130,824 research outputs found
Frequency-Domain Analysis of Linear Time-Periodic Systems
In this paper, we study convergence of truncated representations of the frequency-response operator of a linear time-periodic system. The frequency-response operator is frequently called the harmonic transfer function. We introduce the concepts of input, output, and skew roll-off. These concepts are related to the decay rates of elements in the harmonic transfer function. A system with high input and output roll-off may be well approximated by a low-dimensional matrix function. A system with high skew roll-off may be represented by an operator with only few diagonals. Furthermore, the roll-off rates are shown to be determined by certain properties of Taylor and Fourier expansions of the periodic systems. Finally, we clarify the connections between the different methods for computing the harmonic transfer function that are suggested in the literature
Resonant normal forms as constrained linear systems
We show that a nonlinear dynamical system in Poincare'-Dulac normal form (in
) can be seen as a constrained linear system; the constraints are given
by the resonance conditions satisfied by the spectrum of (the linear part of)
the system and identify a naturally invariant manifold for the flow of the
``parent'' linear system. The parent system is finite dimensional if the
spectrum satisfies only a finite number of resonance conditions, as implied
e.g. by the Poincare' condition. In this case our result can be used to
integrate resonant normal forms, and sheds light on the geometry behind the
classical integration method of Horn, Lyapounov and Dulac.Comment: 15 pages; revised version (with revised title
Dimension increase and splitting for Poincare'-Dulac normal forms
Integration of nonlinear dynamical systems is usually seen as associated to a
symmetry reduction, e.g. via momentum map. In Lax integrable systems, as
pointed out by Kazhdan, Kostant and Sternberg in discussing the Calogero
system, one proceeds in the opposite way, enlarging the nonlinear system to a
system of greater dimension. We discuss how this approach is also fruitful in
studying non integrable systems, focusing on systems in normal form.Comment: 16 page
Stochastic Stability Analysis of Discrete Time System Using Lyapunov Measure
In this paper, we study the stability problem of a stochastic, nonlinear,
discrete-time system. We introduce a linear transfer operator-based Lyapunov
measure as a new tool for stability verification of stochastic systems. Weaker
set-theoretic notion of almost everywhere stochastic stability is introduced
and verified, using Lyapunov measure-based stochastic stability theorems.
Furthermore, connection between Lyapunov functions, a popular tool for
stochastic stability verification, and Lyapunov measures is established. Using
the duality property between the linear transfer Perron-Frobenius and Koopman
operators, we show the Lyapunov measure and Lyapunov function used for the
verification of stochastic stability are dual to each other. Set-oriented
numerical methods are proposed for the finite dimensional approximation of the
Perron-Frobenius operator; hence, Lyapunov measure is proposed. Stability
results in finite dimensional approximation space are also presented. Finite
dimensional approximation is shown to introduce further weaker notion of
stability referred to as coarse stochastic stability. The results in this paper
extend our earlier work on the use of Lyapunov measures for almost everywhere
stability verification of deterministic dynamical systems ("Lyapunov Measure
for Almost Everywhere Stability", {\it IEEE Trans. on Automatic Control}, Vol.
53, No. 1, Feb. 2008).Comment: Proceedings of American Control Conference, Chicago IL, 201
Data-Driven Approximation of Transfer Operators: Naturally Structured Dynamic Mode Decomposition
In this paper, we provide a new algorithm for the finite dimensional
approximation of the linear transfer Koopman and Perron-Frobenius operator from
time series data. We argue that existing approach for the finite dimensional
approximation of these transfer operators such as Dynamic Mode Decomposition
(DMD) and Extended Dynamic Mode Decomposition (EDMD) do not capture two
important properties of these operators, namely positivity and Markov property.
The algorithm we propose in this paper preserve these two properties. We call
the proposed algorithm as naturally structured DMD since it retains the
inherent properties of these operators. Naturally structured DMD algorithm
leads to a better approximation of the steady-state dynamics of the system
regarding computing Koopman and Perron- Frobenius operator eigenfunctions and
eigenvalues. However preserving positivity properties is critical for capturing
the real transient dynamics of the system. This positivity of the transfer
operators and it's finite dimensional approximation also has an important
implication on the application of the transfer operator methods for controller
and estimator design for nonlinear systems from time series data
Generic super-exponential stability of invariant tori in Hamiltonian systems
In this article, we consider solutions starting close to some linearly stable
invariant tori in an analytic Hamiltonian system and we prove results of
stability for a super-exponentially long interval of time, under generic
conditions. The proof combines classical Birkhoff normal forms and a new method
to obtain generic Nekhoroshev estimates developed by the author and L.
Niederman in another paper. We will mainly focus on the neighbourhood of
elliptic fixed points, the other cases being completely similar
Bottlenecks to vibrational energy flow in OCS: Structures and mechanisms
Finding the causes for the nonstatistical vibrational energy relaxation in
the planar carbonyl sulfide (OCS) molecule is a longstanding problem in
chemical physics: Not only is the relaxation incomplete long past the predicted
statistical relaxation time, but it also consists of a sequence of abrupt
transitions between long-lived regions of localized energy modes. We report on
the phase space bottlenecks responsible for this slow and uneven vibrational
energy flow in this Hamiltonian system with three degrees of freedom. They
belong to a particular class of two-dimensional invariant tori which are
organized around elliptic periodic orbits. We relate the trapping and
transition mechanisms with the linear stability of these structures.Comment: 13 pages, 13 figure
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