1,049 research outputs found
Complex Bifurcation from Real Paths
A new bifurcation phenomenon, called complex bifurcation, is studied. The basic idea is simply that real solution paths of real analytic problems frequently have complex paths bifurcating from them. It is shown that this phenomenon occurs at fold points, at pitchfork bifurcation points, and at isola centers. It is also shown that perturbed bifurcations can yield two disjoint real solution branches that are connected by complex paths bifurcating from the perturbed solution paths. This may be useful in finding new real solutions.
A discussion of how existing codes for computing real solution paths may be trivially modified to compute complex paths is included, and examples of numerically computed complex solution paths for a nonlinear two point boundary value problem, and a problem from fluid mechanics are given
Polynomial normal forms of Constrained Differential Equations with three parameters
We study generic constrained differential equations (CDEs) with three
parameters, thereby extending Takens's classification of singularities of such
equations. In this approach, the singularities analyzed are the Swallowtail,
the Hyperbolic, and the Elliptic Umbilics. We provide polynomial local normal
forms of CDEs under topological equivalence. Generic CDEs are important in the
study of slow-fast (SF) systems. Many properties and the characteristic
behavior of the solutions of SF systems can be inferred from the corresponding
CDE. Therefore, the results of this paper show a first approximation of the
flow of generic SF systems with three slow variables.Comment: This is an updated and revised version. Minor modifications mad
Lie-Poincare' transformations and a reduction criterion in Landau theory
In the Landau theory of phase transitions one considers an effective
potential whose symmetry group and degree depend on the system
under consideration; generally speaking, is the most general
-invariant polynomial of degree . When such a turns out to be too
complicate for a direct analysis, it is essential to be able to drop
unessential terms, i.e. to apply a simplifying criterion. Criteria based on
singularity theory exist and have a rigorous foundation, but are often very
difficult to apply in practice. Here we consider a simplifying criterion (as
stated by Gufan) and rigorously justify it on the basis of classical
Lie-Poincar\'e theory as far as one deals with fixed values of the control
parameter(s) in the Landau potential; when one considers a range of values, in
particular near a phase transition, the criterion has to be accordingly
partially modified, as we discuss. We consider some specific cases of group
as examples, and study in detail the application to the
Sergienko-Gufan-Urazhdin model for highly piezoelectric perovskites.Comment: 32 pages, no figures. To appear in Annals of Physic
Differential equations for the cuspoid canonical integrals
Differential equations satisfied by the cuspoid canonical integrals I_n(a) are obtained for arbitrary values of n≥2, where n−1 is the codimension of the singularity and a=(ɑ_1,ɑ_2,...,ɑ_(n−1)). A set of linear coupled ordinary differential equations is derived for each step in the sequence I_n(0,0,...,0,0) →I_n(0,0,...,0,ɑ_(n−1)) →I_n(0,0,...,ɑ_(n−2),ɑ_(n−1)) →...→I_n(0,ɑ_2,...,ɑ_(n−2),ɑ_(n−1)) →I_n(ɑ_1,ɑ_2,...,ɑ_n−2,ɑ_(n−1)). The initial conditions for a given step are obtained from the solutions of the previous step. As examples of the formalism, the differential equations for n=2 (fold), n=3 (cusp), n=4 (swallowtail), and n=5 (butterfly) are given explicitly. In addition, iterative and algebraic methods are described for determining the parameters a that are required in the uniform asymptotic cuspoid approximation for oscillating integrals with many coalescing saddle points. The results in this paper unify and generalize previous researches on the properties of the cuspoid canonical integrals and their partial derivatives
Bistability in Apoptosis by Receptor Clustering
Apoptosis is a highly regulated cell death mechanism involved in many
physiological processes. A key component of extrinsically activated apoptosis
is the death receptor Fas, which, on binding to its cognate ligand FasL,
oligomerize to form the death-inducing signaling complex. Motivated by recent
experimental data, we propose a mathematical model of death ligand-receptor
dynamics where FasL acts as a clustering agent for Fas, which form locally
stable signaling platforms through proximity-induced receptor interactions.
Significantly, the model exhibits hysteresis, providing an upstream mechanism
for bistability and robustness. At low receptor concentrations, the bistability
is contingent on the trimerism of FasL. Moreover, irreversible bistability,
representing a committed cell death decision, emerges at high concentrations,
which may be achieved through receptor pre-association or localization onto
membrane lipid rafts. Thus, our model provides a novel theory for these
observed biological phenomena within the unified context of bistability.
Importantly, as Fas interactions initiate the extrinsic apoptotic pathway, our
model also suggests a mechanism by which cells may function as bistable
life/death switches independently of any such dynamics in their downstream
components. Our results highlight the role of death receptors in deciding cell
fate and add to the signal processing capabilities attributed to receptor
clustering.Comment: Accepted by PLoS Comput Bio
Decoherence of Schrodinger cat states in a Luttinger liquid
Schrodinger cat states built from quantum superpositions of left or right
Luttinger fermions located at different positions in a spinless Luttinger
liquid are considered. Their decoherence rates are computed within the
bosonization approach using as environments the quantum electromagnetic field
or two or three dimensionnal acoustic phonon baths. Emphasis is put on the
differences between the electromagnetic and acoustic environments.Comment: 22 pages revtex4, 7 figures in a separate PS fil
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