Expand all Collapse all | Results 1 - 10 of 10 |
1. CJM Online first
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, non-compact symmetry groups, singular reduction Categories:37J15, 70H33, 70F99, 37C80, 34C14, , 20G20 |
2. CJM 2012 (vol 65 pp. 808)
On Hessian Limit Directions along Gradient Trajectories Given a non-oscillating 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 eigen-vector 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 sub-analytic. 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, non-oscillation, limit of Hessian directions, limit of secants, trajectories at infinity Categories:34A26, 34C08, 32Bxx, 32Sxx |
3. CJM 2004 (vol 56 pp. 310)
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 16-th
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 algebro-geometric concepts of divisor and
zero-cycle 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 integer-valued 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 |
4. CJM 2003 (vol 55 pp. 724)
Sturm-Liouville Problems Whose Leading Coefficient Function Changes Sign For a given Sturm-Liouville equation whose leading coefficient
function changes sign, we establish inequalities among the eigenvalues
for any coupled self-adjoint boundary condition and those for two
corresponding separated self-adjoint boundary conditions. By a recent
result of Binding and Volkmer, the eigenvalues (unbounded from both
below and above) for a separated self-adjoint 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 self-adjoint
boundary condition. Under this indexing scheme, we determine the
discontinuities of each eigenvalue as a function on the space of such
Sturm-Liouville problems, and its range as a function on the space of
self-adjoint 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 |
5. CJM 2002 (vol 54 pp. 1038)
Bifurcations of Limit Cycles From Infinity in Quadratic Systems We investigate the bifurcation of limit cycles in one-parameter
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 well-known 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 (Bogdanov-Takens 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 |
6. CJM 2000 (vol 52 pp. 248)
Spectral Problems for Non-Linear Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions |
Spectral Problems for Non-Linear Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions The nonlinear Sturm-Liouville 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 |
7. CJM 1998 (vol 50 pp. 497)
Morse index of approximating periodic solutions for the billiard problem. Application to existence results |
Morse index of approximating periodic solutions for the billiard problem. Application to existence results This paper deals with periodic solutions for the billiard problem in a
bounded open set of $\hbox{\Bbbvii R}^N$ which are limits of regular
solutions of Lagrangian systems with a potential well. We give a
precise link between the Morse index of approximate solutions
(regarded as critical points of Lagrangian functionals) and the
properties of the bounce trajectory to which they converge.
Categories:34C25, 58E50 |
8. CJM 1997 (vol 49 pp. 583)
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
algebro-geometric 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 |
9. CJM 1997 (vol 49 pp. 212)
Differential equations defined by the sum of two quasi-homogeneous 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
quasi-homogeneous 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 non-linearities.
Categories:34C05, 58F21 |
10. CJM 1997 (vol 49 pp. 338)
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
Christopher-Lloyd 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 |