The Bohl spectrum for nonautonomous differential equations

We develop the Bohl spectrum for nonautonomous linear differential equation on a half line, which is a spectral concept that lies between the Lyapunov and the Sacker--Sell spectrum. We prove that the Bohl spectrum is given by the union of finitely many intervals, and we show by means of an explicit example that the Bohl spectrum does not coincide with the Sacker--Sell spectrum in general. We demonstrate for this example that any higher-order nonlinear perturbation is exponentially stable, although this not evident from the Sacker--Sell spectrum. We also analyze in detail situations in which the Bohl spectrum is identical to the Sacker-Sell spectrum

Stability radius and internal versus external stability in Banach spaces: an evolution semigroup approach

In this paper the theory of evolution semigroups is developed and used to provide a framework to study the stability of general linear control systems. These include time-varying systems modeled with unbounded state-space operators acting on Banach spaces. This approach allows one to apply the classical theory of strongly continuous semigroups to time-varying systems. In particular, the complex stability radius may be expressed explicitly in terms of the generator of a (evolution) semigroup. Examples are given to show that classical formulas for the stability radius of an autonomous Hilbert-space system fail in more general settings. Upper and lower bounds on the stability radius are provided for these general systems. In addition, it is shown that the theory of evolution semigroups allows for a straightforward operator-theoretic analysis of internal stability as determined by classical frequency-domain and input-output operators, even for nonautonomous Banach-space systemsComment: Also at http://www.math.missouri.edu/~stephen/preprint

Matter-wave solitons with a periodic, piecewise-constant nonlinearity

Motivated by recent proposals of ``collisionally inhomogeneous'' Bose-Einstein condensates (BECs), which have a spatially modulated scattering length, we study the existence and stability properties of bright and dark matter-wave solitons of a BEC characterized by a periodic, piecewise-constant scattering length. We use a ``stitching'' approach to analytically approximate the pertinent solutions of the underlying nonlinear Schr\"odinger equation by matching the wavefunction and its derivatives at the interfaces of the nonlinearity coefficient. To accurately quantify the stability of bright and dark solitons, we adapt general tools from the theory of perturbed Hamiltonian systems. We show that solitons can only exist at the centers of the constant regions of the piecewise-constant nonlinearity. We find both stable and unstable configurations for bright solitons and show that all dark solitons are unstable, with different instability mechanisms that depend on the soliton location. We corroborate our analytical results with numerical computations.Comment: 16 pages, 7 figures (some with multiple parts), to appear in Physical Review

A Holographic Path to the Turbulent Side of Gravity

We study the dynamics of a 2+1 dimensional relativistic viscous conformal fluid in Minkowski spacetime. Such fluid solutions arise as duals, under the "gravity/fluid correspondence", to 3+1 dimensional asymptotically anti-de Sitter (AAdS) black brane solutions to the Einstein equation. We examine stability properties of shear flows, which correspond to hydrodynamic quasinormal modes of the black brane. We find that, for sufficiently high Reynolds number, the solution undergoes an inverse turbulent cascade to long wavelength modes. We then map this fluid solution, via the gravity/fluid duality, into a bulk metric. This suggests a new and interesting feature of the behavior of perturbed AAdS black holes and black branes, which is not readily captured by a standard quasinormal mode analysis. Namely, for sufficiently large perturbed black objects (with long-lived quasinormal modes), nonlinear effects transfer energy from short to long wavelength modes via a turbulent cascade within the metric perturbation. As long wavelength modes have slower decay, this lengthens the overall lifetime of the perturbation. We also discuss various implications of this behavior, including expectations for higher dimensions, and the possibility of predicting turbulence in more general gravitational scenarios.Comment: 24 pages, 10 figures; v2: references added, and several minor change

Linear and nonlinear information flow in spatially extended systems

Infinitesimal and finite amplitude error propagation in spatially extended systems are numerically and theoretically investigated. The information transport in these systems can be characterized in terms of the propagation velocity of perturbations $V_p$. A linear stability analysis is sufficient to capture all the relevant aspects associated to propagation of infinitesimal disturbances. In particular, this analysis gives the propagation velocity $V_L$ of infinitesimal errors. If linear mechanisms prevail on the nonlinear ones $V_p = V_L$. On the contrary, if nonlinear effects are predominant finite amplitude disturbances can eventually propagate faster than infinitesimal ones (i.e. $V_p > V_L$). The finite size Lyapunov exponent can be successfully employed to discriminate the linear or nonlinear origin of information flow. A generalization of finite size Lyapunov exponent to a comoving reference frame allows to state a marginal stability criterion able to provide $V_p$ both in the linear and in the nonlinear case. Strong analogies are found between information spreading and propagation of fronts connecting steady states in reaction-diffusion systems. The analysis of the common characteristics of these two phenomena leads to a better understanding of the role played by linear and nonlinear mechanisms for the flow of information in spatially extended systems.Comment: 14 RevTeX pages with 13 eps figures, title/abstract changed minor changes in the text accepted for publication on PR

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

Asymptotic methods for delay equations.

Asymptotic methods for singularly perturbed delay differential equations are in many ways more challenging to implement than for ordinary differential equations. In this paper, four examples of delayed systems which occur in practical models are considered: the delayed recruitment equation, relaxation oscillations in stem cell control, the delayed logistic equation, and density wave oscillations in boilers, the last of these being a problem of concern in engineering two-phase flows. The ways in which asymptotic methods can be used vary from the straightforward to the perverse, and illustrate the general technical difficulties that delay equations provide for the central technique of the applied mathematician. © Springer 2006