)
 Grandjean, Vincent

On Hessian Limit Directions along Gradient Trajectories
Given a nonoscillating gradient trajectory $\gamma$ of a real analytic function $f$,
we show that the limit $\nu$ of the secants at the limit point
$\mathbf{0}$
of $\gamma$ along the trajectory
$\gamma$ is an eigenvector of the limit of the direction of the
Hessian matrix $\operatorname{Hess} (f)$ at $\mathbf{0}$
along $\gamma$. The same holds true at infinity if the function is globally subanalytic. We also deduce
some interesting estimates along the trajectory. Away from the ends of the ambient space, this property is
of metric nature and still holds in a general Riemannian analytic setting.
Keywords:gradient trajectories, nonoscillation, limit of Hessian directions, limit of secants, trajectories at infinity Categories:34A26, 34C08, 32Bxx, 32Sxx 

4. CJM 2004 (vol 56 pp. 310)
 Llibre, Jaume; Schlomiuk, Dana

The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order
In this article we determine the global geometry of the planar
quadratic differential systems with a weak focus of third order. This
class plays a significant role in the context of Hilbert's 16th
problem. Indeed, all examples of quadratic differential systems with
at least four limit cycles, were obtained by perturbing a system in
this family. We use the algebrogeometric concepts of divisor and
zerocycle to encode global properties of the systems and to give
structure to this class. We give a theorem of topological
classification of such systems in terms of integervalued affine
invariants. According to the possible values taken by them in this
family we obtain a total of $18$ topologically distinct phase
portraits. We show that inside the class of all quadratic systems
with the topology of the coefficients, there exists a neighborhood of
the family of quadratic systems with a weak focus of third order and
which may have graphics but no polycycle in the sense of \cite{DRR}
and no limit cycle, such that any quadratic system in this
neighborhood has at most four limit cycles.
Categories:34C40, 51F14, 14D05, 14D25 

5. CJM 2003 (vol 55 pp. 724)
 Cao, Xifang; Kong, Qingkai; Wu, Hongyou; Zettl, Anton

SturmLiouville Problems Whose Leading Coefficient Function Changes Sign
For a given SturmLiouville equation whose leading coefficient
function changes sign, we establish inequalities among the eigenvalues
for any coupled selfadjoint boundary condition and those for two
corresponding separated selfadjoint boundary conditions. By a recent
result of Binding and Volkmer, the eigenvalues (unbounded from both
below and above) for a separated selfadjoint boundary condition can
be numbered in terms of the Pr\"ufer angle; and our inequalities can
then be used to index the eigenvalues for any coupled selfadjoint
boundary condition. Under this indexing scheme, we determine the
discontinuities of each eigenvalue as a function on the space of such
SturmLiouville problems, and its range as a function on the space of
selfadjoint boundary conditions. We also relate this indexing scheme
to the number of zeros of eigenfunctions. In addition, we
characterize the discontinuities of each eigenvalue under a different
indexing scheme.
Categories:34B24, 34C10, 34L05, 34L15, 34L20 

6. CJM 2002 (vol 54 pp. 1038)
 Gavrilov, Lubomir; Iliev, Iliya D.

Bifurcations of Limit Cycles From Infinity in Quadratic Systems
We investigate the bifurcation of limit cycles in oneparameter
unfoldings of quadractic differential systems in the plane having a
degenerate critical point at infinity. It is shown that there are
three types of quadratic systems possessing an elliptic critical point
which bifurcates from infinity together with eventual limit cycles
around it. We establish that these limit cycles can be studied by
performing a degenerate transformation which brings the system to a
small perturbation of certain wellknown reversible systems having a
center. The corresponding displacement function is then expanded in a
Puiseux series with respect to the small parameter and its
coefficients are expressed in terms of Abelian integrals. Finally, we
investigate in more detail four of the cases, among them the elliptic
case (BogdanovTakens system) and the isochronous center
$\mathcal{S}_3$. We show that in each of these cases the
corresponding vector space of bifurcation functions has the Chebishev
property: the number of the zeros of each function is less than the
dimension of the vector space. To prove this we construct the
bifurcation diagram of zeros of certain Abelian integrals in a complex
domain.
Categories:34C07, 34C05, 34C10 

7. CJM 2000 (vol 52 pp. 248)
 Binding, Paul A.; Browne, Patrick J.; Watson, Bruce A.

Spectral Problems for NonLinear SturmLiouville Equations with Eigenparameter Dependent Boundary Conditions
The nonlinear SturmLiouville equation
$$
(py')' + qy = \lambda(1  f)ry \text{ on } [0,1]
$$
is considered subject to the boundary conditions
$$
(a_j\lambda + b_j) y(j) = (c_j\lambda + d_j) (py') (j), \quad j =
0,1.
$$
Here $a_0 = 0 = c_0$ and $p,r > 0$ and $q$ are functions depending
on the independent variable $x$ alone, while $f$ depends on $x$,
$y$ and $y'$. Results are given on existence and location of sets
of $(\lambda,y)$ bifurcating from the linearized eigenvalues, and
for which $y$ has prescribed oscillation count, and on completeness
of the $y$ in an appropriate sense.
Categories:34B24, 34C23, 34L30 

8. CJM 1998 (vol 50 pp. 497)
9. CJM 1997 (vol 49 pp. 583)
 Pal, Janos; Schlomiuk, Dana

Summing up the dynamics of quadratic Hamiltonian systems with a center
In this work we study the global geometry of planar quadratic
Hamiltonian systems with a center and we sum up the dynamics of
these systems in geometrical terms. For this we use the
algebrogeometric concept of multiplicity of intersection
$I_p(P,Q)$ of two complex projective curves $P(x,y,z) = 0$,
$Q(x,y,z) = 0$ at a point $p$ of the plane. This is a
convenient concept when studying polynomial systems and it
could be applied for the analysis of other classes of nonlinear
systems.
Categories:34C, 58F 

10. CJM 1997 (vol 49 pp. 212)
 Coll, B.; Gasull, A.; Prohens, R.

Differential equations defined by the sum of two quasihomogeneous vector fields
In this paper we prove, that under certain hypotheses,
the planar differential equation: $\dot x=X_1(x,y)+X_2(x,y)$,
$\dot y=Y_1(x,y)+Y_2(x,y)$, where $(X_i,Y_i)$, $i=1$, $2$, are
quasihomogeneous vector fields, has at most two limit cycles.
The main tools used in the proof are the generalized polar
coordinates, introduced by Lyapunov to study the stability of degenerate
critical points, and the analysis of the derivatives of the Poincar\'e
return map. Our results generalize those obtained for polynomial
systems with homogeneous nonlinearities.
Categories:34C05, 58F21 

11. CJM 1997 (vol 49 pp. 338)
 Rousseau, C.; Toni, B.

Local bifurcations of critical periods in the reduced Kukles system
In this paper, we study the local bifurcations of critical periods
in the neighborhood of a nondegenerate centre of the reduced Kukles
system. We find at the same time the isochronous systems. We show
that at most three local critical periods bifurcate from the
ChristopherLloyd centres of finite order, at most
two from the linear isochrone and at most one critical period from the
nonlinear isochrone. Moreover, in all cases, there exist
perturbations which lead to the maximum number of critical
periods. We determine the isochrones, using the method of Darboux:
the linearizing transformation of an isochrone is derived from the
expression of the first integral.
Our approach is a combination of computational algebraic techniques
(Gr\"obner bases, theory of the resultant, Sturm's algorithm), the
theory of ideals of noetherian rings and the transversality theory
of algebraic curves.
Categories:34C25, 58F14 
