275 research outputs found
Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations
In this paper we perform the parameter-dependent center manifold reduction
near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf
bifurcations in delay differential equations (DDEs). This allows us to
initialize the continuation of codimension one equilibria and cycle
bifurcations emanating from these codimension two bifurcation points. The
normal form coefficients are derived in the functional analytic perturbation
framework for dual semigroups (sun-star calculus) using a normalization
technique based on the Fredholm alternative. The obtained expressions give
explicit formulas which have been implemented in the freely available numerical
software package DDE-BifTool. While our theoretical results are proven to apply
more generally, the software implementation and examples focus on DDEs with
finitely many discrete delays. Together with the continuation capabilities of
DDE-BifTool, this provides a powerful tool to study the dynamics near
equilibria of such DDEs. The effectiveness is demonstrated on various models
Intra-chain correlation functions and shapes of homopolymers with different architectures in dilute solution
We present results of Monte Carlo study of the monomer-monomer correlation
functions, static structure factor and asphericity characteristics of a single
homopolymer in the coil and globular states for three distinct architectures of
the chain: ring, open and star. To rationalise the results we introduce the
dimensionless correlation functions rescaled via the corresponding mean-squared
distances between monomers. For flexible chains with some architectures these
functions exhibit a large degree of universality by falling onto a single or
several distinct master curves. In the repulsive regime, where a stretched
exponential times a power law form (de Cloizeaux scaling) can be applied, the
corresponding exponents and have been obtained. The exponent
is found to be universal for flexible strongly repulsive coils
and in agreement with the theoretical prediction from improved higher-order
Borel-resummed renormalisation group calculations. The short-distance exponents
of an open flexible chain are in a good agreement with the
theoretical predictions in the strongly repulsive regime also. However,
increasing the Kuhn length in relation to the monomer size leads to their fast
cross-over towards the Gaussian behaviour. Likewise, a strong sensitivity of
various exponents on the stiffness of the chain, or on the number
of arms in star polymers, is observed. The correlation functions in the
globular state are found to have a more complicated oscillating behaviour and
their degree of universality has been reviewed. Average shapes of the polymers
in terms of the asphericity characteristics, as well as the universal behaviour
in the static structure factors, have been also investigated.Comment: RevTeX 12 pages, 10 PS figures. Accepted by J. Chem. Phy
On Local Bifurcations in Neural Field Models with Transmission Delays
Neural field models with transmission delay may be cast as abstract delay
differential equations (DDE). The theory of dual semigroups (also called
sun-star calculus) provides a natural framework for the analysis of a broad
class of delay equations, among which DDE. In particular, it may be used
advantageously for the investigation of stability and bifurcation of steady
states. After introducing the neural field model in its basic functional
analytic setting and discussing its spectral properties, we elaborate
extensively an example and derive a characteristic equation. Under certain
conditions the associated equilibrium may destabilise in a Hopf bifurcation.
Furthermore, two Hopf curves may intersect in a double Hopf point in a
two-dimensional parameter space. We provide general formulas for the
corresponding critical normal form coefficients, evaluate these numerically and
interpret the results
Conformations of dendrimers in dilute solution
Conformations of isolated homo- dendrimers of G=1-7 generations with D=1-6
spacers have been studied in the good and poor solvents, as well as across the
coil-to-globule transition, by means of a version of the Gaussian
self-consistent (GSC) method and Monte Carlo (MC) simulation in continuous
space based on the same coarse-grained model. The latter includes harmonic
springs between connected monomers and the pair-wise Lennard-Jones potential
with a hard core repulsion. The scaling law for the dendrimer size, the degrees
of bond stretching and steric congestion, as well as the radial density, static
structure factor, and asphericity have been analysed. It is also confirmed that
while smaller dendrimers have a dense core, larger ones develop a hollow domain
at some separation from the centre.Comment: RevTeX, 14 pages, 19 PS figures, Accepted for publication in J. Chem.
Phy
Numerical periodic normalization for codim 2 bifurcations of limit cycles : computational formulas, numerical implementation, and examples
Explicit computational formulas for the coefficients of the periodic normal forms for codimension 2 (codim 2) bifurcations of limit cycles in generic autonomous ODEs are derived. All cases (except the weak resonances) with no more than three Floquet multipliers on the unit circle are covered. The resulting formulas are independent of the dimension of the phase space and involve solutions of certain boundary-value problems on the interval [0, T], where T is the period of the critical cycle, as well as multilinear functions from the Taylor expansion of the ODE right-hand side near the cycle. The formulas allow one to distinguish between various bifurcation scenarios near codim 2 bifurcations of limit cycles. Our formulation makes it possible to use robust numerical boundary-value algorithms based on orthogonal collocation, rather than shooting techniques, which greatly expands its applicability. The implementation is described in detail with numerical examples, where numerous codim 2 bifurcations of limit cycles are analyzed for the first time
Periodic Center Manifolds for Nonhyperbolic Limit Cycles in ODEs
In this paper, we deal with a classical object, namely, a nonhyperbolic limit
cycle in a system of smooth autonomous ordinary differential equations. While
the existence of a center manifold near such a cycle was assumed in several
studies on cycle bifurcations based on periodic normal forms, no proofs were
available in the literature until recently. The main goal of this paper is to
give an elementary proof of the existence of a periodic smooth locally
invariant center manifold near a nonhyperbolic cycle in finite-dimensional
ordinary differential equations by using the Lyapunov-Perron method. In
addition, we provide several explicit examples of analytic vector fields
admitting (non)-unique, (non)--smooth and (non)-analytic periodic
center manifolds.Comment: 35 pages, 4 figure
Periodic Center Manifolds for Nonhyperbolic Limit Cycles in ODEs
In this paper, we deal with a classical object, namely, a nonhyperbolic limit cycle in a system of smooth autonomous ordinary differential equations. While the existence of a center manifold near such a cycle was assumed in several studies on cycle bifurcations based on periodic normal forms, no proofs were available in the literature until recently. The main goal of this paper is to give an elementary proof of the existence of a periodic smooth locally invariant center manifold near a nonhyperbolic cycle in finite-dimensional ordinary differential equations by using the Lyapunov-Perron method. In addition, we provide several explicit examples of analytic vector fields admitting (non)-unique, (non)-C∞-smooth and (non)-analytic periodic center manifolds.</p
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