799 research outputs found
The Construction of Finite Difference Approximations to Ordinary Differential Equations
Finite difference approximations of the form ÎŁ^(si)_(i=-rj)d_(j,i)u_(j+i)=ÎŁ^(mj)_(i=1) e_(j,if)(z_(j,i)) for the numerical solution of linear nth order ordinary differential equations are analyzed. The order of these approximations is shown to be at least r_j + s_j + m_j - n, and higher for certain special choices of the points Z_(j,i). Similar approximations to initial or boundary conditions are also considered and the stability of the resulting schemes is investigated
Computation of periodic solution bifurcations in ODEs using bordered systems
We consider numerical methods for the computation and continuation of the three generic secondary periodic solution bifurcations in autonomous ODEs, namely the fold, the period-doubling (or flip) bifurcation, and the torus (or Neimark–Sacker) bifurcation. In the fold and flip cases we append one scalar equation to the standard periodic BVP that defines the periodic solution; in the torus case four scalar equations are appended. Evaluation of these scalar equations and their derivatives requires the solution of linear BVPs, whose sparsity structure (after discretization) is identical to that of the linearization of the periodic BVP. Therefore the calculations can be done using existing numerical linear algebra techniques, such as those implemented in the software AUTO and COLSYS
A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions
Combinig the harmonic balance method (HBM) and a continuation method is a
well-known technique to follow the periodic solutions of dynamical systems when
a control parameter is varied. However, since deriving the algebraic system
containing the Fourier coefficients can be a highly cumbersome procedure, the
classical HBM is often limited to polynomial (quadratic and cubic)
nonlinearities and/or a few harmonics. Several variations on the classical HBM,
such as the incremental HBM or the alternating frequency/time domain HBM, have
been presented in the literature to overcome this shortcoming. Here, we present
an alternative approach that can be applied to a very large class of dynamical
systems (autonomous or forced) with smooth equations. The main idea is to
systematically recast the dynamical system in quadratic polynomial form before
applying the HBM. Once the equations have been rendered quadratic, it becomes
obvious to derive the algebraic system and solve it by the so-called ANM
continuation technique. Several classical examples are presented to illustrate
the use of this numerical approach.Comment: PACS numbers: 02.30.Mv, 02.30.Nw, 02.30.Px, 02.60.-x, 02.70.-
Numerical Continuation of Bound and Resonant States of the Two Channel Schr\"odinger Equation
Resonant solutions of the quantum Schr\"odinger equation occur at complex
energies where the S-matrix becomes singular. Knowledge of such resonances is
important in the study of the underlying physical system. Often the
Schr\"odinger equation is dependent on some parameter and one is interested in
following the path of the resonances in the complex energy plane as the
parameter changes. This is particularly true in coupled channel systems where
the resonant behavior is highly dependent on the strength of the channel
coupling, the energy separation of the channels and other factors. In previous
work it was shown that numerical continuation, a technique familiar in the
study of dynamical systems, can be brought to bear on the problem of following
the resonance path in one dimensional problems and multi-channel problems
without energy separation between the channels. A regularization can be defined
that eliminates coalescing poles and zeros that appear in the S-matrix at the
origin due to symmetries. Following the zeros of this regularized function then
traces the resonance path. In this work we show that this approach can be
extended to channels with energy separation, albeit limited to two channels.
The issue here is that the energy separation introduces branch cuts in the
complex energy domain that need to be eliminated with a so-called
uniformization. We demonstrate that the resulting approach is suitable for
investigating resonances in two-channel systems and provide an extensive
example
Network Inoculation: Heteroclinics and phase transitions in an epidemic model
In epidemiological modelling, dynamics on networks, and in particular
adaptive and heterogeneous networks have recently received much interest. Here
we present a detailed analysis of a previously proposed model that combines
heterogeneity in the individuals with adaptive rewiring of the network
structure in response to a disease. We show that in this model qualitative
changes in the dynamics occur in two phase transitions. In a macroscopic
description one of these corresponds to a local bifurcation whereas the other
one corresponds to a non-local heteroclinic bifurcation. This model thus
provides a rare example of a system where a phase transition is caused by a
non-local bifurcation, while both micro- and macro-level dynamics are
accessible to mathematical analysis. The bifurcation points mark the onset of a
behaviour that we call network inoculation. In the respective parameter region
exposure of the system to a pathogen will lead to an outbreak that collapses,
but leaves the network in a configuration where the disease cannot reinvade,
despite every agent returning to the susceptible class. We argue that this
behaviour and the associated phase transitions can be expected to occur in a
wide class of models of sufficient complexity.Comment: 26 pages, 11 figure
Global invariant manifolds in the transition to preturbulence in the Lorenz system
AbstractWe consider the homoclinic bifurcation of the Lorenz system, where two primary periodic orbits of saddle type bifurcate from a symmetric pair of homoclinic loops. The two secondary equilibria of the Lorenz system remain the only attractors before and after this bifurcation, but a chaotic saddle is created in a tubular neighbourhood of the two homoclinic loops. This invariant hyperbolic set gives rise to preturbulence, which is characterised by the presence of arbitrarily long transients.In this paper, we show how and where preturbulence arises in the three-dimensional phase space. To this end, we consider how the relevant two-dimensional invariant manifolds — the stable manifolds of the origin and of the primary periodic orbits — organise the phase space of the Lorenz system. More specifically, by means of recently developed and very robust numerical methods, we study how these manifolds intersect a suitable sphere in phase space. In this way, we show how the basins of attraction of the two attracting equilibria change topologically in the homoclinic bifurcation. More specifically, we characterise preturbulence in terms of the accessible boundary between the two basins, which accumulate on each other in a Cantor structure
Continuation of connecting orbits in 3D-ODEs: (I) Point-to-cycle connections
We propose new methods for the numerical continuation of point-to-cycle
connecting orbits in 3-dimensional autonomous ODE's using projection boundary
conditions. In our approach, the projection boundary conditions near the cycle
are formulated using an eigenfunction of the associated adjoint variational
equation, avoiding costly and numerically unstable computations of the
monodromy matrix. The equations for the eigenfunction are included in the
defining boundary-value problem, allowing a straightforward implementation in
AUTO, in which only the standard features of the software are employed.
Homotopy methods to find connecting orbits are discussed in general and
illustrated with several examples, including the Lorenz equations. Complete
AUTO demos, which can be easily adapted to any autonomous 3-dimensional ODE
system, are freely available.Comment: 18 pages, 10 figure
Bifurcations of two coupled classical spin oscillators
Two classical, damped and driven spin oscillators with an isotropic exchange
interaction are considered. They represent a nontrivial physical system whose
equations of motion are shown to allow for an analytic treatment of local
codimension 1 and 2 bifurcations. In addition, numerical results are presented
which exhibit a Feigenbaum route to chaos.Comment: 16 pages, .dvi and postscrip
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