3,162 research outputs found
Nonlinear normal modes and spectral submanifolds: Existence, uniqueness and use in model reduction
We propose a unified approach to nonlinear modal analysis in dissipative
oscillatory systems. This approach eliminates conflicting definitions, covers
both autonomous and time-dependent systems, and provides exact mathematical
existence, uniqueness and robustness results. In this setting, a nonlinear
normal mode (NNM) is a set filled with small-amplitude recurrent motions: a
fixed point, a periodic orbit or the closure of a quasiperiodic orbit. In
contrast, a spectral submanifold (SSM) is an invariant manifold asymptotic to a
NNM, serving as the smoothest nonlinear continuation of a spectral subspace of
the linearized system along the NNM. The existence and uniqueness of SSMs turns
out to depend on a spectral quotient computed from the real part of the
spectrum of the linearized system. This quotient may well be large even for
small dissipation, thus the inclusion of damping is essential for firm
conclusions about NNMs, SSMs and the reduced-order models they yield.Comment: To appear in Nonlinear Dynamic
Initial Conditions for Models of Dynamical Systems
The long-time behaviour of many dynamical systems may be effectively
predicted by a low-dimensional model that describes the evolution of a reduced
set of variables. We consider the question of how to equip such a
low-dimensional model with appropriate initial conditions, so that it
faithfully reproduces the long-term behaviour of the original high-dimensional
dynamical system. Our method involves putting the dynamical system into normal
form, which not only generates the low-dimensional model, but also provides the
correct initial conditions for the model. We illustrate the method with several
examples.
Keywords: normal form, isochrons, initialisation, centre manifoldComment: 24 pages in standard LaTeX, 66K, no figure
Computation of maximal local (un)stable manifold patches by the parameterization method
In this work we develop some automatic procedures for computing high order
polynomial expansions of local (un)stable manifolds for equilibria of
differential equations. Our method incorporates validated truncation error
bounds, and maximizes the size of the image of the polynomial approximation
relative to some specified constraints. More precisely we use that the manifold
computations depend heavily on the scalings of the eigenvectors: indeed we
study the precise effects of these scalings on the estimates which determine
the validated error bounds. This relationship between the eigenvector scalings
and the error estimates plays a central role in our automatic procedures. In
order to illustrate the utility of these methods we present several
applications, including visualization of invariant manifolds in the Lorenz and
FitzHugh-Nagumo systems and an automatic continuation scheme for (un)stable
manifolds in a suspension bridge problem. In the present work we treat
explicitly the case where the eigenvalues satisfy a certain non-resonance
condition.Comment: Revised version, typos corrected, references adde
Delay Equations and Radiation Damping
Starting from delay equations that model field retardation effects, we study
the origin of runaway modes that appear in the solutions of the classical
equations of motion involving the radiation reaction force. When retardation
effects are small, we argue that the physically significant solutions belong to
the so-called slow manifold of the system and we identify this invariant
manifold with the attractor in the state space of the delay equation. We
demonstrate via an example that when retardation effects are no longer small,
the motion could exhibit bifurcation phenomena that are not contained in the
local equations of motion.Comment: 15 pages, 1 figure, a paragraph added on page 5; 3 references adde
Visualizing the geometry of state space in plane Couette flow
Motivated by recent experimental and numerical studies of coherent structures
in wall-bounded shear flows, we initiate a systematic exploration of the
hierarchy of unstable invariant solutions of the Navier-Stokes equations. We
construct a dynamical, 10^5-dimensional state-space representation of plane
Couette flow at Re = 400 in a small, periodic cell and offer a new method of
visualizing invariant manifolds embedded in such high dimensions. We compute a
new equilibrium solution of plane Couette flow and the leading eigenvalues and
eigenfunctions of known equilibria at this Reynolds number and cell size. What
emerges from global continuations of their unstable manifolds is a surprisingly
elegant dynamical-systems visualization of moderate-Reynolds turbulence. The
invariant manifolds tessellate the region of state space explored by
transiently turbulent dynamics with a rigid web of continuous and discrete
symmetry-induced heteroclinic connections.Comment: 32 pages, 13 figures submitted to Journal of Fluid Mechanic
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