30 research outputs found
Spatial hidden symmetries in pattern formation
No full textGomes, Gabriela M.; Labouriau, Isabel S.; Pinho, Eliana M.. (1998). Spatial hidden symmetries in pattern formation. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/3240
Note on the unfolding of degenerate Hopf bifurcation germs
No full textAbstractIn an article published in this journal (J. Differential Equations 41 (1981), 375–415) M. Golubitsky and W. F. Langford provide a classification by codimension of local Hopf bifurcation problems which do not satisfy the classical nondegeneracy conditions. They describe all the stable perturbations of a oneparameter family of codimension 3 problems. The stable bifurcation diagrams for this family are presented in Fig. (4.6) of the above article, where one of the cases is indicated as N.A.—not applicable. It is shown here that this case is applicable, and the missing bifurcation diagram is provided
Limit cycles for -equivariant systems without infinite equilibria
No full textWe analyze the dynamics of a class of -equivariant differential
equations of the form
,
where z is complex, the time t is real, while p and s
are complex parameters. This study is the generalisation to
of previous works with and symmetry.
We reduce the problem of finding limit cycles to an Abel equation, and
provide criteria for proving in some cases uniqueness and hyperbolicity of
the limit cycle that surrounds either 1, 2n+1 or 4n+1 equilibria,
the origin being always one of these points
Applications of singularity theory to neurobiology
No full textThe nervous system of animals contains certain specialized
cells, called nerve cells or neurones, that are responsible for
the transmission of information within the animal's body. Those
cells consist of an enlarged part containing the nucleus, and
cytoplasmatic processes extending from if. The
processes are classified by hystological and physiological
criteria as axons or dendrites. Neurones of vertebrates
usually have only one axon, and it rarely branches except at
its termination
Instant Chaos Is Chaos in Slow Motion
No full textAbstractInstant chaos is the onset of chaotic behaviour as a local bifurcation directly from a trivial steady state. We describe a systematic method for constructing examples of instant chaos, by scaling spatial variables and time. In this way we generalize properties of examples previously studied by other authors. We show that whenever a chaotic attractor of limited amplitude is obtained using a scaling property then it appears in slow motion—for any setStransverse to the vector field, the return time toStends to infinity as we approach the bifurcation point. When instant chaos appears for a family of vector fields with a nontrivial scaling property, if it is not in slow motion then the amplitude of the chaotic attractor becomes arbitrarily large around the bifurcation point. We use this method to obtain the Lorenz attractor in a bifurcation directly from an asymptotically stable equilibrium
Dense heteroclinic tangencies near a Bykov cycle
No full textThis article presents a mechanism for the coexistence of hyperbolic and non-hyperbolic dynamics arising in a neighbourhood of a Bykov cycle where trajectories turn in opposite directions near the two nodes - we say that the nodes have different chirality. We show that in the set of vector fields defined on a three-dimensional manifold, there is a class where tangencies of the invariant manifolds of two hyperbolic saddle-foci occur densely. The class is defined by the presence of the Bykov cycle, and by a condition on the parameters that determine the linear part of the vector field at the equilibria. This has important consequences: the global dynamics is persistently dominated by heteroclinic tangencies and by Newhouse phenomena, coexisting with hyperbolic dynamics arising from transversality. The coexistence gives rise to linked suspensions of Cantor sets, with hyperbolic and non-hyperbolic dynamics, in contrast with the case where the nodes have the same chirality. We illustrate our theory with an explicit example where tangencies arise in the unfolding of a symmetric vector field on the three-dimensional sphere
High frequency forcing of an attracting heteroclinic cycle
No full textThis article is concerned with the effect of time-periodic forcing on a
vector field exhibiting an attracting heteroclinic network. We show that as the
forcing frequency tends to infinity, the dynamics reduces to that of a network
under constant forcing, the constant being the average value of the forcing
term. We also show that under small constant forcing the network breaks up into
an attracting periodic solution that persists for periodic forcing of high
frequency.Comment: 15 pages, 3 figure
Global Saddles for Planar Maps
No full textWe study the dynamics of planar diffeomorphisms having a unique fixed point that is a hyperbolic local saddle. We obtain sufficient conditions under which the fixed point is a global saddle. We also address the special case of (Formula presented.)-symmetric maps, for which we obtain a similar result for (Formula presented.) homeomorphisms. Some applications to differential equations are also given. © 2016 Springer Science+Business Media New Yor
Global generic dynamics close to symmetry
No full textAbstractOur object of study is the dynamics that arises in generic perturbations of an asymptotically stable heteroclinic cycle in S3. The cycle involves two saddle-foci of different type and is structurally stable within the class of (Z2⊕Z2)-symmetric vector fields. The cycle contains a two-dimensional connection that persists as a transverse intersection of invariant surfaces under symmetry-breaking perturbations.Gradually breaking the symmetry in a two-parameter family we get a wide range of dynamical behaviour: an attracting periodic trajectory; other heteroclinic trajectories; homoclinic orbits; n-pulses; suspended horseshoes and cascades of bifurcations of periodic trajectories near an unstable homoclinic cycle of Shilnikov type. We also show that, generically, the coexistence of linked homoclinic orbits at the two saddle-foci has codimension 2 and takes place arbitrarily close to the symmetric cycle
