3,742 research outputs found
Wave propagation in one-dimensional nonlinear acoustic metamaterials
The propagation of waves in the nonlinear acoustic metamaterials (NAMs) is
fundamentally different from that in the conventional linear ones. In this
article we consider two one-dimensional NAM systems featuring respectively a
diatomic and a tetratomic meta unit-cell. We investigate the attenuation of the
wave, the band structure and the bifurcations to demonstrate novel nonlinear
effects, which can significantly expand the bandwidth for elastic wave
suppression and cause nonlinear wave phenomena. Harmonic averaging approach,
continuation algorithm, Lyapunov exponents are combined to study the frequency
responses, the nonlinear modes, bifurcations of periodic solutions and chaos.
The nonlinear resonances are studied and the influence of damping on
hyper-chaotic attractors is evaluated. Moreover, a "quantum" behavior is found
between the low-energy and high-energy orbits. This work provides an important
theoretical base for the further understandings and applications of NAMs
Finite-time Lagrangian transport analysis: Stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents
We consider issues associated with the Lagrangian characterisation of flow
structures arising in aperiodically time-dependent vector fields that are only
known on a finite time interval. A major motivation for the consideration of
this problem arises from the desire to study transport and mixing problems in
geophysical flows where the flow is obtained from a numerical solution, on a
finite space-time grid, of an appropriate partial differential equation model
for the velocity field. Of particular interest is the characterisation,
location, and evolution of "transport barriers" in the flow, i.e. material
curves and surfaces. We argue that a general theory of Lagrangian transport has
to account for the effects of transient flow phenomena which are not captured
by the infinite-time notions of hyperbolicity even for flows defined for all
time. Notions of finite-time hyperbolic trajectories, their finite time stable
and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields
and associated Lagrangian coherent structures have been the main tools for
characterizing transport barriers in the time-aperiodic situation. In this
paper we consider a variety of examples, some with explicit solutions, that
illustrate, in a concrete manner, the issues and phenomena that arise in the
setting of finite-time dynamical systems. Of particular significance for
geophysical applications is the notion of "flow transition" which occurs when
finite-time hyperbolicity is lost, or gained. The phenomena discovered and
analysed in our examples point the way to a variety of directions for rigorous
mathematical research in this rapidly developing, and important, new area of
dynamical systems theory
Least Squares Shadowing sensitivity analysis of chaotic limit cycle oscillations
The adjoint method, among other sensitivity analysis methods, can fail in
chaotic dynamical systems. The result from these methods can be too large,
often by orders of magnitude, when the result is the derivative of a long time
averaged quantity. This failure is known to be caused by ill-conditioned
initial value problems. This paper overcomes this failure by replacing the
initial value problem with the well-conditioned "least squares shadowing (LSS)
problem". The LSS problem is then linearized in our sensitivity analysis
algorithm, which computes a derivative that converges to the derivative of the
infinitely long time average. We demonstrate our algorithm in several dynamical
systems exhibiting both periodic and chaotic oscillations.Comment: submitted to JCP in revised for
Linear Hamilton Jacobi Bellman Equations in High Dimensions
The Hamilton Jacobi Bellman Equation (HJB) provides the globally optimal
solution to large classes of control problems. Unfortunately, this generality
comes at a price, the calculation of such solutions is typically intractible
for systems with more than moderate state space size due to the curse of
dimensionality. This work combines recent results in the structure of the HJB,
and its reduction to a linear Partial Differential Equation (PDE), with methods
based on low rank tensor representations, known as a separated representations,
to address the curse of dimensionality. The result is an algorithm to solve
optimal control problems which scales linearly with the number of states in a
system, and is applicable to systems that are nonlinear with stochastic forcing
in finite-horizon, average cost, and first-exit settings. The method is
demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with
system dimension two, six, and twelve respectively.Comment: 8 pages. Accepted to CDC 201
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