10,765 research outputs found
The determination of asymptotic and periodic behavior of dynamic systems arising in control system analysis Final report
Asymptotic and periodic behavior prediction for nonlinear control system with mathematical model of rigid body vehicl
Quasiperiodic Dynamics in Bose-Einstein Condensates in Periodic Lattices and Superlattices
We employ KAM theory to rigorously investigate quasiperiodic dynamics in
cigar-shaped Bose-Einstein condensates (BEC) in periodic lattices and
superlattices. Toward this end, we apply a coherent structure ansatz to the
Gross-Pitaevskii equation to obtain a parametrically forced Duffing equation
describing the spatial dynamics of the condensate. For shallow-well,
intermediate-well, and deep-well potentials, we find KAM tori and Aubry-Mather
sets to prove that one obtains mostly quasiperiodic dynamics for condensate
wave functions of sufficiently large amplitude, where the minimal amplitude
depends on the experimentally adjustable BEC parameters. We show that this
threshold scales with the square root of the inverse of the two-body scattering
length, whereas the rotation number of tori above this threshold is
proportional to the amplitude. As a consequence, one obtains the same dynamical
picture for lattices of all depths, as an increase in depth essentially only
affects scaling in phase space. Our approach is applicable to periodic
superlattices with an arbitrary number of rationally dependent wave numbers.Comment: 29 pages, 6 figures (several with multiple parts; higher-quality
versions of some of them available at
http://www.its.caltech.edu/~mason/papers), to appear very soon in Journal of
Nonlinear Scienc
Rate of Decay of Stable Periodic Solutions of Duffing Equations
In this paper, we consider the second-order equations of Duffing type. Bounds
for the derivative of the restoring force are given that ensure the existence
and uniqueness of a periodic solution. Furthermore, the stability of the unique
periodic solution is analyzed; the sharp rate of exponential decay is
determined for a solution that is near to the unique periodic solution.Comment: Key words: Periodic solution; Stability; Rate of deca
Extensivity of two-dimensional turbulence
This study is concerned with how the attractor dimension of the
two-dimensional Navier--Stokes equations depends on characteristic length
scales, including the system integral length scale, the forcing length scale,
and the dissipation length scale. Upper bounds on the attractor dimension
derived by Constantin--Foias--Temam are analysed. It is shown that the optimal
attractor-dimension estimate grows linearly with the domain area (suggestive of
extensive chaos), for a sufficiently large domain, if the kinematic viscosity
and the amplitude and length scale of the forcing are held fixed. For
sufficiently small domain area, a slightly ``super-extensive'' estimate becomes
optimal. In the extensive regime, the attractor-dimension estimate is given by
the ratio of the domain area to the square of the dissipation length scale
defined, on physical grounds, in terms of the average rate of shear. This
dissipation length scale (which is not necessarily the scale at which the
energy or enstrophy dissipation takes place) can be identified with the
dimension correlation length scale, the square of which is interpreted,
according to the concept of extensive chaos, as the area of a subsystem with
one degree of freedom. Furthermore, these length scales can be identified with
a ``minimum length scale'' of the flow, which is rigorously deduced from the
concept of determining nodes.Comment: No figures, 14 page
Isochronous Partitions for Region-Based Self-Triggered Control
In this work, we propose a region-based self-triggered control (STC) scheme
for nonlinear systems. The state space is partitioned into a finite number of
regions, each of which is associated to a uniform inter-event time. The
controller, at each sampling time instant, checks to which region does the
current state belong, and correspondingly decides the next sampling time
instant. To derive the regions along with their corresponding inter-event
times, we use approximations of isochronous manifolds, a notion firstly
introduced in [1]. This work addresses some theoretical issues of [1] and
proposes an effective computational approach that generates approximations of
isochronous manifolds, thus enabling the region-based STC scheme. The
efficiency of both our theoretical results and the proposed algorithm are
demonstrated through simulation examples
Exponential stabilization of driftless nonlinear control systems using homogeneous feedback
This paper focuses on the problem of exponential stabilization of controllable, driftless systems using time-varying, homogeneous feedback. The analysis is performed with respect to a homogeneous norm in a nonstandard dilation that is compatible with the algebraic structure of the control Lie algebra. It can be shown that any continuous, time-varying controller that achieves exponential stability relative to the Euclidean norm is necessarily non-Lipschitz. Despite these restrictions, we provide a set of constructive, sufficient conditions for extending smooth, asymptotic stabilizers to homogeneous, exponential stabilizers. The modified feedbacks are everywhere continuous, smooth away from the origin, and can be extended to a large class of systems with torque inputs. The feedback laws are applied to an experimental mobile robot and show significant improvement in convergence rate over smooth stabilizers
Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application
With the goal of providing the first example of application of a recently proposed method, thus demonstrating its ability to give results in principle, global stability of a version of the rotating Couette flow is examined. The flow depends on the Reynolds number and a parameter characterising the magnitude of the Coriolis force. By converting the original Navier-Stokes equations to a finite-dimensional uncertain dynamical system using a partial Galerkin expansion, high-degree polynomial Lyapunov functionals were found by sum-of-squares-of-polynomials optimization. It is demonstrated that the proposed method allows obtaining the exact global stability limit for this flow in a range of values of the parameter characterising the Coriolis force. Outside this range a lower bound for the global stability limit was obtained, which is still better than the energy stability limit. In the course of the study several results meaningful in the context of the method used were also obtained. Overall, the results obtained demonstrate the applicability of the recently proposed approach to global stability of the fluid flows. To the best of our knowledge, it is the first case in which global stability of a fluid flow has been proved by a generic method for the value of a Reynolds number greater than that which could be achieved with the energy stability approach
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