807 research outputs found
A new approach to the vakonomic mechanics
The aim of this paper is to show that the Lagrange-d'Alembert and its
equivalent the Gauss and Appel principle are not the only way to deduce the
equations of motion of the nonholonomic systems. Instead of them, here we
consider the generalization of the Hamiltonian principle for nonholonomic
systems with nonzero transpositional relations.
By applying this variational principle which takes into the account
transpositional relations different from the classical ones we deduce the
equations of motion for the nonholonomic systems with constraints that in
general are nonlinear in the velocity. These equations of motion coincide,
except perhaps in a zero Lebesgue measure set, with the classical differential
equations deduced with d'Alembert-Lagrange principle.
We provide a new point of view on the transpositional relations for the
constrained mechanical systems: the virtual variations can produce zero or
non-zero transpositional relations. In particular the independent virtual
variations can produce non-zero transpositional relations. For the
unconstrained mechanical systems the virtual variations always produce zero
transpositional relations.
We conjecture that the existence of the nonlinear constraints in the velocity
must be sought outside of the Newtonian model.
All our results are illustrated with precise examples
On the use of blow up to study regularizations of singularities of piecewise smooth dynamical systems in
In this paper we use the blow up method of Dumortier and Roussarie
\cite{dumortier_1991,dumortier_1993,dumortier_1996}, in the formulation due to
Krupa and Szmolyan \cite{krupa_extending_2001}, to study the regularization of
singularities of piecewise smooth dynamical systems
\cite{filippov1988differential} in . Using the regularization
method of Sotomayor and Teixeira \cite{Sotomayor96}, first we demonstrate the
power of our approach by considering the case of a fold line. We quickly
recover a main result of Bonet and Seara \cite{reves_regularization_2014} in a
simple manner. Then, for the two-fold singularity, we show that the regularized
system only fully retains the features of the singular canards in the piecewise
smooth system in the cases when the sliding region does not include a full
sector of singular canards. In particular, we show that every locally unique
primary singular canard persists the regularizing perturbation. For the case of
a sector of primary singular canards, we show that the regularized system
contains a canard, provided a certain non-resonance condition holds. Finally,
we provide numerical evidence for the existence of secondary canards near
resonance.Comment: To appear in SIAM Journal of Applied Dynamical System
Integrability, degenerate centers, and limit cycles for a class of polynomial differential systems
AbstractWe consider the class of polynomial differential equations x˙ Pn(x,y)+Pn+1(x,y)+Pn+2(x,y), y˙=Qn(x,y)+Qn+1(x,y)+Qn+2(x,y), for n ≥ 1 and where Pi and Qi are homogeneous polynomials of degree i These systems have a linearly zero singular point at the origin if n > 2. Inside this class, we identify a new subclass of Darboux integrable systems, and some of them having a degenerate center, i.e., a center with linear part identically zero. Moreover, under additional conditions such Darboux integrable systems can have at most one limit cycle. We provide the explicit expression of this limit cycle
On the Dynamics of the Unified Chaotic System Between Lorenz and Chen Systems
PublishedA one-parameter family of differential systems that bridges the gap between the Lorenz and the Chen systems was proposed by Lu, Chen, Cheng and Celikovsy. The goal of this paper is to analyze what we can say using analytic tools about the dynamics of this one-parameter family of differential systems. We shall describe its global dynamics at infinity, and for two special values of the parameter a we can also describe the global dynamics in the whole ℝ3 using the invariant algebraic surfaces of the family. Additionally we characterize the Hopf bifurcations of this family.The first author is partially supported by a
MINECO/FEDER grant MTM2008-03437 and
MTM2013-40998-P, an AGAUR grant number
2014SGR-568, an ICREA Academia, the grants
FP7-PEOPLE-2012-IRSES 318999 and 316338,
and UNAB 13-4E-1604
Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields
In the present study we consider planar piecewise linear vector fields with
two zones separated by the straight line . Our goal is to study the
existence of simultaneous crossing and sliding limit cycles for such a class of
vector fields. First, we provide a canonical form for these systems assuming
that each linear system has center, a real one for and a virtual one for
, and such that the real center is a global center. Then, working with a
first order piecewise linear perturbation we obtain piecewise linear
differential systems with three crossing limit cycles. Second, we see that a
sliding cycle can be detected after a second order piecewise linear
perturbation. Finally, imposing the existence of a sliding limit cycle we prove
that only one adittional crossing limit cycle can appear. Furthermore, we also
characterize the stability of the higher amplitude limit cycle and of the
infinity. The main techniques used in our proofs are the Melnikov method, the
Extended Chebyshev systems with positive accuracy, and the Bendixson
transformation.Comment: 24 pages, 7 figure
Further considerations on the number of limit cycles of vector fields of the form X(v) = Av + f(v) Bv
AbstractIn Gasull, Llibre, and Sotomayor. (J. Differential Equations, in press) we studied the number of limit cycles of planar vector fields as in the title. The case where the origin is a node with different eigenvalues, which then resisted our analysis, is solved in this paper
Quadratic vector fields with a weak focus of third order
We study phase portraits of quadratic vector fields with a weak focus of third order at the origin. We show numerically the existence of at least 20 different global phase portraits for such vector fields coming from exactly 16 different local phase portraits available for these vector fields. Among these 20 phase portraits, 17 have no limit cycles and three have at least one limit cycle
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